ae b ˘ e af e c doctor of philosophy ˇb ba › home › feamit › pdfs › thesis.pdf · i thank...

Thesis for the degree Doctor of Philosophy

By Amit Aronovitch

Advisor: Uzy Smilansky

December 2010

Submitted to the Scientific Council of the Weizmann Institute of Science

Rehovot, Israel

ABCBDEFEBDEFA EEDB

Nodal domains of billiard eigenfunctions on their boundary

AFECEAEB BBA

EF CAEF

DEEC

DFEBDEFFB FBFF

ECABAD

F BF

i

The maker of Bonnets ferociously planned

A novel arrangement of bows:

While the Billiard-marker with quivering hand

Was chalking the tip of his nose.

Lewis Carroll,The Hunting of the Snark, 1874

ii

Abstract

Given a Dirichlet eigenfunction of a 2D quantum billiard, the boundary intersec-

tions (BI) are the points where the nodal lines intersect theboundary. We inves-

tigate the density and distribution of these points in the semi-classical limit. In

particular, we derive trace formulae for the boundary domain count, which is the

integer sequence generated by the number of BI for each eigenfunction, sorted

according to the energy. Analogously to the well known traceformulae of the

spectral counting function, this can be used as theoreticaltools for explaining the

numerically observed dependence of the BI count distributions on the stability of

the classical motion. We describe the analytical derivation of such formulae for

integrable systems. For chaotic billiards, we derive a trace formula which is based

on a random waves hypothesis, and verify its correctness numerically. This is the

first time such a trace formula is given for chaotic billiardsin the context of nodal

domains.

תקציר

Boundary) השפה חיתוכי דיריכלה, מסוג שפה תנאי עם דו־מימדי ביליארד של עצמית פונקציה עבור

מחקר הביליארד. שפת עם (nodal lines) הצומת קוי של החיתוך נקודות הן (Intersections—BI

התקבלו בפרט, הסמי־קלאסי. בגבול שלהן, והצפיפות השפה על אלה נקודות של בהתפלגות עוסק זה

האנרגיה לפי ממויינים שלמים, מספרים של סדרה שהוא השפה, חיתוכי מניין עבור עקבה׳׳ ׳׳נוסחאות

הספקטראלית, המניין פונקצית עבור הידועות העקבה לנוסחאות בדומה המתאימה. הגל פונקצית של

בין נומרי, באופן שנצפתה התלות, את להסביר כדי שישמש תאורטי כלי להוות יכולה זה מסוג נוסחה

עקבה נוסחאות המתאימה. הקלאסית התנועה של היציבות לבין השפה חיתוכי מספר של ההתפלגות

עבור המקובלות. בשיטות אנאליטית, בצורה התקבלו אינטגרביליות למערכות השפה חיתוכי למניין

את ווידאנו אקראיים, גלים בהנחת שימוש ע׳׳י מתאימה נוסחה קיבלנו כאוטיים, ביליארדים של המקרה

כאוטיים, לביליארדים זה מסוג עקבה נוסחת שמתקבלת הראשונה הפעם זוהי נומרי. באופן נכונותה

הצומת. לאזורי הקשור גודל עבור

iii

Acknowledgements

I would like to thank the people who have made this work possible, and have made

the last five years as enjoyable and educational as they were for me.

I am grateful to my Ph.D. supervisor, Professor Uzy Smilansky, whose con-

tribution was far more than the direct professional guidance. Besides his vast

knowledge and sharp intuitions, which were a major source for ideas, he managed

to encourage me to pave my own path, while keeping a close eye on my progress,

ready to lend a hand where needed. With his confident manner and cheerful atti-

tude, he was always a pleasant and interesting company. Thanks Uzy, for teaching

me to choose where to concentrate my efforts and, more importantly, where not

to.

I thank Sven Gnutzmann who was the first to introduce me to the group’s field

of research, provided assistance and ideas, and was always friendly and help-

ful. My fellow group members, Rami Band, Yehonatan Elon, Idan Oren, Nir

Auerbach, Amit Godel and Fabien Piotet, have made this period fun with their

friendship, laughter, lunch breaks, and many discussions about ideas and thoughts

related and unrelated to our research.

Klaus Hornberger deserves direct credit for helping me resolve one of the key

issues required for the main result of this thesis, and I would also like to thank

him for his kind hosting at the Max Planck Institute during myvisit to Dresden.

I thank the Weizmann Institute for being a great place to learn and conduct

research. In particular, I would like to thank Yossi Drier, Perla Zalcberg and

Rachel Goldman for taking care of all our needs and making thedepartment feel

like home.

My deepest thanks to my parents, that educated me, made me whoI am today,

and still are, as always, happy to provide help and support. To the rest of my

family, and Kineret’s family, for helping out, babysitting, providing quiet places

to learn and generally being supportive. To Tamir and Omer, for their endless

curiosity and for being the wonderful kids that they are, a stable source of joy and

inspiration to their parents.

To Kineret, who actively encouraged me to follow my dream andgo “back

to school” at this stage of my life, despite the extra burden it would personally

iv

imply for her. I do not know how you managed, with such a demanding career,

and without ever compromising quality time with the kids, tomuster the required

energy and time for orchestrating the family effort and pulling endless strings so

I could always find enough time for learning an research. Thisthesis is, in a very

real sense, a product or your efforts as well. I consider myself lucky for having

you by my side.

Contents

1 Introduction 1

1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Statistical Models 9

2.1 The Monochromatic Random Wave model . . . . . . . . . . . . . 10

2.2 Boundary adjusted models . . . . . . . . . . . . . . . . . . . . . 13

2.3 Extracting the BI statistics . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 BI statistics in terms ofG(r, r′) . . . . . . . . . . . . . . 16

2.3.2 BI Density of the CRW model . . . . . . . . . . . . . . . 18

2.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 BI Density . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.2 Numerical comparison . . . . . . . . . . . . . . . . . . . 21

2.4.3 Correlation functions and form factors . . . . . . . . . . . 23

2.4.4 Comparison to other distributions . . . . . . . . . . . . . 25

3 Integrable Billiards 29

3.1 Separable problems . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 The rectangle billiard . . . . . . . . . . . . . . . . . . . . 33

3.1.2 The circle billiard . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Removing the spectral dependence . . . . . . . . . . . . . . . . . 37

3.2.1 Spectral inversionk(n) . . . . . . . . . . . . . . . . . . . 38

3.2.2 Applying spectral inversion . . . . . . . . . . . . . . . . 44

3.3 A non separable billiard . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 BIC as a function ofk . . . . . . . . . . . . . . . . . . . 47

v

vi CONTENTS

3.3.2 Common form with the separable TF . . . . . . . . . . . 49

3.3.3 Combining with the spectral inversion . . . . . . . . . . . 51

4 Chaotic Billiards 55

4.1 BI density and boundary functions . . . . . . . . . . . . . . . . . 56

4.1.1 Expansion by cumulants . . . . . . . . . . . . . . . . . . 58

4.1.2 Numerical verification of Gaussianity . . . . . . . . . . . 59

4.2 Correlations of the boundary function . . . . . . . . . . . . . . .62

4.2.1 The boundary Green function . . . . . . . . . . . . . . . 62

4.2.2 Explicit expansion forg . . . . . . . . . . . . . . . . . . 65

4.2.3 Semi-classical power densities . . . . . . . . . . . . . . . 67

4.3 Trace formula for the BIC . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Taking the trace . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . 74

5 Discussion 81

5.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 81

5.2 Potential future developments . . . . . . . . . . . . . . . . . . . 83

A The BIC of separable systems 87

A.1 The Weyl term . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A.2 The oscillating part . . . . . . . . . . . . . . . . . . . . . . . . . 90

B Counting BI for the triangle 93

C Computing Cη(k) for the triangle 97

D Expansion of the boundary Green function 101

D.1 The integral equation . . . . . . . . . . . . . . . . . . . . . . . . 101

D.2 Computation ofg1 . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Abbreviations and Symbols 109

Bibliography 113

Author’s Declaration 119

Chapter 1

Introduction

1.1 Framework

Quantum chaos and billiards

The research in quantum chaos centres around the connections between quantum

properties of a system and the properties of the corresponding classical motion.

In particular, the classical features of interest are the ones related to the dynami-

cal stability of the motion and its topological structure inphase space. The focus

of the field is the ways in which these features are manifestedin purely quantum

properties, such as distributions of energy spectra and wave functions. A model

system which is very useful in this context is the billiard problem. While re-

maining relatively simple to analyse both classically and quantum mechanically,

it can generate all types of dynamics, from integrable motion, whose phase space

is foliated by invariant tori, to the fully chaotic case, where all periodic orbits

are unstable and separated in the “chaotic sea” that fills thephase space. Many

important discoveries in the field, such as the universalityof chaotic spectra [1]

and scarring of wave-functions by unstable periodic orbits[2] were first made in

quantum billiards.

A (2D) billiard is defined by the motion of a free particle which is confined

inside a bounded two dimensional domainΩ ⊂ R2. Classically, the particle moves

freely inside the domain, and reflected specularly from the boundary∂Ω. In the

quantum counterpart, the stationary wave-functions satisfy the free Schrodinger

1

2 CHAPTER 1. INTRODUCTION

equation

−∆ψ(r) = k2ψ(r) (1.1)

(a.k.a. the Helmholtz equation) forr ∈ Ω \ ∂Ω, and fulfil some boundary

conditions forr ∈ ∂Ω. In this thesis we consider only the Dirichlet bound-

ary conditionψ|∂Ω = 0. We choose an orthogonal basis forL2(Ω), consisting

of eigenfunctions (stationary states)ψn∞n=1 which are real and normalized to

unity∫Ω|ψn(r)|2d2r = 1. The eigenfunctions are ordered by increasing energy

En+1 ≥ En (for a particle of massm, we use units where~2/(2m) = 1, so the

eigenvalues are exactly the energiesEn = kn2). The semi-classical limit,k → ∞,

corresponds to eigenfunctionsψn whose wavelength1/kn is much smaller than

the smallest relevant “typical length” of the billiard’s shape. This could be, for ex-

ample, the length of the boundaryL = |∂Ω|, or the minimal radius of curvature.

The average density of energy levels in high lying energy intervals is known by

Weyl’s law [3], and depends, asymptotically, only on the billiard’s areaA = |Ω|.However, the fluctuations around this average depend on the details of the sys-

tem. Although there are known cases of different billiards with identical spectrum

(a.k.a. isospectral drums) [4], substantial information can be extracted from the

spectral sequence [5]. For generic integrable systems, Berry and Tabor [6] had

shown that the level spacing distribution matches the spacing of the uncorrelated

Poisson process. As opposed to that, it was observed, experimentally and numer-

ically, that the distribution of the spacing between consecutive energy levels in

many irregular systems matches the universal distributions of random matrix the-

ory (RMT) [7]. Bohigas, Giannoni and Schmit (BGS) [1] conjectured that this is a

general property of systems which are classically chaotic.In [8], an intermediate

model is suggested for the case of systems with partially chaotic phase space.

In order to provide theoretical support for such phenomena and explicitly re-

veal the classical information stored in discrete quantum sequences, we need a

semi-classical approximation for the sequence in question, which depends only

on parameters of the limiting classical motion. This comes in the form of “trace1

formulae”, which consist of a “smooth part” representing the global average of

1The name “trace formula” comes from the fact that these formulae are usuallyderived bycomputing the trace of certain operators.

1.1. FRAMEWORK 3

the sequence and an “oscillating part” representing the local deviations from the

mean. The oscillating part is expressed as a sum over classical orbits. For the case

of billiard spectra, trace formulae for the spectral density d(k) =∑∞

n=1 δ(k− kn)

(whereδ(x) is the Dirac delta function) take the form

d(k) ∼ Ak2π

− L4π

+ kα∑

γ

Cγ sin(kLγ + φγ), (1.2)

whereγ enumerates classical periodic orbits,Lγ is the length of the orbit,φγ and

Cγ are parameters which depend on certain classical features of the orbitγ, andα

depends on the dynamical category of the system. For a general derivation of such

formulae, see [9, 10]. It should be noted that the infinite sumin (1.2) is divergent

in most cases. Hence, equation (1.2) should be understood asshorthand notation,

indicating that if we choose a proper smoothing kernel, whose width is negligible

for k ≫ 1, and convolve it term by term with each side of the equation, the results

would agree.

For integrable systems, the trace formula, derived by Berryand Tabor [11],

hasα = 12

andγ enumerates tori whose orbits are periodic. For chaotic systems,

where all periodic orbits are unstable, the relevant trace formula was derived by

Gutzwiller [12]. It hasα = 0 and the coefficientCγ depends on the stability

(Lyapunov exponent) of the orbitγ. Since only classical parameters appear in

the trace formula, it enables utilizing knowledge from classical mechanics for

studying the spectrum. In particular, this has been used to make progress towards

validating the BGS conjecture [13, 14, 15].

Nodal domains and boundary intersections

Moving on to consider the eigenfunctions themselves, one might argue that the

most intuitive geometrical characteristic of such functions is the pattern displayed

by their “hills and valleys”, namely thenodal domains. Recently, the study of

nodal patterns witnessed a remarkable renaissance, and attracted the active in-

terest of scientists from very diverse fields—quantum chaos, acoustics, optics,

spectral theory, percolation and more [16]. For a real continuous functionψ, the

nodal domains are connected components on whichψ(r) 6= 0. The nodal set


(nodal lines in the 2D case) is the zero set ofψ(r). For quantum billiards,νn,

the number of nodal domains of thenth solution (ordered by increasing eigen-

value) has the Courant upper boundνn ≤ n, but this limit is only reached for a

finite number of eigenfunctions [17]. In 2002, Blum et al. [18] have shown that

the nodal count sequenceνn∞n=1 of quantum billiards, much like the spectrum,

contains fingerprints of the underlying classical system. The information in this

sequence differs from the information stored in the spectrum, and it was shown to

resolve several known cases of isospectral systems [19, 20]. It was also shown that

the sequence uniquely determines a single problem within certain classes (inverse

nodal problem) [21, 22]. Like the spectrum, the nodal count sequence clearly dis-

tinguishes between separable and chaotic dynamics [18]. While for the separable

case the nodal pattern is well understood, current understanding of the chaotic

case relies on statistical models. Away from boundaries, the eigenfunctions are

modelled by Berry’s random wave (RW) model [23]. For statistics like the nodal

count, which depend on the detailed topology of the function, Bogomolny and

Schmit [24] proposed a critical percolation model. This model successfully pre-

dicts the asymptotic spectral averages ofνn, as well as other nodal statistics. In

particular, it predicts that in the high energy limit, the nodal lines should behave

like SLE6 curves, which was also numerically confirmed [25]. However,it fails

to predict some other statistical measures of the random wave model [26, 27].

Formulating an analytical expression for the exact nodal count (which could

be used as a starting point for semi-classical analysis) seems to be a hard prob-

lem, even in two dimensions. In one dimension the situation is much simpler,

because in this case the number of nodal points uniquely determine the number

of nodal domains, without requiring further topological information. Hence it can

be expressed in terms of purely local quantities. Explicitly, the number of nodal

domains for aC1 functionu(s) on a curve of lengthL is given by

η =

∫ L

0

δ(u(s))|u(s)| ds.

There is no equivalent formula for the number of nodal domains in more than

one dimension. In [27] we restricted 2D fields to a reference curve, and used the

resulting nodal statistics to characterize the fields.

1.1. FRAMEWORK 5

In the case of billiard eigenfunctions, restriction to the boundary is particularly

attractive. For Dirichlet eigenfunctions, the boundary functionun(s) is defined

as the scaled normal derivative ofψn at the boundary, i.e.un(s) = kn−1n(s) ·

∇nψ(r(s)), wherer(s) is the natural parametrization of∂Ω, s is the arc-length

andn(s) is the outwards-pointing normal derivative ofψn atr(s). Sinceun com-

pletely determines the eigenfunction, it should hold all the relevant information.

In particular, the “boundary intersections” (BI), intersections of nodal lines with

the boundary (which are just the nodal points of the boundaryfunction), pro-

vide restrictions on the possible configurations of 2D nodaldomains. Toth and

Zelditch [28] have recently shown that the number of boundary domains (a.k.a. the

boundary intersections count (BIC)) isO(√n). This was used by Polterovich [29]

to obtain an upper limit to the number of nodal domains in Neumann billiards.

Further than the upper bound of [28], the average BIC is also believed to be pro-

portional to√n. This is based on random wave models [18] and analysis of ran-

dom combinations of eigenfunctions [30]. In addition to theinformation that the

BIC sequenceηn∞n=1 provides on the nodal domains count, it is shown in [18]

that the BIC contains the same fingerprints of classical dynamics (i.e. clear dis-

tinction between the separable and chaotic case) thatνn does. Hence, the BIC

is a relevant feature that is worth studying in the context ofquantum chaos.

Goals and results

The goal of my research was to derive the required theoretical basis for under-

standing the distribution of nodal domains (and nodal points) on the boundaries

of 2D quantum billiards, and in particular its connections with the geometry of

the domain and properties of the underlying classical motion.

As a starting point, we considered the statistical model which was used in [18]

to derive the expected average BIC for the case of chaotic billiards. With a proper

choice of refined model, to account for the effect of a curved boundary, the next

order correction was found. Furthermore, using the Gaussianity of the relevant

models, it was possible to derive detailed predictions about the distribution of

nodal intersections (NI) with an arbitrary reference curve. This distribution, which

was found to be different than other, well known, point processes, can be used to


characterize the RW field. Comparing it to the correspondingnumerical distri-

bution in chaotic billiards yielded supportive evidence tothe validity of the RW

model.

In analogy to the case of the spectrum, where the Gutzwiller trace formula was

used to gain theoretical support for BGS conjecture, a similar trace formula for the

BIC should provide a powerful theoretical tool for understanding that sequence

beyond the average “Weyl-like” term. For separable systems, where the nodal

pattern is simple and predictable, we derived a trace formula by using the methods

of [11] and [31]. To sample the case of integrable systems which are not separable,

we derived a trace formula for the BIC of the right angled isosceles triangle. In

this case, although the nodal pattern has a complex structure, there exists a closed

formula, expressing the BIC in terms of the quantum numbers.The resulting trace

formula was verified to high level of accuracy. It was found tobe similar to the

separable case, while differences do exist in higher order corrections.

Finally, we derived a trace formula for the chaotic case. This result is not

entirely rigorous, as it employs some assumption of Gaussianity (analogous to

the assumption which led to the RW model). Nevertheless, it was successfully

supported by the BI statistics of numerically computed eigenfunctions.

1.2 Structure of the thesis

In chapter 2, we derive a formula2 that predicts the distribution of the points where

nodal lines of a given Gaussian field intersect a reference curve. This was em-

ployed on various random fields which are commonly used to model eigenfunc-

tions of chaotic billiards. The new universal point statistics obtained in this way

were found to be qualitatively different than other, well known point statistics.

Chapter 3 deals with integrable billiards. Using semi-classical torus quantiza-

tion, we derived an asymptotic trace formula for the BIC of separable systems, and

verified it for the circle and rectangle billiards. As an example for a system which

is integrable but not separable, we examined the right isosceles triangle billiard,

which is one of the few non-separable systems that have simple explicit solutions.

2A generalized form of a formula that was first discovered by Rice [32].

1.2. STRUCTURE OF THE THESIS 7

We found the BIC as a function of the quantum numbers, and proceeded to derive

a trace formula for the BIC sequence of this billiard as well.The result has the

same form as the formula for the separable case. The qualitative differences occur

in higher order corrections.

Chapter 4 contains a derivation of a trace formula for the BICof chaotic bil-

liards. Based on numerical results, we conjectured that thelimiting distribution of

the ensembleUσ(k) = un(s) | |kn−k| . σ ask → ∞ is Gaussian if the billiard

is chaotic. Given this conjecture, it is possible to derive the BIC from the correla-

tions ofu. Using a semi-classical expansion of the boundary Green function [33],

we derived a trace formula for amplitude correlations inu. This, combined with

the conjecture, provided the desired trace formula for the BIC, which was also

confirmed numerically.

Chapter 5 contains conclusions and discussion of the results.

Chapter 2

Statistical models for nodal

intersections with a curve

The nodal domains of 2D billiards have, in the general case, acomplex topo-

logical structure, and not much is known about their properties. However, for

the case where the classical motion is fully chaotic, statistical methods (based on

Berry’s random wave model [23]) had a remarkable success in producing results

that could be verified numerically ([24], [18], [34]). In this chapter we assume

such statistical models for the eigenfunction, and derive the expected statistics for

the distributions that their nodal lines induce on a reference curve. In particu-

lar, the prediction for the value of the BIC, averaged over eigenfunctions whose

ordinal number is aroundn, is found to be

〈η〉(n) ∼ L2πq +

L2 − 6πA4πA , (2.1)

whereq =√4πn/A is the leading asymptotic estimate for thenth wavenumber

kn. The leading term was reported in [18], while the second termrequires a refined

model, as described below. This result matches the smooth part (first two terms)

of the semi-classical trace formula, equation (4.32), which is derived in chapter 4

based on a semi-classical expansion.

While the main focus of this thesis isη and the relevant local density of in-

tersectionsb (related byη =∫∂Ωb(s) ds), the models used in this chapter were

9

10 CHAPTER 2. STATISTICAL MODELS

simple enough to compute more detailed features of the distribution. Since the

pattern of BI is a one dimensional point process, we can compare it to known

universal distributions which are often used to model otherpoint processes (such

as the spectrum). For this purpose, a universal two-point correlation function is

computed for each model by taking the limitk → ∞ while keeping the mean

spacing normalized to 1. From the correlation function, other two-point statistics

can be extracted, and compared to the corresponding data from other distribu-

tions. The universal distributions produced by the random wave models displayed

unique features which do not appear in distributions that are used to model energy

spectra. Before going on to a description of the results (section 2.4), we start by

describing the statistical models we used (sections 2.1-2.2) and the formulae that

were used to extract the required statistics (section 2.3).

2.1 The Monochromatic Random Wave model

In [23], Berry had computed semi-classical approximationsfor the averaged prob-

ability densityΠ(r) = 〈|ψ(r)|2〉 and the correlationG(r, r′) = 〈ψ(r)ψ∗(r′)〉,whereψ is an eigenfunction of a Hamiltonian representing some bound classical

motion. The angled brackets denote averages taken over an ensemble of eigen-

functions1 in a small spectral window around a specified energyk2 ≫ 1. For

chaotic 2D billiards, assumingr andr′ are bounded away from the boundary,

the approximated probability density is constant (Π ∼ 1/A), and the correlation

function was found to be

GRW(r, r′) = J0(k|r − r′|) · Π (2.2)

(J0 is Bessel’s function [35] of order 0). He then suggested thatfor chaotic classi-

cal motion, the contributions that constituted these results should be uncorrelated,

and therefore (by a generalized central limit theorem) the field ψ(r) should be

Gaussian.

1As opposed to the averages used in this thesis, Berry had useda fixed eigenfunction, and acoordinate-space average over some region aroundr andr′. However, the results quoted hereremain the same.

2.1. THE MONOCHROMATIC RANDOM WAVE MODEL 11

A random fieldψ(r) is Gaussian if for any vectoru whose elementsu1, u2,

. . .un are field amplitudesui = ψ(ri) or partial derivatives of the field at some

pointsr1, . . . , rn, the probability distribution ofu is multivariate normal. The

probability density of a multivariate normal random vectoru, with mean valueu0

is given by

p(u) =1

(2π)n/2√detC

exp

[−1

2(u− u0)

TC−1(u− u0)

], (2.3)

whereC is the covariance matrixCi,j = Cov(ui, uj). Assuming smoothness of

the field, means and covariances of the field’s spacial derivatives can be calculated

by deriving the appropriate amplitude statistics. Combining that with the fact that

all the elements ofu0 andC are single point means or two-point covariances, it

follows that all statistical properties of a Gaussian fieldψ(r) are completely deter-

mined by its mean value〈ψ(r)〉 (which we assume to be zero) and the two-point

covariance functionG(r, r′) = Cov(ψ(r), ψ(r′)). The Gaussian field whose cor-

relation function is given byJ0(k|r− r′|), as in equation (2.2) (the normalization

constant does not matter because scaling does not affect thenodal pattern), is

called Berry’s monochromatic random wave model. This random wave field can

be realized as a linear superposition of plane waves:

√2

N

N∑

j=1

cos(kjr + φj) (2.4)

(with |kj | = k andN ≫ 1), wherekj/|k| andφj are distributed uniformly and

independently on the unit circle. Equivalently, one can also use random superpo-

sitions of solutions of the wave equation (2.6) in polar coordinates(r, θ):

a0J0(kr) + 2∑

l>0

alJl(kr) cos(lθ + φl) (2.5)

with real coefficientsal, which are identically and independently distributed Gaus-

sian variables, and where the phasesφl are independent and uniformly distributed

on [0, 2π).

With these representations it becomes evident that each realization of the field


Figure 2.1: Sign plots: billiard eigenfunction (left) compared to random waves (right).k = 60.0in both cases.

is a solution of the wave equation

−∆ψ(r) = k2ψ(r) (2.6)

in R2. Hence, its Fourier transformψ(k) is limited to the circle|k| = k, which is

the reason we call it “monochromatic”. It is also evident that it is invariant under

translations and rotations. In fact, the random wave model is the unique (up to the

normalization factorΠ) Gaussian distribution which is both monochromatic and

invariant.

Figure 2.1 is a sign-plot (the colours in the plot encode the sign of the func-

tion) showing the nodal domains of an eigenfunction of a chaotic billiard (the

Africa billiard, see chapter 4) next to the nodal domains of arealization of the

RW distribution, simulated using equation (2.4). It can be seen that the nodal

domains have a similar structure in the bulk (i.e. in regionswhich are not too

close to the boundary). This visual observation was quantitatively supported by

comparing the predicted value for various measurable features to the numerically-

computed equivalents in eigenfunctions of chaotic billiards. Examples for such

features are avoidance (measure for suppression of near crossings) [36], average

area and number of nodal domains [18, 24], and morphologicalfactors [34]. For

comparison, Foltin et al. [26] suggested a different Gaussian field, which is also

2.2. BOUNDARY ADJUSTED MODELS 13

invariant under rotations and translations, but has a shortrange (exponentially de-

caying) correlation function (short range field, SRF model). In section 2.4.4, we

shall add a reference line inside the billiard, and use the statistics of the generated

1D point process as further means for comparing such 2D models and testing the

validity of Berry’s random wave model.

2.2 Boundary adjusted models

The RW model only applies for positionsr whose distance from the boundary∂Ω

is large compared to the wavelength1/k. To get meaningful results regarding the

BI, we need a model that works all the way to the boundary. In fact, the region

which is relevant for our purpose is the limit wherer approaches the boundary.

Since any smooth boundary can be approximated by a straight line when the wave-

length is much smaller than the radius of curvature, our firstapproximation should

be a model which is defined on a half-plane, and satisfies the proper conditions on

the boundary line.

A Gaussian field which is defined on the half planey > 0 and satisfies the

Dirichlet boundary condition on thex axis can be constructed by symmetrization

of the unbound random wave field:

ψBRW(x, y) = ψ(x, y)− ψ(x,−y) (2.7)

(ψ = ψRW is distributed according to the RW model andψBRW is the boundary-

adjusted random wave (BRW) model). Applying this symmetrization to equa-

tion (2.4) one gets a Gaussian model which is also monochromatic (satisfying

equation (2.6)) and identically zero on the “boundary”y = 0. In [37], Berry uses

this model to derive various statistics. Instead of following this approach, we no-

tice (from equation (2.7)) that the value of the field at distanceǫ from the boundary

is

ψBRW(x, ǫ) ∼ 2ǫ∂ψ

∂y

∣∣∣∣(x,0)

.

Hence, on a line which is parallel to the boundary at a very short distance from it,

the field is proportional to the normal derivative of the “generating” RW field


ψ across the boundary. This suggests that the naıve way to extend the BRW

model to a general billiard boundary would be to study the normal derivative of

a monochromatic RW field across the boundary. This normally derived random

wave (NRW) field is defined on the boundary as

ψNRW(s) = −n(s) ·∇ψ(r(s)) , (2.8)

wheren(s) is the outwards pointing normal vector atr(s), andψ is distributed

according to the RW model.

With the NRW model, the first order estimate for the BI densityis 〈b(s)〉 ∼k/(2π), which matches2 equation (2.1). However, the correction due to curva-

ture isO(k−1) (see in [18] and in section 2.4), and does not match the second

term of (2.1). The reason for that is that in this model the field ψ is indepen-

dent of the boundary. The only way the curvature can affect the distribution is by

changing the direction in which the derivative is measured.The symmetrized RW

model (2.7) on which it is based has a straight boundary and cannot predict the

direct effect of the curvature on the field.

In [38], Wheeler constructed a Gaussian model satisfying Dirichlet boundary

conditions on a curved boundary. The domain considered is the outer part of a

circle of radiusR (the domain is concave, with curvatureκ = −1/R). In order

to build such a model, consider the polar representation of the RW model, equa-

tion (2.5). When the origin is excluded from the domain, the singular solutions

of (2.6),Yl(kr) cos(lθ + φl) (which diverge at the origin), become relevant, and

they should be added to the expansion. It is possible to choose their coefficients

such that on the boundaryr = R, eachYl term cancels the contribution of the cor-

respondingJl term. The result is a concrete Gaussian model which is monochro-

matic and satisfies the boundary condition. This might be referred to as the curved

BRW (or CRW) model. To get the results needed for computing the BI density,

we used Wheeler’s results in the limitr → R+, as described in section 2.3.2. The

resulting BI density in this model is〈b(s)〉 ∼ (k− 32κ(s))/(2π), which reproduces

equation (2.1). Note that the CRW model has negative curvature, and the validity

of the result to convex domains is not rigorously proved. However, it is consistent

2See section 2.4.1 for a discussion regarding the connectionbetween〈b(s)〉 and〈η〉(n).

2.3. EXTRACTING THE BI STATISTICS 15

with numerical tests and with the semi-classical results ofchapter 4.

2.3 Extracting the BI statistics

In order to derive BI statistics from a given 2D statistical model (such as RW,

BRW or CRW), one needs to derive the required statistics for the restriction of

the field to the relevant reference curve3. Given a realization of the fieldψ(r),

the restricted 1D field isf(s) = ψ(r(s)), wherer(s) is the natural (arc-length)

parametrization of the curve. The Gaussianity off (which follows trivially from

the Gaussianity ofψ) greatly simplifies the calculations, and allows derivation of

closed-form formulae for both the BI density

〈b(s)〉 =⟨∑

i∈BI

δ(s− si)

⟩=⟨δ(f(s))|f(s)|

⟩

and the BI two-point correlation function

R(s, s′) =

⟨∑

i 6=jδ(s− si)δ(s

′ − sj)

⟩

=⟨δ(f(s))δ(f(s′))|f(s)f(s′)|

⟩− δ(s− s′)〈b(s)〉.

Such formulae were first derived by Rice [32]. They depend on the correlations

of the restricted fieldf and its derivative along the curvef(s) ≡ df/ds:

C0(s, s′) ≡ Cov(f(s), f(s′))

C1(s, s′) ≡ Cov

(f(s), f(s′)

), C1(s, s

′) ≡ Cov(f(s), f(s′)

)

C2(s, s′) ≡ Cov

(f(s), f(s′)

)(2.9)

We will also use lowercasec to denote the single point limit of these, i.e.c0(s) ≡C0(s, s) = Var(f(s)) etc.

To get explicit expressions for〈b(s)〉 andR(s, s′), one needs to express the co-

variances (2.9) in terms of the respective model’s 2D correlation functionG(r, r′),

3In the case of the NRW, no restriction is needed, as the field isalready 1D.


and then substitute the result in the relevant “Rice formulae” (which are given

in section 2.3). In [39] Longuet-Higgins applies a similar process to compute the

“zero crossing rate” of random sea waves. In what follows, wedescribe the for-

mulae that were used in this process. The derivation was published in [27]. It

follows the same methods that were used in [40] and [39]. However, our results

are more general, because as opposed to the case that was investigated in [39], our

reference line (e.g. the billiard boundary) can not be assumed to be a straight line.

Hence, the 2-point statistics of the restricted field are nottranslation invariant, and

the Rice formulae [32] had to be generalized as well. The computation of the BI

density for the CRW model is described in more details, sinceit does not appear

in [27].

2.3.1 Formulae for expressing BI statistics in terms ofG(r, r′)

Before we write down expressions forC0, C1 andC2, let us take note of spe-

cial properties of these functions, which follow from the symmetries of our mod-

els. For fields which are translationally invariant, the mean and the variance of

the field are constant over the domain. This allows us to applya constant lin-

ear transformationψ = Aψ + B, normalizing the field so that〈ψ(r)〉 = 0 and

c0 = 〈ψ2 (r)〉 = 1 for all r (equations (2.4) and (2.5) are normalized in this way).

Furthermore, wheneverc0 is constant over the reference curve, one can show that

for |s− s′| ≪ 1 we haveC1(s, s′) ∼ −C1(s, s

′), and in particularc1 = 0.

For isotropic fields (such as the RW model), the correlation function depends

only on the distanced = |r − r′|, so we use a single parameter functionG(d) =

〈ψ(r)ψ(r′)〉. In terms of this function, the correlation functions are:

C0 = G(d)

C1 = −r′ · d G(d), C1 = r · d G(d)

C2 = −∑

i,j

ri

[(G− G

d

)didjd2

+G

dδi,j

]r′j, (2.10)

wherer′ ≡ r(s′), d = r − r′, andd ≡ d/d (the dots onr denote derivative with

respect to the arc-lengths, while G andG are derivatives by the 2D distance).


For the NRW, the “restricted” field isg(s) = −n(s) ·ψ(r(s)), wheren(s) is

the unit vector normal to the curve at the pointr(s). Here as well, we can use the

isotropy of the generating fieldψ and get formulae for the restricted correlations

C0–C2 on a general curve. These are given in [27]. Here we only quotethe results

for the case where the reference curve has constant curvatureκ. In this case, we

denoteα = 12κ · (s− s′), d = 2 sin(α)/κ, and the correlations are given by:

C0 = 〈gg′〉 = − cos2 αG(d)

d+ sin2 αG(d)

C1 = 〈gg′〉 = cosα(1− 3 sin2 α

)(G

d− G

d2

)

− sin2 α cosα

(...G+ 4

G

d2

)

C2 = 〈gg′〉 = −2

(cos(2α) +

7

8sin2(2α)

)1

d2

(G− G

d

)

+

(1− 3

2sin2(2α)

) ...G

d

− sin2 α

(cos2 αG(4) + 4 cos(2α)

G

d3

), (2.11)

where as before, primed quantities are taken ats′. The correlations for the straight

line are easily derived from equations (2.11) by taking the limit α → 0.

In terms of the correlationsC0–C2 (and theirs → s′ limit c0–c2), the density

of intersections is given by

〈b(s)〉 = 1

π

(∂2 logC0(s, s

′)

∂s∂s′

)1/2∣∣∣∣∣s′=s

=1

π

√c2c0 − c12

c02, (2.12)

which in the cases wherec0 is constant over the curve, reduces to Rice’s original

formulab = 1/π√c2/c0.

For the two-point correlationR(s, s′), we write down the covariance matrix of


the four variables(f, f ′, f , f ′) (with f ≡ f(s), f ′ ≡ f(s′))

M =

c0 C0 c1 C1

C0 c0 C1 c1

c1 C1 c2 C2

C1 c1 C2 c2

≡(A C

CT B

)

(A,B,C stand for the2 × 2 sub-matrices). In terms of these parameters, and

assumingc1 = 0, the two-point correlation function is given by

R(s, s′) =1

π2

a

|A|3/2(√

1− c2 + c arcsin(c)),

wherea = c2|A| − c0|C|, c = (C2|A| − C0|C|)/a, |A| = c02 − C0

2 and|C| =−C1C1 (here|A| and|C| stand for determinants—not absolute value).

The normalized correlation functionR = R/(bb′) − 1 is another parameter,

which is useful for computing the statistics in section 2.4.For the case wherec0is constant over the curve, it is given by

R =c0c2

a

|A|3/2(√

1− c2 + c arcsin(c))− 1. (2.13)

Due to the normalization,R is dimensionless, so the lowest order approximation

should be independent ofk. In other words, takingk → ∞ produces a “uni-

versal” limiting distribution (i.e. a distribution that does not depend on the scale

introduced by the wavelength of the field under consideration).

2.3.2 BI Density of the CRW model

The CRW model satisfies (2.6) inR2 \ CR, whereCR =(x, y)

∣∣|x2 + y2| < R

,

with Dirichlet boundary condition on the boundary. Using polar coordinates(r, θ),

consider the circler = R + ǫ, with ǫ ≪ 1, which is parallel and very close to the

boundary. The arc-length parameter iss = (R + ǫ)θ, the restricted field is given

by

f(s) = ψ(R + ǫ, θ) = ǫ∂ψ

∂r

∣∣∣∣(R,s/(R+ǫ))

,


and

f(s) =ǫ

R + ǫ

∂2ψ

∂r∂θ

∣∣∣∣(R,s/(R+ǫ))

.

Using equation (2.12), and takingǫ→ 0 we see that

〈b(s)〉 = 1

πR

√〈ψrθ2〉〈ψr2〉

, (2.14)

where the indices denote partial derivatives (e.g.ψr = ∂ψ/∂r).

In [38], Wheeler uses the following representation of the CRW field

√2

N

∑

j=1...Nl∈Z

[(Jl(kr)Jl(kR) + Yl(kr)Yl(kR)

)cos(l(θ + ϕj) + φj)

−(Jl(kr)Yl(kR)− Yl(kr)Jl(kR)

)sin(l(θ + ϕj) + φj)

]

· −Jl(kR)Jl(kR)2 + Yl(kR)2

+Jl(kr) · cos(l(θ + ϕj) + φj)

,

where Gaussianity is achieved by summing upN ≫ 1 “planar” solutions (as

in (2.4)). The directionsϕj and phasesφj are distributed uniformly in[0, 2π).

This representation allows computation of second order moments by utilizing the

orthogonality relations of the trigonometrical functions. In particular〈ψr2〉 was

computed in [38]:

k2∞∑

l=−∞

Jl

′(kr)2 − Jl(kR)

Jl(kR)2 + Yl(kR)

2

[Jl(kR)

(Jl

′(kr)2 − Yl

′(kr)2)

+ 2Yl(kR)Jl′(kr)Yl

′(kr)].

(2.15)

The other needed moment,⟨ψrθ

2⟩, can be computed using the same method:

k2∞∑

l=−∞l2Jl

′(kr)2 − Jl(kR)

Jl(kR)2 + Yl(kR)

2

[Jl(kR)

(Jl

′(kr)2 − Yl

′(kr)2)

+ 2Yl(kR)Jl′(kr)Yl

′(kr)].

(2.16)


Next, we need to estimate the asymptotics of equations (2.15)–(2.16) askr =

kR → ∞. The estimates that were used in [38] cannot be used, since they are

singular atr = R, so other methods must be used. The result for the required ratio

is〈ψrθ2〉〈ψr2〉

=(kR)2

4

(1 +

3

kR+

1

(kR)2+O

((kR)−3)

).

Finally, substituting this in equation (2.14) we get

〈b(s)〉 = 1

2π

(k − 3

2κ− 5

8

κ2

k+O(k−2)

), (2.17)

whereκ = −1/R.

2.4 Summary of results

2.4.1 BI Density

As described in section 2.3, the expected density of intersections on a reference

curve〈b(s)〉 can be computed using equation (2.12). For the models discussed

in section 2.2 the results are given in table 2.1 (whereκ denotes the curvature at

the reference pointr(s)).

Table 2.1: BI Density for Gaussian models.

Model : RW/SRF† NRW MRW‡/CRW

Density:k√2π

,1

2π

√k2 + (2κ)2 ,

k

2π− 3κ

4π− 5κ2

16πk+O(k−2)

†By construction, the SRF model has the sameb as RW.‡For MRW, substituteκ = 0.

The averaged BIC can be computed by integrating over the boundary 〈η〉 =∮〈b(s)〉ds. For the CRW, this gives

〈η〉(k) ∼ kL2π

− 3

2+ O(k−1), (2.18)

where for the second term we have used the Gauss-Bonnet formula∮κ(s) ds =

2.4. SUMMARY OF RESULTS 21

Figure 2.2: Density of the nearest neighbour level spacing.

2π. Finally, to express this in terms ofn, we invert the Dirichlet-corrected Weyl

approximation forn(k):

k(n) ∼√

4πn

A +L2A .

Inserting this in (2.18) yields〈η〉(n), equation (2.1).

2.4.2 Numerical comparison

When considering the statistics of a sequence of points, thenearest neighbour

spacing distribution (where the spacing is measured in units of the mean spacing)

is perhaps the most natural, and easy to evaluate experimentally. In figure 2.2 we

show the nearest neighbour distributions for the zeros of the RW and NRW mod-

els on a straight reference curve. The distributions were generated by numerical

simulations, and are compared with the corresponding statistics of the random

(Poisson) ensemble and the Wigner surmise (corresponding to the GOE ensem-

ble) [41]. At short ranges, both ensembles display linear level repulsion similar to

the GOE distribution, but with different slopes. The most conspicuous differences


Figure 2.3: Nearest neighbour level spacing (log density). Inset showspersistent oscillationsrelative to the mean decaying curve.

appear at large spacings, and to get a clearer impression, weshow in figure 2.3,

the same data in semi-log plot. We observe two important differences.

1. On average, both distributions decay exponentially withapproximately the

same ratep(s) ∼ exp(−1.4s). This decay is faster than theexp(−s) Pois-

son decay, but slower than the “Semi Poisson” [42] distribution, which de-

cays likeexp(−2s).

2. The overall exponential decay is decorated by persistentoscillations, which

are clearly seen in the inset of figure 2.3. The oscillations have slowly de-

caying amplitudes and their frequencies are∼ 1.4π for RW and∼ 2π for

NRW.

The results above demonstrate that the nodal intersection statistics are signif-

icantly different than those of other, well known, point processes. However, the

nearest neighbour statistic is not easily amenable to analytic derivation. Rather,

we use the two-point correlation function, for which we haveexplicit formulae, as

described in section 2.3. For the asymptotics ≪ 1 limit, this statistic coincides


Figure 2.4: Normalized Correlation: RW and NRW models, compared to GOE.

with the nearest neighbour density. Furthermore, other statistics with well defined

physical meaning, such as the form factor and number variance can be expressed

in terms of the two-point correlation function.

2.4.3 Correlation functions and form factors

The normalized two-point correlation functions for the RW and NRW models are

shown in figure 2.4. Equation (2.13) gives an exact expression for these functions.

However, since they have a very complicated form, [27] contains asymptotic ex-

pansions for small and large arguments. Nears = 0 the functions rise linearly.

The slope for the RW model isπ2/16, and for the NRW it isπ2/8. Fors≫ 1, the

leading terms are given by

RRW ∼ 1

πωRWs(1 + 9 sin(ωRWs)) RNRW ∼ 2

π(2πs)3(9− 25 sin(ωNRWs)),

whereωRW = 2π√2, andωNRW = 4π.

The form factor (scaled power spectrum) for the NI is the Fourier transform of


Figure 2.5: Form factor for the NI of normally derived RW.

the scaled two-point correlation function (up to the subtraction of aδ(τ) term),

K(τ) =

∫

Γ

ei2πτs〈b(x)b(x+ s)〉

〈b(x)〉2ds− δ(τ)

= 1 +

∫

Γ

ei2πτsR(s) ds, (2.19)

whereΓ is the reference curve.

In figure 2.5 the form factor of the RW and NRW models are plotted. As op-

posed to the GOE (and other RMT ensembles), whereK(0) = 0, the NRW model

has a finite value atτ = 0, while the RW exhibits a logarithmic divergence. The

inset exhibits the effect of a curved reference line: oscillations whose frequency

is increased as we increasek.

The number variance functionΣ2(L) = Var(ηL), whereηL is the number of

NI on a segment of lengthL around some point on the reference curve, can also

be expressed in terms of the normalized correlation [41]:

Σ2(L) = L+ 2

∫ L

0

(L− s)R(s) ds. (2.20)


[27] contains asymptotic expansions for the number variance of the Gaussian

models, as well as plots comparing them to the correspondingfunctions based

on other models.

2.4.4 Comparison to other distributions

Various statistics for the NI distributions of the Gaussianfields discussed above

are summarized in table 2.2, and compared to their analoguesin the well known

random matrix and Poisson ensembles. For short distances, the normalized cor-

relation function for all the distributions considered (except of the Poisson distri-

bution) approaches−1, so the leading term of1 +R(s) is listed, fors = ǫ ≪ 1.

Similarly, the number variance for short intervals is asymptoticallyL−L2 (again,

except of the Poisson case), so the leading term ofR2 = Σ2(L) − L + L2 =

〈ηL(ηL−1)〉 is listed. (forL = ǫ≪ 1, this is also twice the probability for finding

two points in the interval). For large distances, we list theleading terms ofΣ2 and

R. For the form factor, we list the “asymptotic rigidity”K(0) (which is directly

related to the number varianceΣ2 and rigidity∆ in large intervals [41],[43]), the

singular frequencies whereK(τ) or its derivatives diverge and the type of di-

vergence (number of the first diverging derivative) in the first non zero singular

frequency.

Table 2.2: Comparison of NI statistics.

ProcessCorrelation Form Factor Number Variance

1+R(ǫ) R(s≫ 1) K(0) Divergences† R2(ǫ) Σ2(L≫ 1)

Poisson 1 0 1 — ǫ2 L

SRF 2.467ǫ 48.7s4e−(πs)2 0.572 — 0.822ǫ3 0.572LNRW 1.234ǫ 0.023s−3 0.484 (3)2, 4, . . . 0.411ǫ3 0.484L

RW 0.617ǫ 0.036s−1 ∞ (1)√2, 2

√2, . . . 0.206ǫ3 0.072L logL

GOE 1.645ǫ −0.101s−2 0 (4)1 0.548ǫ3 0.203 logLGUE 3.290ǫ2 −0.051s−2 0 (2)1 0.548ǫ4 0.101 logLGSE 11.55ǫ4 0.25s−1 cos(ωs) 0 (0)1, 2 0.770ǫ6 0.051 logL

†Nonzero values ofτ , for whichK(τ) or one of its derivatives diverge. The number in parentheses

specifies the diverging derivative at the first of these points.


(a) Stadium billiard

Line 2

Line 1

(b) Sinai billiard

Figure 2.6: Internal reference curves in the quantum billiards

It should be noted that while for short distances the NI repellinearly, like

the levels in the GOE do, the long range behaviour is quite different. The NI

counts in long segments do not exhibit “spectral rigidity” (low variance) like the

RMT ensembles. In fact, the variance grows likeL logL, which is faster than the

Poisson model (which is the prediction of the critical percolation model).

Finally, as a test for the validity of Berry’s random wave conjecture, some of

the statistics considered above were evaluated numerically for eigenfunctions of

desymmetrized Bunimovich stadium (0.5 × 1 rectangle joined with a quarter of

a circle of radius1) and desymmetrized Sinai billiard (1.2 × 1 rectangle, with a

quarter of a circle of radius0.5 cut out of one of the corners)—both with Dirichlet

boundary conditions. For the Stadium, 1500 eigenfunctionswere taken, with wave

numbers ranging fromk = 110 to k = 165. For the Sinai billiard, we used 10000

wave functions, with wave numbers from 350 to 500.

Fixed reference curves were chosen in the interior of the billiards, as shown

in figure 2.6. For each wave function, the sequence of intersections of the nodal

lines with these curves was calculated, and normalized to unit average spacing

according to the corresponding wavenumber. As shown in figure 2.7, the nearest

neighbour distribution agrees very well with the predictions of the random waves

model. The form factor is also very close to the expected curve.

The number variance of nodal intersections on segments of the reference curve

is plotted in figure 2.8. For short segments, the results are close to the predictions

of the random waves model, but deviate considerably from this model as we move

to segments of large normalized length, especially in the case of the Sinai billiard.


(a) NN spacing (b) Form factor

Figure 2.7: Billiards: NI with internal line.

This deviation happens when the (unscaled) length of the segment is comparable

to the billiard dimensions. On such scales, it seems likely that the geometrical

details of the billiard would have an effect on the statistical properties of the wave

functions.


Figure 2.8: Number variance of NI with segments inside chaotic billiards.

Chapter 3

Integrable Billiards

Before going on to consider chaotic billiards, we examine several simpler cases,

where trace formulae can be obtained in a straightforward way. Specifically, we

consider some integrable billiards, where exact analytical expressions are known

for both the wavenumberk and the BICη in terms of the quantum numbersl,m.

In such cases, an expression for the BIC densityη terms ofn, the position in the

energy sortedsequence of eigenfunctions, can be derived in two stages. First, the

spectral BIC densitydη(k) =∑

n ηnδ(k−kn) is written as a sum overl andm, and

evaluated using Poisson’s summation formula (PSF). Second, the known formula

for n(k) is inverted and substituted in the expression, yielding an expression for

dη(n) =∑

m ηmδ(n −m), which is simply related toηn. This method was used

in [31] to compute trace formulae for the number of nodal domains νn. In [18] a

simplified variant of this method was used to compute the limiting distribution for

ηn/√n. However, trace formulae forηn in these simple cases were not previously

published.

In section 3.1 we provide trace formulae fordη(k) in separable billiards, where

the nodal pattern is a simple checker-board (see figure 3.1).We give explicit

formulae for the rectangle and circle billiards. These formulae were derived as

special cases of a more general formula, for counting nodal intersections in a

wider class of separable problems. In section 3.2 we follow the second stage

of the procedure described above, to get the required formulae fordη(n). Then,

in section 3.3, we consider a problem which is integrable butnot separable. It

29

30 CHAPTER 3. INTEGRABLE BILLIARDS

Figure 3.1: Separable billiards: sign plots for the rectangle and circle,k ∼ 60.

is interesting to compare this to the separable case, because the nodal patterns of

such problems are much more complex than the patterns in separable problems,

despite the fact that the underlying classical motions are basically equivalent. We

find that the lowest order terms of the formulae have the same structure.

3.1 Separable problems

One class of problems whose nodal pattern is particularly simple is the class of

integrable systems which are also separable. As mentioned in [18], these include

the rectangle and the elliptic billiards, as well as surfaces of revolution and Liou-

ville surfaces. Within this class, only the rectangle and the ellipse match the scope

of this thesis, which is flat billiards with a boundary. However, it is still beneficial

to derive a more general formula that holds for the whole class and covers these

two problems as special cases. The resulting formula is alsomore general in the

sense that it gives the number of intersections with any set of curves∂Ω that sat-

isfies a certain condition (described below)—it does not necessarily have to be a

boundary. As such, it is also applicable to problems that do not have a boundary

at all (such as tori and surfaces of revolution).

For separable quantum problems, there exists a coordinate systemqi(x, y)

(i = 1, 2), and an orthogonal basisψl,ml,m∈N for L2(Ω), whereψl,m are eigen-

functions which can be written in a factorized formψl,m(q1, q2) = φl(q1)ϕm(q2).

3.1. SEPARABLE PROBLEMS 31

Let q(0)1 be a point satisfyingφl(q(0)1 ) = 0, then the line(q(0)1 , q2) will be called aq1

nodal line ofψl,m. Similarly (q1, q(0)2 ) is aq2 (nodal) line ofψl,m if ϕm(q

(0)2 ) = 0.

The nodal pattern is generally a mesh grid ofq1 lines andq2 lines. In the case

of Dirichlet billiards, the boundary∂Ω is also composed of one or moreqi nodal

lines. Furthermore, in the class of problems considered, the following condition

holds:

Condition1. There exist integers(τ1, τ2) such that eachqi nodal line intersects

the set∂Ω at exactlyτi points.

The factor functionsφl(q1) of the separable problems under consideration sat-

isfy the Sturm oscillation theorem. Hence,φl has exactlyl zeros, and the mesh

contains exactlyl q1 nodal lines. Similarly, there arem q2 lines, and by condi-

tion 1, the number of NI ofψl,m with ∂Ω is lτ1 + mτ2 (since we have not used

any other property of∂Ω, the result holds for any set of curves∂Ω that satisfies

condition 1).

From this, we can proceed to derive an expression for the BIC spectral density

dη(k), using the quantum conditions of Einstein, Brillouin and Keller (EBK). The

derivation, which is explained in appendix A, follows the footsteps of [11] (see

also [21]). As expected, the result depends on classical parameters, which we

now describe. For a review on semi-classical quantization,see [44]. Letp1, p2be the canonical momenta corresponding toq1, q2, andI1, I2 the corresponding

actions (Ii =∫pi dqi on a trajectory that takesqi andpi to their initial value). The

classical motion on a torus with fixed energyE(I1, I2) is characterized by angular

frequenciesωi = ∂E∂Ii

, where the value of each coordinateqi is periodic with period

Ti = (2π)/ωi. For each torus, we define the “partial BIC contribution”η(I1, I2) ≡τ1I1 + τ2I2. We will also use the Maslov indices(α1, α2), which take account

of the divergences encountered at turning points of the respective coordinates.

Turning points that correspond to reflection from a Dirichlet boundary contribute

2 to the Maslov index [45], as demonstrated in sections 3.1.1-3.1.2. Periodic

tori (PT) are tori whose angular frequencies are rationallydependent(ω1, ω2) =

(µ, ν)2πT , whereµ, ν are co-prime integers. In this case, orbits on the torus close

up, and the torus becomes a continuous family of periodic orbits, all of which have

periodT . The pair(µ, ν) is called the topological index of the torus (see [11]).

Factorizable pairs(M,N) = (rµ, rν) correspond tor repetitions of a(µ, ν) orbit.


When using(M,N) to refer to a “generalized periodic torus” (GPT) , we mean

the collection of orbits, each of which isr repetitions of an orbit from the torus

(µ, ν) (hence, its period and length arer times longer).

Since we are considering a single particle system, the effect of the particle’s

massm on physical quantities such as the energy and momenta is trivial. For

simplicity, we choose the mass such that~2/(2m) = 1. Furthermore, as in [18],

we will assume that the HamiltonianH(I1, I2) is homogeneous of degree 2. This

allows us to scale the problem to the unit energy surface, which in our case is

the curveH(I1, I2) = 1. The curvatureκ of this curve, at the point(I1, I2) also

appears in the formula. With these notations, the trace formula for the BIC spectral

density is given by

dη(k) ∼3V η

4π2k2 (3.1)

+k3/21

π

∑

γ∈PT

∞∑

r=1

2Tγηγ|µγ|3/2

√r|κγ|

cos(r(2kTγ −

π

2µγ ·α)− σγ

π

4

),

whereγ enumerates periodic tori (PT) on the unit energy surface,I(γ)i , µγ, Tγ

are, respectively, the actions, topological index, and period of a representative

orbit on the torus,κγ is the curvature of the energy surface at(I(γ)1 , I

(γ)2 ), ηγ ≡

τ1I(γ)1 + τ2I

(γ)2 is the BIC contribution of the torus,α = (α1, α2) are the Maslov

indices, andσγ is the sign ofκγ . In the first, Weyl-like term,V is the volume of

phase space upto unit energy, andη is the average BIC contribution in that volume

η =1

V

∫η(q,p)Θ(1−H(x,p)) d2q d2p,

(whereΘ(x) is the Heaviside step function), The coefficient of theO(k2) “Weyl

term” is a sum of two partial contributions3V η/(4π2) = τ1A1 + τ2A2, where the

“Weyl coefficients”Ai are given by

Ai =3

4π2

∫Ii(q,p) Θ(1−H(x,p)) d2q d2p. (3.2)


Equivalently, they may be described as an integral over the unit energy surface

Ai =

∫ A

0

2 ds

ω(s)Ii(s), (3.3)

where(I1(s), I2(s)) is the arc-length parametrization of the surface,ω is the mag-

nitude of angular frequenciesω(s) =√ω1

2 + ω22, andA is the total area (arc-

length) of the surface (see appendix A). Note that due to the homogeneous scal-

ing, actions measured on the unit energy surface (or scaled to fit in the unit energy

volume) acquire extra dimension ofE−1/2. Hence, in our units,V has units of

area, and the BIC contributionsη, ηγ have units of length.

For flat billiards, the particle moves at constant speed ofp/m = 2k. Hence on

the unit energy surface we haveT = L/2 (whereL is the length of the orbit. The

volume of phase-space upto unit energy is∫q∈Ω,p2<1

d2q d2p = πA (whereA is

the area of the billiard). Substituting these in equation (3.1), we finally get

dη(k) ∼3Aη4π

k2 (3.4)

+k3/2

π

∑

γ∈PT

∞∑

r=1

Lγ ηγ

|µγ|3/2√r|κγ|

cos(r(kLγ −

π

2µγ ·α)− σγ

π

4

),

whereLγ is the length of a periodic orbit on the torusγ. As expected, this has the

same form, sans the partial BIC’sηγ andη, as Berry and Tabor’s trace formula for

the spectrum [11], adapted to billiards

dη(k) ∼A2πk (3.5)

+

√k

π

∑

γ∈PT

∞∑

r=1

Lγ

|µγ|3/2√r|κγ|

cos(r(kLγ −

π

2µγ ·α)− σγ

π

4

).

3.1.1 The rectangle billiard

The rectangle billiard is the simplest, most basic case of anintegrable 2D billiard.

The Cartesian coordinates and momenta have simple connections to the standard

action-angle parametrization of integrable phase space. In this section we derive

the trace formula fordη(k) of the rectangle, by applying (3.4).


Consider a rectangle of widtha and heightb. Quantum mechanically, the

problem is separated in the Cartesian coordinates(x, y) whose axes are parallel to

the edges, and it is easy to see thatτx = τy = 2. Classically, for a motion which at

t = 0 has momentum(px, py), the actions can be easily computed by integrating

along the path:Ix = aπ|px|, Iy = b

π|py|. The corresponding Maslov indices are

αx = αy = 4, since each coordinate encounters two Dirichlet reflections per

cycle. The Weyl coefficientAx is computed as the phase space integral ofIx

Ax =3

4π2

∫a

π|px|Θ(1− p2) d2x d2p

=3a2b

4π3

∫ 1

0

dp

∫ 2π

0

dθ p2| cos θ| = a2b

π3.

SimilarlyAy = ab2/π3, andη = 8(a+ b)/(3π2).

In terms of the actions, the Hamiltonian is written:

H = π2

(Ix

2

a2+Iy

2

b2

).

To parametrize the unit energy surfaceH(Ix, Iy) = 1, we define0 ≤ θ < π/2,

such thattan(θ) = |py|/|px|, and the actions are given byIx = (a/π) cos θ,

Iy = (b/π) sin θ. The arc length element on the surface is given by

ds = dθab

π

√cos2 θ

a2+

sin2 θ

b2.

With this parametrization, we find the angular frequenciesωx = 2π cos(θ)/a,

ωy = 2π sin(θ)/b and the curvature

κ =π

(ab)2

(cos2 θ

a2+

sin2 θ

b2

)−3/2

,

which is positive (σγ = 1).

Periodic orbits are enumerated by(M,N) = r(µ, ν) ∈ N2 \ (0, 0), where the

corresponding point on the unit energy surface is given by(cos θµ,ν , sin θµ,ν) ∝(µa, νb). The length of the orbit isLµ,ν = 2

√(µa)2 + (νb)2. In terms of the

topological index(µ, ν) (and the lengthLµ,ν), the curvature is given byκµ,ν−1 =


8(ab)2(µ2 + ν2)3/2/(L3π) and the the partial BIC contribution isηµ,ν = 4(µa2 +

νb2)/(πLµ,ν). Substituting these in equation (3.4), we finally get

dη(k) ∼2ab(a + b)

π3k2 (3.6)

+k3/2(2

π

)5/2 ∑

(µ,ν),r

2ab(µa2 + νb2)

Lµ,ν3/2√r

cos(rkLµ,ν −π

4).

3.1.2 The circle billiard

The second kind of flat 2D billiard which is separable is the ellipse. However,

the circle billiard suffices for demonstrating the application of equation (3.4) to a

problem which is not as trivial as the rectangle, while avoiding complications due

to implicit elliptic integrals.

We consider a circle of radiusa. The problem is separable in polar coordinates

(r, θ), and we find thatτθ = 2 andτr = 0. Hence, for the Weyl term it suffices to

computeAθ. The Hamiltonian inside the billiard is given by

H = pr2 +

pθ2

r2,

and we find thatpθ is constant andIθ = |pθ| (also, the Maslov indexαθ is 0,

becauseθ is cyclic). We use this to compute

Aθ =3

4π2

∫ 2π

0

dθ

∫ a

0

dr

∫dpθ

∫dpr |pθ|Θ

(1− pr

2 − pθ2

r2

)

=3

2π

∫ a

0

dr

∫ 1

0

dx

∫ 2π

0

dϕ (rx)2| cosϕ| = 2a3

3π,

where we have used the substitutionpθ/r = x cosϕ, pr = x sinϕ. Henceη =

τθAθ · 4π/(3A) = 8(2πa)/(9π2).

To find the radial action for a given orbit, letψ be the bounce angle of the orbit

(ψ is also the angle subtended by each chord of the orbit, or halfthe central angle

leaning on it). From standard classical mechanics, we find that for a particle with


energyk2, Iθ = |pθ| = ka cosψ, and that motion on the chord is characterized by

r cos θ = Iθ/k = a cosψ, (3.7)

whereθ = θ−θ0 varies from−ψ toψ, andθ0 is the radial direction at the centre of

the chord. Thus, for each chord,r oscillates froma to a minimal value ofa cosψ

and back toa. The Maslov indexαr = 3, because each cycle involves a turning

point at the minimal value and a Dirichlet reflection atr = a. Equation (3.7) can

be used to change variable toθ in the action integral and find

Ir =2

2π

∫ a

Iθ/k

√k2 − pθ2

r2dr

=Iθπ

∫ ψ

0

√1− cos2 θ

sin θ

cos2 θdθ =

Iθπ(tanψ − ψ).

Using this connection, we can now write the Hamiltonian in terms of the actions:

H =Iθ

2

a2 cos2[ψ(π IrIθ

)] = 1

a2

(Iθ

2 +[ψ(πIrIθ

)Iθ + πIr

]2),

where in this expression,ψ(x) is simply the inverse of (the monotonically increas-

ing function)x = tanψ−ψ. Note that the mapping from the space spanned by the

actions(Iθ, Ir) and their corresponding angle variables to the original phase space

(r, θ, pr, pθ) is bi-valued, since replacingpθ with −pθ yields the same actions. In

what follows we consider only the counter-clockwise part ofphase spacepθ > 0,

and correct for the missing part using the fact that the computed coefficients for a

reversed orbit remain the same.

Parametrize the unit energy surface using0 ≤ ψ < π/2, Iθ = a cosψ, Ir =

a/π(sinψ − ψ cosψ). The arc length element is given by

ds = dψa sinψ

π

√π2 + ψ2

The angular frequencies are given byωθ = 2ψ/(a sinψ), ωr = 2π/(a sinψ), and

3.2. REMOVING THE SPECTRAL DEPENDENCE 37

the curvature (which is always negative) is

κ =−π2

a sinψ(ψ2 + π2)3/2.

Periodic orbits are represented by values ofψ for which the angular frequencies

are rationally dependent. They are enumerated by(M,N) = r(µ, ν) ∈ N2 \ (0, 0)

with N ≥ 1 and0 ≤ M ≤ N/2. The inscribed angle isψµ,ν = π µν. The orbit

with µ = 0 is the “whispering gallery” orbit, which is a singular case that we do

not consider here. The length of the orbit isLµ,ν = 2νa sinψ. To account for the

‘clockwise’ part of phase space (pθ < 0), we take each orbit twice by considering

negative values ofM , except for the caseM = N/2, which yieldsψ = π/2 ⇒pθ = 0 (pθ = 0 is the surface on which the two parts of phase space are joined).

This is equivalent to identifyingM = N/2 with M = −N/2 (however, it makes

no difference because the partial BIC contributionηµ,ν = 2I(µ,ν)θ = 0 in this case).

Substituting the results stated above in equation (3.4), weget

dη(k) =4a3

3πk2 (3.8)

+k3/2a

√2

π

∑

M,N

Lµ,ν3/2 cosψµ,ν√rν2

cos(r(kLµ,ν − ν3π

2) +

π

4)).

3.2 Removing the spectral dependence

Equation (3.4) provides an explicit expression for the quantum BIC in terms of

the classical dynamics. However, the spectral BIC densitydη(k) combines infor-

mation from the BIC sequenceηn with information from the spectral sequencekn(which is already known to depend on classical periodic orbits in a similar man-

ner). A direct formula forηn (or equivalentlydη(n)) would allow separating out

the BIC information. In particular, it could be used to answer “inverse nodal”

questions [21, 22], and for resolving isospectrality [19, 20].

For the calculation ofdη(n), we need the inverse functionk(n). As was done

in [31], we compute this by formally invertingn(k). The derivation given here is

done in a more detailed way, and yields higher order terms. These higher order

corrections are not required for the separable examples above, but we use them


in section 3.3, where the theoretical trace formula for the right triangle billiard is

numerically tested to high accuracy.

3.2.1 Spectral inversionk(n)

We will assume that the spectrum of the integrable billiard satisfies a trace formula

of the following form

n(k) =A4πk2 − L

4πk +D0 (3.9)

+√k∑

γ∈GPT

Cγ sin(kLγ + φγ) +∑

γ∈SPODγ sin(kLp + ϕp) + O(k−1/2),

where GPT is the set of generalized (possibly repeating) periodic tori, Lγ the

length of a representative orbit (including multiplicity)andCγ andφγ are other

parameters which depend on classical features of the orbit.Similarly, SPO is a

set of “special”, isolated, periodic orbits, withDp andϕp depending on classical

features of the orbit. An example for such a formula is the trace formula for the

right isosceles triangle, equation (3.18), given in section 3.3. In that example,

the generalized periodic tori are parametrized byγ = (M,N) ∈ Z2∗ (where

Z2∗ ≡ Z × Z \ (0, 0)), with LM,N = 2a

√M2 +N2, CM,N = 2A(2πLM,N)

−3/2

andφM,N = −π/4. The SPO in that example consist of repetitions of two special

orbits, of lengths2a and√2a, with N ≥ 1 the number of repetitions,DN =

1/(2πN) andϕN = π. In general,Cγ andφγ can be derived from equation (3.5)

Cγ =1

|Mγ|3/2√

|κγ|φγ = −π

2Mγ ·α− σγ

π

4, (3.10)

whereMγ = rµγ is the topological index, including multiplicity, of the orbit γ.

Sincen(k) is not really invertible, we introduce a smoothed functionnσ(k)

which is monotonously increasing. We choose some positive and symmetric

smoothing kernelρσ(k′) of typical widthσ (decaying fast enough for|k′| ≫ σ),

and such that the Fourier transformρσ(x) decays exponentially for largex. Con-


volving both sides of (3.9) withρσ, we get

nσ(k) =A4πk2 − L

4πk +Dσ

0 (3.11)

+√k∑

γ∈GPT

Cσγ sin(kLγ + φγ) +∑

γ∈SPODσp sin(kLp + ϕp) + O(k−1/2)

wherenσ = n ∗ ρσ, Dσ0 = D0 +

∫∞−∞ k2ρσ(k) dk · A/(4π), Cσγ = Cγ ρσ(Lγ)

andDσγ = Dγ ρσ(Lγ). The smoothed step functionnσ is now monotonously in-

creasing and invertible1. We try to derive an asymptotic trace formula for the

inverse functionkσ ≡ (nσ)−1 based on (3.11). To simplify notation, we will use

q =√

4πn/A as the parameter of the inverted function, instead ofn. Denote also

fσ1 (k) =∑

GPT Cσγ sin(kLγ + φγ) andfσ2 (k) =∑

SPO Dσγ sin(kLγ + ϕγ).

The exponential decay ofρσ ensures thatMσC

≡ ∑GPT |Cσγ | andMσ

D≡∑

SPO |Dσγ | converge, so they can be used as a uniform bounds to|fσ1 (k)| and

|fσ2 (k)| respectively. Hence, the fourth and fifth terms of equation (3.11) are

O(√k) andO(1) respectively. With this, we can formally invert the expansion,

and get

A2πkσ(q) =

A2πq +

L4π

+

( L2

16πA −Dσ0

)q−1

−q−1/2fσ1 (k)− q−1fσ2 (k) + O(q−3/2

). (3.12)

To eliminate thek dependence from the right hand side of equation (3.12), we

will use the fact that thek dependent terms in (3.12) vanish for highq. Denoting

q ≡ q+L/(2A) andδ ≡ −q−1/2(2π/A)fσ1 (k), we have from (3.12)kσ = q+ δ+

O(q−1). Choose a proper truncation lengthLM , so that∑

Lγ>LMCσγ ≪ 1. For q

large enough,q ≫ (MσCLM/A)2, we have|δ| < q−1/22πMσ

C/A ≪ 2π/LM , which

is smaller than the period2π/Lγ of the sine term corresponding to orbitγ, for all

orbits that have a significant contribution tofσ1 (k). Hence we can approximate

1Note that theDσ

0 as defined above might be ill defined in cases whereρσ has no finite variance(one example is the Lorentzian smoothing). In such cases we subtract the Weyl term from bothsides, and add it back after convolving withρσ (effectively settingDσ

0 = D0). With this operationwe lose the monotonicity ofnσ(k), but only for small values ofk.


the sine by the first order Taylor expansion inδ:

sin(kLγ + ϕγ) ∼ sin(qLγ + ϕγ)

−q−1/2 2πLγA

∑

γ′∈GPT

Cσγ′ cos(qLγ + φγ) sin(qLγ′ + φγ′).

Inserting this into (3.12), we finally get

A2πkσ(q) =

A2πq +

L4π

+

( L2

16πA −Dσ0

)q−1 (3.13)

−q−1/2∑

γ∈GPT

Cσγ sin(qLγ + φγ)

−q−1∑

γ∈SPODσp sin(qLγ + ϕγ)

−q−1 π

2A∑

γ,γ′∈GPT

Cσγ Cσγ′(Lγ − Lγ′) sin[q(Lγ − Lγ′) + φγ − φγ′]

+q−1 π

2A∑

γ,γ′∈GPT

Cσγ Cσγ′(Lγ + Lγ′) sin[q(Lγ + Lγ′) + φγ + φγ′ ]

+O(q−3/2).

While equations (3.11) and (3.12) are correct for anyσ, we have only shown

the correctness of (3.13) for values ofq which are large compared to aσ-dependent

lower bound,qm ≡ (MσCLM/A)2. The bound might increase to infinity asσ → 0

(for example, in the case of the triangle, with Lorentzian smoothing ρσ(x) =

exp(−2σ|x|), we getMC ∼ σ−1/2 and qm ∼ σ−1). This is not sufficient for

our case, wherekσ(n) is used as an approximation fork(n). To keep the error

small,nσ must be close ton, so the influence of neighbouring steps ofn(k) must

be suppressed by the convolution, i.e. the smoothing lengthσ should be smaller

than the level spacing. Forq ≫ 2π/(Aσ), the above derivation of (3.13) works,

but the step structure is wiped out by the smoothing, andnσ loses accuracy as

an approximation ofn. To remedy this, one needs to find a tighter bound for

fσ1 (k). In actual problems, we expect such a bound to exist. For example, in

the case of the triangle,fσ1 (k) converges atσ = 0 for everyk, and numerics

suggest thatf 01 (k) isO(1) in k. One should be able to find a lower limitqm which


does not depend onσ, and show (3.13) for smoothing tight enough to keep the

step structure. However, instead of doing that, we demonstrate the validity of

equation (3.13) numerically, for different values ofσ.

The q−1/2 term of (3.13) containsfσ1 (q), the usual sum over periodic tori.

Define

ksmσ = q +L2A +

L2 − 16πADσ0

8A2q−1, koscσ (q) = kσ(q)− ksmσ (q).

For numerical verification of theq−1/2 term, we calculatef1(q) ≡ q1/2koscσ (q) ·W∆(q − q0), wherekoscσ is calculated from the numerical spectrum (usingkσ =

(ρσ ∗ n(k))−1), andW∆(q − q0) is a smooth spectral window of width∆, centred

aroundq0 (we used a Gaussian, which has a more localized effect on the Fourier

transformf1 than the naıve cutoffΘ(qmax−q), introduced by the fact that we only

consider a finite number,nmax, of eigenfunctions). From the right hand side of

equation (3.13), we see that the Fourier transform off1 can be approximated by

g1(x) =π

iA∑

γ∈GPT

Cσγ e−i[φγ+LγL/(2A)] · W∆(x− Lγ)eiq0(x−Lγ)

whereg1(q) ≡ −(2π/A)fσ1 (q) ·W∆ is theO(q−1/2) term of (3.13), cut off with

the windowW , andg1(x) is its Fourier transform. As can be seen in figure 3.2,

we get an accurate match between the numerics and this prediction. We can see

that the heights and positions of the peaks match the periodic tori as predicted by

equation (3.13). Peaks of higher order are also observable there: in particular, the

SPO peak atLγ = 2, and the pair-sum peak atLγ + Lγ′ = 2 + 2√2. The inset

shows a complex-valued magnification of the peak corresponding to the torus with

Lγ = 2. This demonstrates the accuracy of the match and the validity of the phase

φγ.

To verify theO(q−1) terms, we subtract the known sum of periodic tori, and

proceed in a similar manner. Definef2(q) ≡ q[koscσ (q)− q−1/2g1(q)] ·W∆(q− q0),

then by equation (3.13) this can be estimated byg2(q) = gSPO

2 + gdiff2 + gsum2 , with


Figure 3.2: Fourier transform ofkoscσ

(q)—periodic orbit contribution.q0 = 840, ∆ = 280(Gaussian windowW∆), σ = 0.225 (Lorentzian smoothingρσ).

gSPO

2 = −(2π/A)fσ2 (q) ·W∆,

gdiff2 =−π2

A2

∑

γ,γ′∈GPT

Cσγ Cσγ′(Lγ − Lγ′) sin[q(Lγ − Lγ′) + φγ − φγ′] ·W∆, and

gsum2 =π2

A2

∑

γ,γ′∈GPT

Cσγ Cσγ′(Lγ + Lγ′) sin[q(Lγ + Lγ′) + φγ + φγ′ ] ·W∆.

Figure 3.3 demonstrates that theO(q−1) contributions are well matched to the

predictions as well. The magnified inset of figure 3.3a shows two peaks: the one

atx = 8 is composed of contributions fromgsum2 , gdiff2 andgSPO

2 , while the one at

2(√22 + 52 −

√2) ∼ 7.94 is purely due to an orbit difference (included ingdiff2 ).

Comparing figure 3.3a with figure 3.3b, we can see that whenσ is decreased, an

increasing amount of orbit-pair differences fromgdiffσ become observable. How-

ever, even atσ = 0.005 the predictions of equation (3.13) are still in good agree-

ment with the numerics. This is demonstrated by the inset of figure 3.3b, which

shows a complex valued magnification of the peak corresponding to the orbit pair

difference atx = 7.94.


(a) σ = 0.225

(b) σ = 0.005

Figure 3.3: Fourier transform ofkoscσ

(q): TheO(q−1) contributions.


Figure 3.4: Exact and approximatek(q), atσ = 0.005.

A smoothing kernel of widthσ = 0.005 is thin enough to preserve the step

structure ofk(q) aroundq0 = 840, which is the centre of the spectral window that

we have used. As can be seen in figure 3.4, the periodic orbits termg1 improves

the estimate of the Weyl term. Given the numerics above, we might also expect

that theO(q−1) term g2 would get us even closer tokσ. However, if we expect

the esimate to be sensitive to fluctuations comparable to theaverage level spacing

at this region,δ = 0.015, we should include orbits and orbit-pairs upto a length

(and length difference, respectively) of2π/δ ∼ 420. This means that a very

large number of orbit-pairs must be considered, and the taskof computingg2numerically becomes unfeasible.

3.2.2 Applying spectral inversion to eliminate thek dependence

In order to transform equation (3.4) into an expression fordη(q), we need to sub-

stitute the expression fork(q) into the formula, and multiply the result by the

3.3. A NON SEPARABLE BILLIARD 45

appropriate derivative as required for changing variablesin density functions:

dη(q) = dη(k(q))dkdq

. Here we will only compute contributions upto orderq3/2.

When we substitute equation (3.13), we only need to expandk(q) uptoO(1) (as

can be seen by formal substitution, the lowest order contribution of theO(q−1/2)

term is in theO(q1/2) term of the result). Hence, we use only the first two terms:

k ∼ q +L2A = q.

The required derivativedk/dq is obtained by deriving equation (3.13) (withCγandφγ from equation (3.10)), and using terms upto orderq−1/2:

dk

dq∼ 1− 2

Aq−

1

2

∑

γ∈PT

∞∑

r=1

Lγ

(µ2 + ν2)3/4√r|κγ|

cos(r(qLγ −

π

2µγ ·α)− σγ

π

4

)..

Combining these with equation (3.4), while making sure to retain the non van-

ishing termL/(2A) in the argument of the cosine, as was done in section 3.2.1,

yields

dη(q) ∼3Aη4π

q2 (3.14)

+q3/2

π

∑

γ∈PT

∞∑

r=1

Lγ(ηγ − 32η)

(µ2 + ν2)3/4√r|κγ|

cos(r(qLγ −

π

2µγ ·α)− σγ

π

4

).

A more accurate analysis is described in section 3.3.3, whereCη(q) =∫ q

dη(q′) dq′

is computed for the (non-separable) case of the right isosceles triangle. That

expansion involves contributions from the oscillating terms of the inversion for-

mula (3.13).

3.3 An example for non-separable billiards: The right

isosceles triangle

Integrable systems which are not separable do not have the simple checker-board-

like nodal pattern that characterizes separable ones (see,for example, the sample

eigenfunctions which are shown in figure 3.5). Nevertheless, we expect that nodal


(a)ψ13,6 (b) ψ12,8

Figure 3.5: Sign-plots of sample eigenfunctions of the isosceles righttriangle.

statistics would still have common features with the separable case, which would

enable us to differentiate between integrable and chaotic systems. The isosceles

right triangle is one of the few cases where despite being non-separable we still

have analytical expressions for the eigenvalues, eigenfunctions, and the BIC (the

number of nodal domains can also be computed using a closed algorithm [46]).

This allows us to follow the same procedure that was used in section 3.1, and

derive a trace formula.

Consider the triangle bounded by the linesx = a, y = 0 andx = y. It has

an areaA = a2/2, and boundary lengthL = a(√2 + 2). The eigenfunctions

are exactly the anti-symmetric combinations constructed from degenerate pairs of

basis functions of thea× a square:

ψl,m(x, y) =

√2

a

[sin(

lπ

ax) sin(

mπ

ay)− sin(

mπ

ax) sin(

lπ

ay)

](3.15)

for all l > m > 0. The eigenvalues are given by

kl,m =π

a

√l2 +m2.

It is not hard to see (see appendix B) that the BIC is given by

ηl,m =

l +m− 3 if l +m = 1 (mod 2)

l +m− 2 if l +m = 0 (mod 2),(3.16)


or equivalentlyηl,m = l +m− 52+ 1

2exp[iπ(l +m)].

The trace formula for the spectral counting function can be found directly by

applying the Poisson summation formula (PSF) on the definition

n(k) =∑

l>m>0

Θ(k − kl,m) =∑

l>m>0

Θ(k − π

a

√l2 +m2

). (3.17)

This evaluates to

n(k) =A4πk2 − L

4πk +

3

8

+∑

(M,N)∈Z2∗

2A√k

(2πLM,N)3/2

sin(kLM,N − π

4)

−∑

N∈N∗

1

2πN

[sin(2Nak) + sin(

√2Nak)

]+ o(1), (3.18)

whereLM,N = 2a√M2 +N2. TheO(

√k) term may be interpreted as a sum

over periodic tori, where the torus corresponding to(M,N) consists of orbits that

bounce from the bottom edge(y = 0) at angleψ with tan(ψ) = N/M , and

whose length isLM,N . Note that ifr = gcd(M,N) and(M,N) = (rµ, rν), then

such an orbit is in factr repetitions of the orbit(µ, ν) (this is just a special case

of the generic semi-classical formula by Berry and Tabor [11]). TheO(1) term

corresponds to isolated orbits.√2Na is the length of the orbit hitting the corner

(x = a, y = 0) at45 (withN repetitions), and2Na is the length of the orbits that

lie on the catheti(y = 0) and(x = a).

3.3.1 BIC as a function ofk

In a similar way to the approach used above, we now combine equation (3.16)

with the PSF, to get a trace formula forCη(k) ≡ ∑kl,m<k

ηl,m =∫ k0dη(k

′) dk′.

In this section, we useCη(k) (which is a monotonously increasing step function)

rather than its derivativedη(k), for compatibility with [31] and to simplify the

description of the procedure that was used for changing the independent variable


from k to q (described in section 3.3.3). Starting with the definition,we have

Cη(k) =∑

l>m>0

(l +m− 52+ 1

2eiπ(l+m))Θ

(k − π

a

√l2 +m2

)(3.19)

Split ηl,m into three partsηl,m = ηBl,m + ηCl,m + ηRl,m: the “bulk” termηB = l +m,

the constant termηC = −52

and the “round off” termηR = 12exp[iπ(l + m)].

Correspondingly, we splitCη(k) = CB(k) + CC(k) + CR(k) and compute each

contribution in turn. Comparing (3.19) to equation (3.17),we immediately get

CC(k) = −52n(k). The contribution of the round-off termCR is a small variation

of the same calculations that were used to derive the spectral trace formula (3.18).

The largest contribution toCR(k) is given by

18

∑

M,N

∫∫ ∞

−∞dl dmηR|l|,|m|Θ(k − k|l|,|m|)e

2πi(|l|M+|m|N)

= 116

∑

M,N

∫∫dl dmΘ(k − π

a

√l2 +m2)e2πi[|l|(M+

12)+|m|(N+

12)]

= 12

∑

M,N

N (M + 12, N + 1

2), (3.20)

whereN (M,N) is the same expression that appears in the in the(M,N) summa-

tion of theO(√k) (periodic tori) term of equation (3.18). The difference is that

here we enumerate half integer values rather than integer ones.

Some notes relevant for the rest of the calculations, and in particular the deriva-

tion of the bulk termCB are given in appendix C. The lowest order terms of the

final result are given by

Cη(k) =(ak)3

3π3− 5π + 8

16π2(ak)2 +

7 + 3√2

6πak

+a3k3

2

∑

(M,N)∈Z2∗

2λM,N

(2πLM,N)5/2

sin(LM,Nk −π

4)

−a2k∑

N∈N∗

1

π22aN

[1

πNcos(2Nak) + sin(2Nak)

]

−a2k∑

N∈N∗

1

π2√2aN

sin(√2Nak) + O(

√k), (3.21)


whereLM,N = 2a√M2 +N2 is the length of the orbit (as before), andλM,N =

2a(|M |+ |N |). As in equation (3.18), we can see the contributions of the periodic

tori and the isolated orbits. The next order corrections areO(√k). As explained

above, they contain (among other terms, involving lengths of periodic tori) the

sum∑

M,N∈Z

A√k

(2πLM,N)3/2

sin(kLM,N − π

4),

whereLM,N ≡ a√[(2M + 1)2 + (2N + 1)2]. This introduces oscillations whose

frequencies do not correspond to lengths of periodic orbits, but rather to “semi

periodic” ones. After moving a length ofLM,N (which is half the length of some

periodic orbit), the particle reaches a point which is the reflection (along the sym-

metry axis of the triangle) of the starting point, and from there follows a path

which is the reflection of the first half of the orbit. As shown in equation (3.20)),

these terms arise due to the contribution of the round-off term ηR. They do not

appear in the spectral trace formula, nor in the BIC formulaeof separable bil-

liards. In figure 3.6 the contribution of such a semi-periodic orbit is observed

at L0,1 =√10. The Fourier analysis here followed the procedure described

in section 3.2.1. LetCosc = Cη − Csm with Csm denoting the terms on the

first line of equation (3.21), thenfσ1(k) = (ρσ ∗ Cosc(k)k−3/2)W∆(k − k0) is the

(scaled and smoothed) numerical oscillation, truncated with a Gaussian of width

∆ = 280 aroundk0 = 840. This is estimated bygσ1 (k), the smoothedO(k3/2)

term on the second line of (3.21). Continuing, we subtractf2 = (f1 − g1)k1/2,

andf3 = (f2 − g2)k1/2, getting increasing level of detail. The corresponding the-

oretical estimatesg2 andg3 correspond to theO(k) andO(√k) terms ofCη(k)

respectively.

3.3.2 Common form with the separable TF

It should be noted that the lowest order terms of equation (3.21) can be fitted to

the form of equation (3.4) (the formula fordη(k) in separable billiards). Most of

the parameters used in equation (3.21) are defined for all integrable systems. The

only exceptions are the coefficients(τ1, τ2), which were defined in terms of the

separated coordinates. However, we shall see that settingτ1 = τ2 = 1 yields a


Figure 3.6: Fourier transform of theO(√k) term ofCσ

η(k) (Gaussian smoothingρσ, with σ =

0.15).

result which is consistent with (3.21).

As in section 3.1, we start with the classical description ofthe system. The

classical actions (computed by integrating the total action over a representative

loop in each homotopy class of the torus) are given byI> = aπmax|px|, |py|

andI< = aπmin|px|, |py|. The corresponding Weyl coefficients are

A> = 83

4π2

a2

2

∫ 1

0

dp p2∫ π

4

0

a

πcos θ dθ =

a3

π3

√2

2

A< = 83

4π2

a2

2

∫ 1

0

dp p2∫ π

4

0

a

πsin θ dθ =

a3

π3

2−√2

2.

The Hamiltonian is given by

H =π2

a2(I>

2 + I<2),

so the curvature of the unit energy surface is constantκ = π/a. Periodic tori are


enumerated using co-prime indicesµ ≥ ν ≥ 0, with

(I>, I<) =a

π√µ2 + ν2

(µ, ν),

and the corresponding orbit lengths areL = 2a√µ2 + ν2. Substituting the above

in (3.4) yields

dη(k) ∼a3k2

π3

(τ>

√2

2+ τ<

2−√2

2

)

+8a3

πk

3

2

∑

r,µ,ν

2a(τ>µ+ τ<ν)

(2πLµ,nu)3/2√r

cos(rLµ,νk −π

4) + O(k).

On the other hand, the analytical result fordη(k), obtained by taking the

derivative of equation (3.21), is

a3k2

π3+a3

πk

3

2

∑

(M,N)∈Z2∗

λM,N

(2πLM,N)3/2

cos(LM,Nk −π

4) + O(k).

Comparing the two formulae, we see that if we setτ> = τ< = 1 and take proper

account of the multiplicity of the mapping(M,N) → (µ, ν, r), they become iden-

tical.

3.3.3 Combining with the spectral inversion

To get the final trace formula forCη(q), one needs to substitute the spectral in-

version formula (3.13) in equation (3.21) (the relevant parametersCγ andDγ are

listed in section 3.2.1). Before we combine the formulae, wewill assume that

the step functionCη(k) is smoothed using a convolution kernelρς(k) of width

ς. DefineCς(k) = (ρς ∗ Cη)(k). We will later useCς,σ(q) ≡ Cς(kσ(q)) as an

approximation forCη(q). Note that the limiting form ofCς,σ(q) asς, σ → 0 de-

pends on the order of the limits. Ifς ≪ σ, the limiting form starts at 0, and jumps

to Cn at q =√[(n − 1

2)4π/A]. If σ ≪ ς, the limiting form starts atC1/2 and

jumps to 12(Cn + Cn+1) at q =

√n · 4π/A. If they go to0 together (keeping

σ = ς ≪ 1), then the limiting form is composed of linear segments connecting


the points(√n · 4π/A, Cn).

A trace formula forCς is easily obtained by applying the convolution on equa-

tion (3.21). As usual, each trigonometric functionsin(Lγk+φγ) on the right hand

side gains a factor ofρς(Lγ). Next, a formula forCς,σ(q) is obtained by formally

substituting (3.13) in the formula forCς(k). When substituting (3.13) fork in the

arguments of the trigonometric functions, we repeat the reasoning of section 3.2.1.

We chooseq high enough to make the oscillating part of (3.13) smaller than the

relevant period, and then expand the the sine to first order aroundq. In particular,

for theO(k3/2) terms of (3.21) we get

ρς(Lγ)2λγ

(2πLγ)5/2

sin(Lγk −π

4) ∼ ρς(Lγ)

2λγ

(2πLγ)5/2

sin(Lγ q −π

4)

−q−1/2∑

γ′∈Z2∗

ρς(Lγ)ρσ(Lγ′)λγ

2π3Lγ3/2Lγ′

3/2cos(Lγ q −

π

4) sin(Lγ′ q −

π

4).

Summing theO(q−1/2) correction overγ ∈ Z2∗, we get, after symmetrization

with respect to exchange of the dummy indicesγ ↔ γ′

q−1/2∑

γ,γ′∈Z2∗

ρσρ′σ

8π3Lγ3/2Lγ′

3/2(λγ

ρςρσ

+ λγ′ρ′ςρ′σ

) cos((Lγ + Lγ′)q)

+q−1/2∑

γ,γ′∈Z2∗

ρσ ρ′σ

8π3Lγ3/2Lγ′

3/2(λγ

ρςρσ

− λγ′ρ′ςρ′σ

) sin((Lγ − Lγ′)q),

where we have used the shorthand notationsρς ≡ ρς(Lγ), ρ′ς ≡ ρς(Lγ′), ρσ ≡ρσ(Lp), andρ′σ ≡ ρσ(Lγ′).

After combining this with the rest of the terms that arise from the substitution


of kσ(q) in Cς(k), we finally get

Cς,σ(q) =(aq)3

3π3+ α2

(aqπ

)2+ ας1

aq

π

+a3q3/2∑

γ∈Z2∗

2(ρςπλγ − ρσ4Lγ)

π(2πLγ)5/2

sin(Lγ q −π

4)

−a2q∑

N∈N∗

1

π32aN

[ρςN

cos(2aNq) + (ρςπ − ρσ4) sin(2aNq)

]

−a2q∑

N∈N∗

ρςπ − ρσ2√2

π3√2aN

sin(√2aNq)

+a2q∑

γ,γ′∈Z2∗

ρσρ′σ

π(λγρςρσ

+ λγ′ρ′ςρ′σ)− 4(Lγ + Lγ′)

8π4Lγ3/2Lγ′

3/2cos((Lγ + Lγ′)q)

+a2q∑

γ,γ′∈Z2∗

ρσρ′σ

π(λγρςρσ

− λγ′ρ′ςρ′σ)− 4(Lγ − Lγ′)

8π4Lγ3/2Lγ′

3/2sin((Lγ − Lγ′)q)

+O(√q), (3.22)

where

α2 =2 +

√2

π− 5π + 8

16

ας1 =3

π2(3 + 2

√2− π

2)− (5π + 8)(2 +

√2)

8π+

7 + 3√2

6

+(aπ

)2 ∫ ∞

−∞ρς(q)q

2dq.

Note that the “semi-periodic” orbit lengths described in section 3.3.1 are of higher

order than the contribution of the orbit-differences, which is badly behaved for

smallσ. Hence they cannot be observed in the numerical data forCη(q).

In figure 3.7 we test the validity of theO(q) terms of equation (3.22). We as-

sume a “uniform smoothing”ρς = ρσ → δ(k). As mentioned above, this amounts

to connecting the points(√n · 4π/A, Cn) linearly (and settingρς = ρσ = 1

in (3.22)). As before, we usefi to denote numerically computed contributions

andgi for the corresponding theoretical estimates. The inset shows the peak at

x = 2(√2− 1) ∼ 0.83, which corresponds to an orbit difference.


Figure 3.7: Fourier transform of theO(q) term ofCη(q) (with uniform smoothing).

Chapter 4

The BIC of chaotic billiards

In chapter 3, trace formulae for the BIC were derived using explicit expressions

for it and for the spectrum in terms of the quantum numbers. For generic systems,

such an expression for the BIC (or even a complete set of quantum numbers) does

not exist, so different approaches must be considered. In chapter 2, we have used

statistical models that are believed to match the behaviourof chaotic eigenfunc-

tions, and adapted them for the region close to the boundary.In this chapter we

approach the problem from another direction. We start with atheoretical descrip-

tion of the problem, expressed in terms of the boundary function un. In [33],

semi-classical formulae were derived for relevant features, and in particular the

correlation function of that function. By extending these results to higher orders

in k, we derive a trace formula for the correlation function (section 4.2). Gen-

erally, the connection between the correlation function ofun and the BI density

bn(s) is not trivial. However, in the case whereun is Gaussian (see section 2.1

for definition), which we conjecture to hold for chaotic billiards, the computation

is greatly simplified, and we can use the results described above to derive a trace

formula forb(s), and finally forη.

The conjecture mentioned above is analogous to the assumptions made in [23],

which led to the RW models that we have used in chapter 2. However, introducing

the conjecture from within the scope of the semi-classical theory of boundary

functions, allows a description which is more specific to theproblem at hand, and

ultimately leads to a trace formula that can be verified numerically.

55

56 CHAPTER 4. CHAOTIC BILLIARDS

4.1 BI density and boundary functions

As mentioned in section 1.1, we use scaled boundary functions un, defined, for

Dirichlet billiards, as

un(s) =1

knn(s) ·∇ψn (r(s)) .

For normalized eigenfunctions∫Ωψn

2d2r = 1, the corresponding boundary func-

tions satisfy the identity

∮

∂Ω

un2(s) r · n(s) ds = 2 (4.1)

(for proof see [47]). The scaling was chosen such that this normalization condition

does not depend onkn. Sinceun(s) is proportional to the value ofψn on a curve

parallel to the boundary at a short, constant, distance inwards from it (see also

in section 2.2), it is evident that the BI are exactly the points whereun changes

sign. Hence, the local density of BI can be written as

bn(s) =

ηn∑

i=1

δ(s− s(n)i ) = δ(un(s)) |un(s)|,

wheres(n)i are the zeros ofun, andun(s) ≡ ddsu(s). The BIC is simply the inte-

gral of this functionηn =∮bn(s) ds. Correspondingly, we can define the spectral

density ofb, db(s; k) ≡ ∑n bn(s)δ(k − kn). To make the discussion more pre-

cise mathematically, we convolve the definitions above witha “smoothing kernel”

ρσ(x), which is negligible for|x| > σ (the smoothing width), and get a “smoothed

density”

dσb (s; k) = ρσ ∗ db(s; k) =∑

n

bn(s)ρσ(k − kn). (4.2)

Unless otherwise stated, we chooseρσ to be a Gaussian, whose widthσ is propor-

tional tok−1/2. With this definition, the spectral interval(k − σ, k+ σ) shrinks to

0 ask → ∞, but the number of levels it contains grows to infinity. We shall con-

sistently use the notationdσX(y) to denote theρσ smoothed density of the quantity

X in the variabley. With this definition,dσb (s; k) ≡ db(s)(k). Also, the smoothed

4.1. BI DENSITY AND BOUNDARY FUNCTIONS 57

spectral density of BIC is given by

dση (k) =∑

n

ηnρσ(k − kn) =

∮dσb (s; k) ds.

The fact that equation (4.2) describes a weighted sum ofbn(s) over a spectral

window can be used to apply techniques from probability theory. To make this

more explicit, we rewrite (4.2) as

dσb (s; k) = dσ(k)〈b(s)〉k,σ, (4.3)

wheredσ(k) ≡ ∑n ρσ(k − kn) is the sum of weights in the spectral window

of width σ aroundk (with our choice ofσ, this sum should effectively contain

O(√k) non negligible elements), and

〈b(s)〉k,σ ≡ dσb (s; k)/dσ(k) =

∑

n

ρσ(k − kn)

dσ(k)bn(s) (4.4)

is the density of intersections ats averaged over the spectral window. Note that

the sum of weightsdσ(k) is exactly the spectral densityd(k), smoothed withρσ,

and there are well known trace formulae which approximate itfor chaotic and

integrable billiards. Since the weights ofbn(s) in equation (4.4) add up to 1,

they could be considered as probabilities, and〈b(s)〉 becomes a statistical mean

value. The statistical ensemble considered is the set of boundary functions whose

eigenvalues fall in the specified spectral windowUσ(k) = un(s) | |kn−k| . σ,

where the functionun is chosen1 with probability ρσ(k − kn)/dσ(k). If u is a

random function chosen from this ensemble then〈b(s)〉k,σ is exactly the statistical

mean value〈δ(u(s))|u(s)|〉. Furthermore, with the choice ofρσ specified above,

the number of functions in the ensemble grows to infinity ask → ∞, so the

discrete random field defined in this way might converge to a continuous limiting

distribution.

1Since the sign ofψn (and hence ofun) may be chosen arbitrarily, we assume that for eachnthe sign was randomly selected, with equal probability, upon choosing the initial basisψn


4.1.1 Expansion by cumulants

As we shall show in section 4.2, semi-classical analysis of the boundary functions

allows us to derive expressions for certain moments of the random fieldu. How-

ever, in the general case, the desired expectation value〈b(s)〉 depends on these

moments in a non trivial way. To see this, consider the Fourier representation of

this variable

〈δ(u(s))|u(s)|〉 = 1

2π2

∫∫ ∞

−∞

⟨eiξu(s)(1− eiχu(s))

⟩ dξdχχ2

. (4.5)

The averaged expression contains terms of the formexp(i∑2

l=1 ξlul), whereulare the random fields (u1 ≡ u, u2 ≡ u). Since〈exp(i∑ ξlul)〉 is exactly the (bi-

variate) cumulant generating function, the “brute force” approach would suggest

writing a cumulant expansion series for the average in equation (4.5), evaluate the

integrals, and substitute semi-classical expansions for the cumulants in the result.

In the expansion, we ignore cumulants of odd order, since theresult cannot depend

on the arbitrary choice of signs for the boundary functionsun. For example, the

cumulant expansion of⟨eiξu(s)(1− eiχu(s))

⟩/χ2 upto 4th order reads

1

χ2

exp[− 1

2c0ξ

2 +1

4!δ0ξ

4]

− exp[− 1

2(c0ξ

2 + 2c1ξχ+ c2χ2)]

· exp[ 14!(δ0ξ

4 + 4δ1ξ3χ+ 6δ2ξ

2χ2 + 4δ3ξχ3 + δ4χ

4)],

(4.6)

wherec0,c1 andc2 are the second order moments〈u2〉, 〈uu〉 and〈u2〉 respectively

(as in section 2.3), theδi are the fourth order cumulants:δ0 = d0−3c02, δ1 = d1−

3c0c1, δ2 = d2− 2c12− c0c2, δ3 = d3− 3c1c2, δ4 = d4− 3c2

2, anddi = 〈u(4−i)ui〉are the fourth order moments. Integrating this expression over ξ andχ, should

yield an expression depending on the momentsci andδi, which we would have to

estimate using semi-classical methods.

We shall see in section 4.2, that the expressions for momentsof u(s) involve

summation over classical orbits passing throughr(s). Analogously to [23] we

assume that for chaotic billiards, these contributions areuncorrelated, such that


some variant of the central limit theorem holds, and all cumulants of higher order

than 2 vanish. In terms of the ensembles defined above, this means thatδi as well

as higher order cumulants areo(1) in k, and the random fieldu (taken from the

limiting distribution ofUσ(k)) is Gaussian (see definition in section 2.1, above

equation (2.3)). With this assumption, the cumulant expansion may be truncated

after the terms involving theci, and the integration in (4.5) yields (just like in [40]

and [27]) the Rice formula—equation (2.12):

〈b(s)〉 = 1

π

√c2c0 − c12

c02.

4.1.2 Numerical verification of Gaussianity

Before going on to compute semi-classical expressions for the required correla-

tionsc0–c2, we present some numerical results, providing direct evidence for the

validity of the conjecture regarding the asymptotic Gaussianity of the boundary

functions of chaotic billiards. The computations were donefor the Africa bil-

liard (see section 4.3.2), for which we have computed the first 20,000 boundary

functions.

In figure 4.1, we plot the single-variable distributions of the values ofu andu.

These functions, normalized to unit varianceun = un ·√A andun = un ·2

√A/kn

(see section 4.2 for the computation of the expected variance), were sampled at a

fixed point on the boundary of the billiard, in a 2000 level window centred around

k = 232. It can be seen that the numerical densities were close to thestandard

normal distribution. The corresponding p-values for the Kolmogorov-Smirnov

(KS) test [48] (i.e. the probability that a true Gaussian distribution would yield

the same or worse value for the KS statistic, than the value computed for the

numerical distribution), were 0.51 foru and 0.18 foru.

Figure 4.2 shows the polar 2D distribution of(un(s), un(s′)) at a fixed pair of

points(s, s′) in the same energy window. The KS p-value for comparingu2 +

u2 to the theoreticalχ2(2) distribution was 0.69. While these tests are not very

extensive, they provide a simple demonstration of Gaussianity, and their results

are consistent with our conjecture.

The Kurtosis,Kurt(x) ≡ 〈x4〉/〈x2〉2−3, provides a measure of non-Gaussianity,


Figure 4.1: Density of normalized boundary functionu and derivativeu of the Africa billiard in a2000 levels window aroundk = 232.

(a) Amplituder =

√u2 + u2

(b) Angle tan(θ) = u/u

Figure 4.2: 2d distribution ofu(0) andu(0.355).


Figure 4.3: Kurtosis—Boundary amplitudes vs Gaussian (random simulation)

and we can use it to measure the relative size of the 4th order cumulants appearing

in equation (4.6) and in particular their decay as a functionof k (which is another

direct test for the conjecture). In figure 4.3, the numericalKurtosis ofu(s) and

u(s) (which correspond, in terms of (4.6), toδ0/c02 andδ4/c22 respectively) are

plotted as a function ofk. Data was sampled at 15 points on the boundary, with

window width proportional tok−1/2. For the highest window, this gives around

900 eigenfunctions and∼ 13000 samples. For comparison, the numerical Kurto-

sis for an equivalent number of samples taken from a simulated normal random

variable is also plotted for each window (as can be seen, evenwith this number of

samples, the fluctuations of the numerical Kurtosis are still high). The results indi-

cate a convergence towards Gaussianity, and a power-law fit forKurt(u(s)) yields

roughly ak−1 law. However, it seems that even at the highest values ofk used in

the numerical computations of section 4.3.2, a residual Kurtosis of∼ −0.05 re-

mains, and is observable in this figure, when compared to the simulated normal

data. The Kurtosis-like parameter corresponding toδ2/(c0c2) was also computed,


yielding similar results.

4.2 Correlations of the boundary function

If asymptotic Gaussianity holds for chaotic billiards, theBI density can be com-

puted directly from the second moments of the boundary functions, using Rice’s

formula (2.12). In this section we derive semi-classical expressions (trace for-

mulae) for these moments. Note that the results of this section do not require

Gaussianity. Hence, they do not depend on any random wave conjecture, and the

derivation is correct for all types of billiards (the trace formulae derived below do

assume that the orbits are isolated in phase space. However,for the case of in-

tegrable billiards, different approximations can be used on the same expressions,

yielding analogous formulae). The main object of interest here is the amplitude

correlation densityduu′(k) =∑

n un(s)un(s′)δ(k − kn). As in section 4.1, if we

convolve this withρσ, we get

dσuu′(k) = ρσ ∗ duu′ = dσ(k)〈u(s)u(s′)〉k,σ,

so the required momentsc0 andc2 are given by

c0(s) =dσu2

dσ=

dσuu′(k)|s′=sdσ(k)

, c2(s) =dσu2

dσ=

1

dσ(k)

∂2dσuu′(k)

∂s∂s′

∣∣∣∣s′=s

. (4.7)

4.2.1 The boundary Green function

Following [33], we develop a semi-classical description ofthe boundary function

using a Green function formalism. For a given Dirichlet billiard, we define the

boundary Green functiong(s, s′) as the infinite sum

g(s, s′; k) =

∞∑

n=1

un(s)un(s′)

kn2 − k2

.

In this definition,k should be assumed to approach the real axis from the upper

half of the complex plain, i.e. usek = Re(k) + iǫ, and take the limitǫ → 0+

after applying any relevant smoothing procedure on the expression (involvingg)

4.2. CORRELATIONS OF THE BOUNDARY FUNCTION 63

considered. With this in mind, we can write the boundary correlation density in

terms ofg

duu′(s, s′; k) =

∞∑

n=1

un(s)un(s′)δ(k − kn) =

2k

πIm g(s, s′; k). (4.8)

In appendix D, we show (closely following the derivation in [33]) that the

boundary Green function satisfies the integral equation

hg = g − g0, (4.9)

whereh is the integral operator with kernelh(s, s′). The functionsh andg0 are

given by

h(s, s′; k) = 2n(s) ·∇r(s)G0(r(s), r(s′); k)

g0(s, s′; k) =

2

k2

∑

i,j

nin′j∂2G0(r(s), r(s

′); k)

∂ri∂r′j, (4.10)

whereG0 is the free Green function in 2D:

G0(r, r′; k) =

i

4H+

0 (k|r − r′|)

(H+n is the ordern Hankel function of the first kind, see [35]).

Taking derivatives ofG0(r, r′; k), the functionsh andg0 can also be expressed

explicitly, in terms of Hankel functions:

h(s, s′; k) = − ik

2(n · d)H+

1 (kd),

g0(s, s′; k) =

i

2

[−(n · d)(n′ · d)H+

2 (kd) +1

kd(n · n′)H+

1 (kd)

],

whered = r − r′, d = |d| and d = d/d. Equivalently, letψ be the final “hit

angle” at which the path fromr′ to r hits the boundary, andψ′ the initial “bounce

angle” from the boundary (i.e.cosψ = d · t andcos ψ′ = d · t′, wheret, t′ are the


tangents ats, s′ respectively), then the functions are given by

h(s, s′; k) = − ik

2sinψH+

1 (kd) (4.11)

g0(s, s′; k) =

i

4

[cos(ψ − ψ′)H+

2 (kd) + cos(ψ + ψ′)H+0 (kd)

]. (4.12)

Examining the limit of these functions ats → s′, we see thath → −κ/(2π)(whereκ is the curvature of the boundary ats), soh might be considered aC1

integral kernel. However, from (4.12) we find thatg0(s′ + δ, s′) ∼ 1πk2δ−2 for

δ ≪ 1. As will be shown below, thisδ−2 divergence occurs for the actual Green

function g as well. Since the divergence occurs only on the real part ofg (the

imaginary parts of bothg andg0 approach14

at s → s′), it causes no problem for

the derivation ofdu2 via equation (4.8). However, for the integral equation (4.9) to

have any meaning, the domain of the integral operatorhmust be extended, and we

must define its operation on functions withδ−2 divergence (a.k.a. hypersingular

functions). As shown in appendix D, the operation ofh in (4.9) should be defined

as

(hg)(s, s′; k) = =

∫

∂Ω

h(s, s1; k)g(s1, s′; k) ds1,

where=∫

stands for Hadamard finite part integration ([49], and see also [50] for an

historical overview).

For a functionf , with a unique second order singularity in[−a, b], located at

0 (i.e. f(δ) ∼ Aδ−2 for δ ≪ 1 and some constantA), the finite part integral can

be defined as follows:

=

∫ b

−af(x) dx =

∫ b

−a

(f(x)− A

x2− B

x

)dx− A(

1

a+

1

b) +B log

b

a, (4.13)

whereA = limx→0 x2f(x), andB = limx→0(xf(x)−A

x). An equivalent definition

(for simplicity, we consider only the symmetric casea = b) is given by :

=

∫ a

−af(x) dx = lim

ǫ→0

(∫ −ǫ

−af(x) dx+

∫ a

ǫ

f(x) dx− 2A

ǫ

)(4.14)


4.2.2 Explicit expansion forg

The basis for the semi-classical derivation is an expansionof g in terms of the

two explicit functions,g0 andh. It is not hard to see that (at least as a formal

expansion) the integral equation (4.9) is satisfied by the series

g =

∞∑

n=0

hng0. (4.15)

To handle the singularities involved in this expansion and make it more amenable

to semi-classical treatment, we choose a large cutoff1 ≪ xC ≪ ka, wherea is

a typical length scale relevant for the billiard (such as thesmallest radius of cur-

vature, or the boundary lengthL), and splitg0 into a “near” (close to the diagonal

s = s′) part and a “far” (off diagonal) part:

g(N)0 =

g0(s, s

′) if |s− s′| ≤ xC/k,

0 else,

g(F )0 =

g0(s, s

′) if |s− s′| > xC/k,

0 else.

Obviously g0 = g(F )0 + g

(N)0 , and we can apply the linear operators in (4.15)

separately on each part. For the far part, we havekd ≫ 1, and by using the

large argument approximation for the Hankel functions in (4.12), we find (for

|s− s′| > xC/k)

g(F )0 (s, s′; k) ∼ sinψ sin ψ′

√2πkd

ei(kd−3

4π). (4.16)

For the near part, denoteg1 = hg(N)0 , and split that into near and far parts as well

g(N)1 =

g1(s, s

′) if |s− s′| ≤ xC/k,

0 else,

andg(F )1 = g1 − g

(N)1 . In appendix D, we show thatg(F )

1 ∼ g(F )0 , and thatg(N)

1 has

a finite,O(k−1) limit at s → s′. Further application ofh is negligible:hg(N)1 ≪


hg(F )0 (for large enoughk). Applying these results, (4.15) is transformed into the

following form

g ∼ g(N)0 + g

(N)1 + 2

∞∑

n=0

hng(F )0 . (4.17)

As opposed to (4.15), all the integrations that appear in this representation are

regular. For any finite separations − s′, the first two terms are semi-classically

irrelevant, since fork large enough we will have|s − s′| > xC/k. However,

when taking the limits → s′, as required for computing the “power densities”

du2(s) anddu2(s) (which, in turn, are required for computing the BI density—

see equation (4.7)), they give the most significant contribution. For this limit, it is

sufficient to approximate them fors = s′+δ whereδ ≪ 1/k. This approximation,

which was explicitly computed in [33] (see also in appendix D) reads

Re(g(N)0 + g

(N)1 ) ∼ 1

πk2δ2− 1

2πlog

kδ

2+ O(δ0)

Im(g(N)0 + g

(N)1 ) ∼ 1

4

(1− κ

k− k2 − 4κk

8δ2)+O(δ4) (4.18)

and (as shown below) ultimately leads to the smooth part ofη, as given in equa-

tion (2.1) (which was computed there using the CRW model).

The third term of equation (4.17) describes the oscillatingpart ofg. It is an

infinite sum, where thenth term involves integration overn intermediate points

s1, s2 . . . sn

gosc(s, s′) = 2

∞∑

n=0

∫ds1ds2 . . . dsn h(s, sn) · · ·h(s2, s1)g(F )

0 (s1, s′). (4.19)

We interpret this as a summation over all possible orbits from s′ to s, where an

orbit is composed of straight chords connecting a sequence of points on the bound-

ary. The bounces at the intermediate points are not necessarily specular (an orbit

that has only specular bounces is called a classical orbit).


4.2.3 Semi-classical power densities

We now combine the expansion (4.17) with equation (4.8) to derive an expansion

for the correlation densityduu′. This shall be used to derive the power densities

du2 anddu2 of equation (4.7) (which are needed for subsequent derivation of the

BI densityb). Following equation (4.17), we split the correlation density function

into two partsduu′ = dsmuu′ + doscuu′. The smooth partdsmuu′, which corresponds to

g(N)0 + g

(N)1 is only relevant when|s − s′| < xC/k. For s − s′ = δ ≪ 1/k, we

have (from equation (4.18))

dsmuu′(s+ δ, s) ∼ k

2π

(1− κ

k− k2 − 4κk

8δ2)+O(δ4).

Hence, the smooth parts of the power densities (see (4.7)) are given by

dsmu2 =k − κ

2π, and dsmu2 =

k3

8π

(1− 4

κ

k

)(4.20)

(the mixed momentduu has a vanishing smooth part, as expected).

The oscillating partdoscuu′ corresponds to the orbit sum of equation (4.19). Since

we are interested in the semi-classical limit, we use equation (4.16) to approximate

g(F )0 . We also use the large argument approximation for the Hankelfunction inh

(equation (4.11))

h(s, s′) ∼√

k

2πdsinψ ei(kd+

3

4π).

This is allowed, because all the integrations involved are regular, and the contribu-

tion from the “near zones” (regions where|s− s′| < xC/k) is negligible for large

enough values ofk. For writing down the orbit summation explicitly, we use the

notation shown in figure 4.4. The chord originating from the intermediate pointsiis labelleddi, the incidence angle atsi is labelledψi, while ψi denotes the angle

of reflection fromsi. We also denotes0 = s′ andsn+1 = s (and correspondingly

ψ′ = ψ0, ψ = ψn+1, d0 = d′ anddn+1 = d). With this notation, the oscillating

partdoscuu′ becomes

4

π

∞∑

n=0

( k2π

)n+1

2

∫ n∏

l=1

dslsin ψ0

∏n+1l=1 sinψl

(∏n

l=0 dl)1/2

cos

(k

n∑

l=0

dl +3(n+ 1)

4π

). (4.21)


n′

ψ1

n1

ψ2

ψ

ψ1

ψ′

n2

d2

d′

d1

ψ2

n

Figure 4.4: Path froms′ to s: notations

The integrals over the intermediate pointss1, . . . sn should now be evaluated us-

ing the stationary phase approximation (SPA). The fast oscillatory phase in this

expression is exactly the actionS = k∑

l dl = kL(s1, . . . sn) (as is usually the

case in semi-classical quantization), so the stationary phase conditionsk ∂L∂si

= 0

for i = 1 . . . n read

∂L

∂si=∂di−1

∂si+∂di∂si

= cosψi − cos ψi = 0.

Hence, the integral becomes a sum over orbits where all the intermediate bounces

are specular reflections—a.k.a. classical orbits froms′ tos. The only non-vanishing

second derivatives ofL are

∂2L

∂si∂si−1

=sin ψi−1 sinψi

di−1

for i = 2, . . . n, and

∂2L

∂s2i= sinψi (ιi − κi) + sin ψi (ιi − κi) for i = 1 . . . n,

whereιi = sinψi/di−1 andιi = sin ψi/di. Hence, the matrix of second derivatives


at a stationary (classical) path is tri-diagonal, and is given by

D =

2S1(ι1 − κ1) S1 · ι2 0

S1 · ι2 2S2(ι2 − κ2) S2 · ι3S2 · ι3 · · ·

. . . Sn−1 · ιn0 Sn−1 · ιn 2Sn(ιn − κn)

whereιi = 12(ιi + ιi) andSi = sinψi (we shall also useSi = sin ψi, but they are

equal for classical orbits). The result fordoscuu′, after applying of the SPA, is

√k

(2

π

)3

2 ∑

γ∈Cl(s,s′)

sin ψ0

∏nγ+1l=1 sinψl√

| detDγ|∏nγ

l=0 dlcos

(kLγ +

3π

4− (2nγ + µγ)

π

2

),

whereγ enumerates classical paths froms′ to s. For each path,nγ is the number

of intermediate bounce pointss1, . . . sn, Lγ is the length of the path,Dγ is the

matrix of second derivatives as specified above andµγ is the number of negative

eigenvalues of that matrix.

This expression may be further simplified using classical analysis of the bil-

liard map. In Birkhoff coordinates, the billiard maps the canonical pair(qi, pi) to

the next one(qi+1, pi+1), whereqi = si andpi = k cosψi (the tangential com-

ponent of the momentum). The equations of motion are generated by the action

Si+1,i = kdi (i.e. ∂S∂qi+1

= pi+1 and− ∂S∂qi

= pi). The tangent matrixTi, mapping

(δqi, δpi/k) to (δqi+1, δpi+1/k) is given by

Ti =−1

ιi+1Si

(Si(ιi − κi) 1

κiκi+1SiSi+1(1− ιiκi− ιi+1

κi+1) Si+1(ιi+1 − κi+1)

).

For the full path, the stability matrix is given byT = ∂(q,(p/k))∂(q′,(p′/k))

= Tn · · ·T1 · T0.This formalism is discussed in [51], where it is shown that the required determi-

nant is given by

detD = (−1)n+1 sin ψ0 sinψn+1

∏nl=1 sin

2 ψi∏nl=0 dl

[T ]1,2


(and[T ]1,2 = k ∂sn+1

∂p0

∣∣∣s0

). Substituting this, we get

doscuu′(s, s′) ∼

(2

π

) 3

2 ∑

γ∈Cl(s,s′)

√sinψ sin ψ′

|∂s/∂p′|s′|cos(kLγ +

3π

4− νx

π

2), (4.22)

whereνγ = 2nγ+µγ is the Maslov index as defined for Dirichlet billiards (number

of conjugate points plus 2 for each Dirichlet bounce [45, 51]). Note that this is

actuallyO(√k), as there is a factor ofk “hidden” in the partial derivative∂s/∂p′

(sincep′ = k cosψ′). To find duu′, duu′ andduu′ we differentiate equation (4.22)

by s ands′. The largest contributions come from the derivatives of theoscillating

phase∂(kL)

∂s= k cosψ and

∂(kL)

∂s′= −k cos ψ′,

and the resulting power densities are

doscuu′(s, s′) ∼ k

(2

π

) 3

2 ∑

γ∈Cl(s,s′)

cos ψ′√

sinψ sin ψ′√

|∂s/∂p′|s′|sin(kLγ +

3π

4− νx

π

2

),

doscuu′(s, s′) ∼ k

(2

π

) 3

2 ∑

γ∈Cl(s,s′)

cosψ

√sinψ sin ψ′

√|∂s/∂p′|s′|

sin(kLγ −

π

4− νx

π

2

)(4.23)

(note: thes = s′ limit of the last two is identical, due to time reversal symmetry),

and finally

doscuu′ ∼ k2(2

π

) 3

2 ∑

γ∈Cl(s,s′)

cosψ cos ψ′√

sinψ sin ψ′√

|∂s/∂p′|s′|cos(kLγ +

3π

4− νx

π

2

). (4.24)

The complete trace formulae for the power densities are found by takings = s′

in equations (4.24) and (4.22), and adding them up with the smooth parts, equa-

tion (4.20). It should be noted that although the classical paths that appear in

the sum are closed (i.e. end up at the point of origins), they are not periodic or-

bits, sincepn+1 6= p0, so the next bounce would lead elsewhere. If, however, we

compute the density of the total power∫Γdu2(s) ds on a finite segmentΓ ⊂ ∂Ω,

then the extra integration introduces another SPA condition, which effectively se-

4.3. TRACE FORMULA FOR THE BIC 71

lects only periodic orbits, following the same procedure that we describe below

(in section 4.3.1) for the case of the BIC density. In fact, using this procedure

on the identity12

∮r · n(s)du2(s; k) ds = d(k) (which follows directly from the

normalization condition (4.1)), we recover exactly the Gutzwiller trace formula

for the spectral densityd(k) of chaotic Dirichlet billiards

d(k) ∼ kA2π

− L4π

+1

π

∑

γ∈PPO

∞∑

r=1

Lγ√| tr[Mγ

r − I]|cos(r(kLγ − νγ

π

2)), (4.25)

whereγ enumerates primitive periodic orbits,r is the number of repetitions, and

Mγ is the monodromy matrix.

4.3 Trace formula for the BIC

In section 4.1, we have seen that a given smoothing functionρσ, induces an en-

sembleUσ(k) of eigenfunctions in a spectral window aroundk, where the mean

BI density is given by〈b(s)〉 = dσb (s; k)/dσ(k). Furthermore, for high enoughk,

we have seen (section 4.1.2) that this ensemble seems to approach Gaussianity.

Hence, the mean BI density should approach the prediction ofthe Rice formula,

equation (2.12). In terms of the moments computed in section4.2, this is written

as

dσb = dσ1

π

√dσu2d

σu2 − (dσuu)

2

(dσu2)2 . (4.26)

Using equation (4.20) and equations (4.22)–(4.24), we write down the trace for-

mulae for the densitiesdu2, du2 andduu. Convolving each withρσ, an oscillating

term corresponding to orbitx gains a factor ofρ(Lγ), which, as in section 3.2.1,

we will assume to decay fast enough such that the orbit summations are effectively

cut off at some maximal length, and the sums are finite. Substituting the results in


equation (4.26), and expanding asymptotically ink, we find

〈b(s)〉k,σ ∼ k

2π− 3κ(s)

4π

+1

4

(2

π

) 3

2 ∑

γ∈Cl(s,s)

ρσ(Lγ) · φ0γ√

|∂s/∂p′|s′|cos

(kLγ +

3π

4− νγ

π

2

)

+∑

γ,γ′∈Cl(s,s)

ρ(Lγ)ρ(Lγ′)P(γ, γ′, k) (4.27)

whereγ enumerates closed classical orbits returning tos = s′ afternγ specular

bounces, andφ0γ = 2(4 cosψ cos ψ′ − 1)

√sinψ sin ψ′ is a trigonometrical factor

depending on the angles ats0 = snγ+1. The second line in (4.27) is of order√k and represents the leading oscillatory terms in the trace formula. The last

term is anO(k0) oscillating contribution that comes from pair products of the

three oscillating terms in the expansions ofdu2 , du2 andduu. Analogously to the

arguments leading to equation (3.13), this leads to summation over pairs of orbits

(γ, γ′), whereP(γ, γ′, k) is a collection of oscillatory terms, whose phases are

k(Lγ + Lγ′) andk|Lγ − Lγ′ |. As we have seen in section 3.2.1, such terms can

cause dense “noise” which is hard to control numerically. However, it is formally

of orderk−1/2 smaller than the main oscillatory contribution.

4.3.1 Taking the trace

Now that we have the BI density, the BIC can be derived by integrating equa-

tion (4.27) over the boundary. The first oscillatory term (O√k) of (4.27) is very

similar todoscuu′ of equation (4.22), and, like its analogue, it is written in aform that

involvess ands′ separately, without assuming them to be equal. This means that

we can think of it as a 2 parameter function〈b(s, s′)〉osck,σ, whose trace is exactly the

BIC oscillations〈η〉osck,σ. The operation of integratingb(s) overs is the analogue of

the trace operation performed in derivations of other traceformulae. The classical

periodic orbits appear once we apply the SPA on this integration.

Instead of integrating directly, we refer again to the similarity with (4.22), and

recall that this was derived via the SPA from equation (4.21). “Rolling back”

〈b〉osc of equation (4.27) to the analogue of (4.21) before applyingthe trace, we


get the following integral representation for〈η〉osc = tr 〈b(s, s′)〉osc

〈η〉osck =1

π

∞∑

n=0

( k2π

)n+1

2

∫ n∏

l=0

dslφ0s

√sin ψ0 sinψn+1

∏nl=1 sinψl

(∏n

l=0 dl)1/2

cos

(k

n∑

l=0

dl +3(n+ 1)

4π

), (4.28)

wheres = (s0, s1, . . . sn) andφ0s = 2(4 cosψn+1 cos ψ0 − 1)

√sinψn+1 sin ψ0

as before. We now apply the SPA over all nodes of the closed orbit s. This

time, the SPA condition yields a summation over classical periodic orbits. The

(n+ 1)× (n+ 1) matrix of second derivatives is now given by

D =

2S0(ι0 − κ0) S0 · ι1 0 Sn · ι0S0 · ι1 2S1(ι1 − κ1) S1 · ι2

S1 · ι2 · · ·. . . Sn−1 · ιn

Sn · ι0 0 Sn−1 · ιn 2Sn(ιn − κn).

The determinant of this matrix is also computed in [51]. It isgiven by

det D = (−1)n+1

∏sinψi

∏sin ψi∏

ditr[Ms − I],

whereMs = Tn · · ·T1 ·T0 is the monodromy matrix corresponding to the periodic

orbit s. We substitute this in the expression we get after applying the SPA on

equation (4.28). The result, written as a sum over primitiveperiodic orbits (PPO)

and the number of repetitionsr is

〈η〉k,σ ∼ kL2π

− 6π

4π

+1

π

∑

γ∈PPO

∞∑

r=1

ρσ(Lγ) · Φγ√| tr[Mγ

r − I]|cos(r(kLγ − νγ

π

2

)), (4.29)

whereΦγ is the following trigonometrical factor depending on thenγ bounce


angles of the orbitx

Φγ =

nx∑

i=1

(4 cos2 ψ(x)i − 1)2 sinψ

(x)i , (4.30)

νγ = 2nγ + µγ is the Maslov index, andMγ is the monodromy matrix.

To get an expression fordη(k), we multiply byd(k), using equation (4.25),

dη(k) ∼AL4π2

k2 − L2 + 6πA8π2

k

+k

2π2

∑

γ∈PPO

∞∑

r=1

LLγ +AΦγ√| tr[Mγ

r − I]|cos(r(kLp − νp

π

2)), (4.31)

Finally, to remove spectral dependence, as in section 3.2.2, we usedη(q) =

dη(k(q))dkdq

. The inversion formula for chaotic billiards is derived from Gutzwiller’s

formula (see equation (4.25)):

k(q) = q +L2A + q−13L2 − 8πA

24A2

− 2

Aq−1∑

r,x

1

r√

| tr[Mγr − I]|

sin(r(qLx − νx

π

2))+O(q−2).

Applying the above, we finally get

dη(q) ∼AL4π2

q2 +L2 − 6πA

8π2q

+A2π2

q∑

γ∈PPO

∞∑

r=1

Φγ√| tr[Mγ

r − I]|cos(r(qLp − νp

π

2)). (4.32)

4.3.2 Numerical results

The results of section 4.3.1 depend on the assumption that the boundary functions

are approximately Gaussian. We have seen numerical evidence for the decay of

the Kurtosis, but we do not have a theoretical estimate for it. Neither did we

compute the expected effect of deviations from Gaussianityon the BIC, and we

do not know how much deviation can be tolerated without invalidating our result.

Due to these uncertainties, a numerical verification is required. For this purpose


Figure 4.5: Africa Billiard: sample periodic orbits

we chose to investigate the Africa billiard of Robnik and Berry[52]. The boundary

of this billiard is smooth (as opposed,e.g. to the Sinai billiard), and is given by

points(x, y) such that

x+ iy = eiθ +Bei2θ + Cei(3θ+φ), (4.33)

where0 ≤ θ < 2π is the curve parameter. The constants we have used are

B = C = 0.2 andφ = π/3, which correspond to a billiard which is believed to

be chaotic (numerical evidence is given in [52]). It is not convex, and the concave

regions are a major source for the instability. Figure 4.5 shows several periodic

orbits (all unstable) of this billiard. Sample eigenfunctions are shown in figure 4.6.

We have computed the lowest 20,000 eigenfunctions (0 < kn < 260) of

the billiard, the corresponding BI count sequenceηn, and the densitydση(q) =∑

n ρσ(q −√4πn/A)ηn (whereρσ(x) is a narrow Gaussian). Subtracting the

predicted smooth partdsmη (q) from equation (4.32) and scaling, we computed

f(q) = (dρη − dsmη )/q ·W (q), whereW is a Gaussian “window function” of width

σ = 50 and centreq0 = 130, used for softening the sharp cutoff due to the finite-

ness of the computed spectrum. The length spectrum, which isthe Fourier trans-

form f(x) of f(q) was compared withgσ(x), the theoretical prediction based on the

oscillating part of (4.32). The latter was computed using 70classical periodic or-

bits (with upto 7 bounce points) and 15 complex periodic orbits whose lengths had

a very small imaginary part. The effect of these complex orbits was observable in


(a) Nodal domains (sign(ψ200)) (b) Probability density (|ψ8000|2)

Figure 4.6: Africa Eigenfunctions

the Fourier transform of the spectral densityd(k) (marked “C” in figure 4.7), but

it was not significant enough to be observable indη(q).

Figure 4.8(a) displays several peaks centred at lengths of periodic orbits which

match quite well with the theoretical predictions. A more detailed comparison is

presented in figure 4.8(b). Due to the form of the trigonometrical factor (4.30) in

equation (4.32), orbits whose angles are close to 60 are inhibited. Indeed, the

triangular periodic orbits of the billiard, whose lengths are in the range 5.07–6.05,

cannot be seen above the background level (although they do appear in the Fourier

transform of spectral density—figure 4.7). The structure inthe range6.3 < x <

6.9 is due to a bunching of periodic orbits that pass very close tothe boundary at

the region of its highest concavity. Some examples are shownin figure 4.9. The

poor agreement between the semi-classical theory and the numerics in this region

is due to penumbra corrections [53], which were not includedin our simulation.

The random background of amplitude∼ 0.04 observed in the plots does not seem

to diminish whenq0 increases (within the range of our study). This means that this

background is not due to theO(1) terms of (4.32) (which are expected to contain

dense contributions from orbit differences, as in figure 3.7). We suspect that the

source for this noise is the limited validity of the Gaussianmodel for finitek.

As explained in section 4.1.1, the fourth order cumulantsδ0–δ4 of equation (4.6)


Figure 4.7: Length spectrum of the Africa billiards—Fourier transformof the spectral densityd(k) (predictions computed according to the Gutzwiller TF).

Figure 4.8: The semi-classical and numerical length spectra (Fourier transform ofdoscη

(q)). (a):Absolute value, (b): Magnified view of 3 prominent peaks.


Figure 4.9: The region6.3 < L < 6.9 contains orbits that pass close to the concave wedge (seetext).

might have a significant effect on the BI density. As we have seen in figure 4.3,

there is a residual Kurtosis even at the highest boundary functions. By the power-

law fit, for 80 < k < 260 the Kurtosis varies from 0.12 to 0.04. To reduce

it to 0.01, one would have to compute around 300,000 eigenfunctions, which is

unfeasible using our current algorithms and hardware.

A more stringent test of the theory is based on the following argument. Since

η =∮b(s) ds, a “partial” trace formula which counts BI located on a prescribed

partΓ ⊂ ∂Ω of the boundary can be similarly derived by integrating the BI density

overΓ alone:dη Γ(k) =∫Γdb(s; k)ds. Since the formula fordb, much like (4.27),

involves summation over orbits starting and ending ats, we conclude that only

periodic orbits that have a bounce point inΓ will contribute to the sum in the

resulting trace formula fordη Γ. By choosing aΓ which is bounded away from the

bounce points of a specific orbit, we can effectively turn offthe effect of that orbit.

Similarly, we expect orbits that have some, but not all of their bounce points in

the excluded regions∂Ω \ Γ, to have reduced amplitude in the length spectrum.

This result is demonstrated in figure 4.10. In figure 4.10(b),Γ is plotted with a

wide line, while the excluded part∂Ω \ Γ is dotted. For demonstration purposes,

the BI of ψ150 are shown on the boundary (totalη150 = 24), and the points to

be excluded fromηΓ are marked with empty circles (ηΓ,150 = 16). Three orbits

are shown, and the corresponding peaks in the length spectrum are also marked


Figure 4.10:Restricting the BIC toΓ ⊂ ∂Ω reduces the amplitudes for orbits hitting the excludedregion. Comparef to fΓ for the 3 marked orbits.

in figure 4.10(a). Orbit 1 has both its bounce points in the excluded regions, so

it completely disappears from the length spectrum corresponding to the partial

countηΓ. Orbit 2, which has both of its bounce points inΓ is not effected by the

exclusion, and orbit 3, which has only 1 out of 4 bounce pointsin Γ, is significantly

inhibited, and drops below the noise level for the numericalcase. This test and the

general agreement between the semi-classical and the numerical length spectra

give credence to the validity of the proposed trace formula.

Chapter 5

Discussion

5.1 Summary and conclusions

As was mentioned in chapter 1, the study of geometrical features of Laplacian

eigenfunctions, and in particular their nodal domains, hasrecently experienced

a renaissance [16]. However, as we described in the introduction, in compari-

son with the study of spectral sequences, the problems involved in the statistical

and semi-classical description of the eigenfunctions are more complicated, and

there are larger gaps in current knowledge. In this context,the 1D nodal struc-

tures induced by 2D eigenfunctions on their boundary are a promising field of

research. In addition to the fact that they contain relevantinformation about the

full 2D nodal structure [29], it was shown [18] that they directly exhibit finger-

prints of the underlying classical dynamics. On the other hand, the fact that they

are one dimensional greatly simplifies their analysis. Furthermore, this simpli-

fication should allow us to adapt techniques from the research of the spectrum,

and apply them to the study of boundary domains. The main conclusion of this

thesis is that this advantage can indeed be utilized, and theresults reveal explicit

connections between the boundary nodal structures and the underlying classical

motion, which are similar to the connections known from the study of the spectral

sequence. For the chaotic case, our results can serve as theoretical tools that en-

able harnessing of knowledge from statistics and classicalmechanics to the study

of chaotic eigenfunctions.

81

82 CHAPTER 5. DISCUSSION

For the separable integrable case, the BI counting problem is rather simple.

The asymptotic distribution of the BIC has been derived in [18]. The BIC of these

billiards were also used in [28], where the authors examinedthe upper bounds for

ηn (and also used the circle billiard as an example to show that there is no lower

bound forη in terms ofk). Trace formulae, which describe the fluctuations around

the mean smooth part in terms of classical parameters, have been derived in [31]

for the nodal sequenceνn, but not forηn. We close this gap in chapter 3, where

we apply the same techniques on the BIC, and derive the relevant trace formula,

equation (3.14). Furthermore, we also use the same method toderive the trace

formula for a system which is not separable. Comparison between that result and

the results for the separable case (section 3.3.2), suggests that it might be possible

to generalize (3.14) to cover a wider class of integrable systems.

For the chaotic case, we took two separate paths. In chapter 2, we have used

statistical models to describe the boundary function. Models that take the bil-

liard boundary into account [37, 38] were originally suggested as corrections for

Berry’s RW model, and the authors had focused on regions which are far from

the boundary. Here we took the opposite limit, where the sampled region ap-

proaches the boundary itself. For the research of boundary domains, the role of

such boundary-restricted models is the analogue of the rolethat Berry’s RW model

plays in the research of nodal domains away from the boundary. The result for the

BI density, equation (2.17), refines the previous result of [18] by taking into ac-

count the effect of a curved boundary. Moreover, since the boundary is only one

dimensional, the nodal intersections constitute a discrete sequence, and we can

apply the rich machinery that had been used in the study of RMTand spectral

research. This allows a more detailed description of the NI distribution, and in

particular computation of statistics such as the scaled correlations and the form

factor.

chapter 4 takes a different path. First, we construct a semi-classical expansion

for the second moments of the boundary functionsun (section 4.2.3), based on

a Green function formalism. This is achieved by extending the results of [33]

to a formal trace formula, in a manner similar to the way in which [10] ex-

tends [9]. Then, employing a “Central Limit”-type conjecture about the distri-

bution of boundary functions within a suitable spectral window, we derive the

5.2. POTENTIAL FUTURE DEVELOPMENTS 83

relevant trace formulae for the BI density (equation (4.27)and the BIC (equa-

tion (4.29)). Since this is the first time such a formula appears in the context of

nodal domains of chaotic systems, it opens up new ways to facilitate the theoreti-

cal description of the nodal pattern and possibly advance the understanding of the

numerically observed connections with the underlying classical dynamics.

5.2 Potential future developments

Due to the fact that the interest in boundary domains is relatively new, currently

there are not many publications on the topic, and each new result opens up ques-

tions for further research. Obviously, some possible extensions for the research

presented here naturally suggest themselves due to the factthat we purposely lim-

ited the discussion. For example, we expect that all major results of this thesis

could be extended to (or replaced with a suitable analogue for) the case of bil-

liards with Neumann and Robin boundary conditions. In what follows, we list

several other directions, which might be less obvious, but still seem to naturally

follow from our results.

• As was shown in section 3.3.2, the result for the right isosceles triangle

matches the formula that was derived for separable systems.This natu-

rally raises the question whether it is possible to generalize the result (equa-

tion (3.14)) to form a trace formula which is valid for all, orat least a

larger subset of, integrable billiards. This might be done using a suitable

generalization for the coefficientsτ1,τ2 and the condition that defines them

(condition 1 of section 3.1, which describes the required properties of the

generalized “boundary”), possibly based on the geometrical structure of the

boundary when transformed into action-angle space. Other non-separable

integrable systems, such as the equilateral triangle, could be used as an “ex-

perimental” test case for this question.

• The statistical approach taken in chapter 2 provides a way toderive some

predictions regarding the two-point statistics of the BI from the distribution

of the boundary function itself. On the other hand, chapter 4yields a de-

tailed description, in terms of trace formulae, both for theboundary function


and for the BI density at a given point. Since the trace formula for the BI

density involves a sum over classical orbits, we expect thatan estimate for

two point statistics such as the form factor could be derivedusing methods

similar to the ones used in [13]. These predictions could then be com-

pared and analyzed numerically using the methods of chapter2. Numerical

confirmation of such results would give further credence to the validity of

the conjectures described in this thesis, and provide further insight into the

structure of the nodal patterns on the boundary.

• In section 4.1.1, the conjecture about the Gaussianity of the boundary func-

tion is presented as a statement regarding the asymptotic behaviour of the

cumulants involved in the expansion of equation (4.5). Numerical evidence

indicate that the Kurtosis, which is probably the next orderterm in the ex-

pansion, does in fact decay for largek. Integration of (4.6) overχ andξ

should lead to a formula that would allow computing the expected effect of

the residual Kurtosis at finite values ofk. However, at present we do not

have any theoretical estimate for 4th order moments ofu. Nevertheless, it

might be possible1, using a diagonal approximation similar to the methods

of [13], to derive such estimates. This would enable quantitative estimates

regarding the rate of convergence to Gaussianity, and wouldconstitute a

step forwards towards proving the conjecture.

• Comparing the results of chapter 4 with those of chapter 3 reveals that the

contributions of invariant tori to〈η〉 is of order√k larger than contributions

of isolated hyperbolic orbits. This is consistent with the analogous com-

parison of the spectral trace formulae. Continuing this analogy, we expect

that even for chaotic billiards, the contribution from continuous families of

bouncing-ball orbits should be of the same order ink as the contribution

of tori in the integrable case (this is a known fact for the spectral trace for-

mulae [45]). Thus, despite the fact that their phase-space measure is 0, they

should dominate the length spectrum and hide the contributions of the unsta-

ble orbits. In our early numerical experiments with BIC sequences (which

we performed to gain evidence for the existence of a trace formula for the

1This idea was suggested by Roman Schubert, University of Bristol.

5.2. POTENTIAL FUTURE DEVELOPMENTS 85

chaotic BIC), we had used eigenfunctions of the Sinai billiard, and did in

fact observe the dominating contributions of bouncing ballorbits. By using

analogous methods to the ones used in the derivation of the spectral formu-

lae, it should be possible to derive the correct contribution of bouncing ball

orbits and confirm this phenomenon theoretically.

• We believe that some progress could be made by combining our techniques

and results with the methods that were recently used by otherresearchers. In

particular, the method used in [28] and [29] to establish bounds for the BIC

sequence is based on compexification of the boundary functions. It seems

different than our approach, but still related to it. Studying the connections

between such theories could provide more rigour to the knownresults and

perhaps even yield new ones. Furthermore, in [30], the authors derive an

estimate for the average BIC of a different asymptotic ensemble, namely

random linear combinations of the eigenfunctions in a givenspectral win-

dow. In this case, Gaussianity is introduced through the choice of the linear

coefficients and there is no need for the random wave conjecture. Such en-

sembles could be used, for example, to construct random wavemodels for

the 2 point statistics of the BI, via the mechanisms described in chapter 2.

Appendix A

Derivation of the BIC for separable

systems

As described in section 3.1, the derivation of the BIC trace formula for separable

systems, equation (3.1) is based on the simple relation between the BIC and quan-

tum numbers of a given eigenfunctionηl,m = lτ1 +mτ2. Once this is known, the

derivation is straightforward, and closely follows [11]. We provide it in this ap-

pendix for completeness, since as far as we know, it was not published elsewhere

before.

According to the EBK quantization rules [54], The actions are given byI1 =

l + α1/4, I2 = m + α2/4 wherel, m are the integer quantum numbers, and the

αi are geometrical indices, counting the multiplicity of singularities encountered

at turning points of the corresponding coordinates along the orbit. Hence, for a

system with HamiltonianH(I1, I2), the BIC density is given by

dη(k) =∑

l,m≥0

(lτ1 +mτ2) δ(k −

√H(l + β1, m+ β2)

),

where we have introducedβi ≡ αi/4. Applying the “half infinite” form of the

Poisson summation formula

∞∑

n=m

f(n) =∑

N∈Z

∫ ∞

m

f(n)ei2πNndn +1

2f(m),

87

88 APPENDIX A. THE BIC OF SEPARABLE SYSTEMS

we get the exact expansion

dη(k) =∑

M,N

∫ ∞

β1

dI1

∫ ∞

β2

dI2 δ(k −√H) η(I1 − β1, I2 − β2) e

i2π(M(I1−β1)+N(I2−β2))

+1

2

∑

M

∫ ∞

β1

dI1 δ(k −√H(I1, β2)) η(I1 − β1, 0) e

i2πM(I1−β1)

+1

2

∑

N

∫ ∞

β2

dI2 δ(k −√H(β1, I2)) η(0, I2 − β2)) e

i2πN(I2−β2)

+1

4δ(k −

√H(β1, β2))η0,0. (A.1)

The last term is 0 for anyk larger than the ground state momentum. The terms

with the single integration reduce to sums over repetitionsof the two special orbits

which represent the limiting case, where one of the actions is minimal (approxi-

mately 0). However, they areO(k), and we wish to concentrate on an expansion

up to the largest oscillating contribution, which isO(k3/2). This leaves us with

the first term.

We now use the homogeneity ofH to reduce the intergral to the unit energy

surface. Lets be the arc length on the surface and parametrize it as(I1, I2) =

(ξ1(s), ξ2(s)), so thatξ21+ ξ22 = 1 (where the dots specify derivative bys). Chang-

ing the coordinates from(I1, I2) to (λ, s) using

I1 = λξ1(s); I2 = λξ2(s),

we find thatdI1 dI2 = dλ ds (λW (s)) whereW = ξ1ξ2 − ξ2ξ1 is the Wronskian.

Furthermore√H(I1, I2) = λ (does not depend ons), soδ(k −

√H) = δ(k −

λ). Applying these to the leading term of equation (A.1), we get, after ignoring

corrections of orderβi/k,

dη(k) ∼ k2∑

M∈Z2

∫ A

0

dsW (s)(τ1ξ1 + τ2ξ2) ei2πkM ·ξ−iπ

2M ·α, (A.2)

whereA is the area (arc length) of the energy surface, andM = (M,N).

Before we go on, we give an equivalent expression for the Wronskian. Let

ωi =∂H∂Ii

be the angular velocities at(ξ1, ξ2), andω =√ω1

2 + ω22. From the

A.1. THE WEYL TERM 89

fact thatH is constant on the surface (H = 0), we getω · ξ = 0. But ξ is a unit

vector, so we find (when choosing the orientation ofs to be counter-clockwise)

that(ω1, ω2) = ω(−ξ2, ξ1). Hence we can write

W (s) = ξ · (−ξ2, ξ1) =ξ · ωω

=2

ω.

The last equality follows from the identity

ω1I1 + ω2I2 = 2λ, (A.3)

which can be verified by taking theλ derivative of both sides of the homogeneity

conditionH(λξ1, λξ2) = λ2.

A.1 The Weyl term

TheM = N = 0 term of (A.2) gives the smooth (Weyl) partk2(τ1A1 + τ2A2),

where theAi are given in equation (3.3). To get the representation in terms of the

phase space volume we write

W (s) ds = ξ1 dξ2 − ξ2 dξ1,

and use this to splitA1 =∫W (s)ξ1(s) ds into two integrals, with different inte-

gration variables

∫ A

0

ξ1(s) ·W (s) ds =

∫ a2

0

ξ12(ξ2) dξ2 −

∫ 0

a1

ξ2(ξ1) ξ1 dξ1,

where (choosing counter-clockwise orientation fors), we assume that the edge

points of thes integration are given by(ξ1(0), ξ2(0)) = (a1, 0) and(ξ1(A), ξ2(A)) =

(0, a2). In the first integral, we substitute

ξ12(ξ2) =

∫ ξ1(ξ2)

0

(2I1)dI1


and get

∫ a2

0

ξ12(ξ2) dξ2 =

∫ a2

0

∫ ξ1(ξ2)

0

(2I1) dI1 dξ1 = 2

∫∫I1Θ(1−H(I1, I2)) dI1 dI2.

Similarly, for the second integral we have

−∫ 0

a1

ξ2(ξ1) ξ1 dξ1 =

∫ a1

0

∫ ξ2(ξ1)

0

dI2 ξ1 dξ1 =

∫∫I1Θ(1−H(I1, I2)) dI2 dI1.

Combining the two parts, we get

A1 = 3

∫∫I1Θ(1−H(I1, I2)) dI1 dI2, (A.4)

and by similar argument we can show the corresponding resultfor A2 as well.

Using the fact thatH does not depend on the corresponding angle variables, we

can integrate over them, and then return toq,p by canonical transformation, i.e.

for any functionF of the actions alone, we have

∫F (I) d2I =

1

4π2

∫F d2I d2θ =

1

4π2

∫F d2q d2p.

Applying this procedure to (A.4) and to the corresponding result forA2, we re-

cover equation (3.3), from which the Weyl term of equation (3.1) follows imme-

diately.

A.2 The oscillating part

Returning to equation (A.2), we now examine the terms where(M,N) 6= (0, 0),

and evaluate them using the SPA. The stationary phase condition readsM · ξ = 0.

Since the angular frequenciesω are also orthogonal to the tangent vector, we find

that they must be proportional toM . Hence they are rationally dependent and

the orbit is periodic. Taking out the common factor,(M,N) = r(µ, ν), we write

(ω1, ω2) = (µ, ν)2π/T , whereT is the period of the orbit.

A.2. THE OSCILLATING PART 91

The phase of the exponent in (A.2) is proportional to the action of the orbit

2π(µξ1 + νξ2) =

∮(p1dq1 + p2dq2) = S,

but in the homogeneous case, we can also apply the identity (A.3) and get

S = 2πµ · ξ = T ω · ξ = 2T .

For (M,N) the action isr · 2T (so we considerr to be the number of repetitions

over the orbit). The second derivative of the phase is proportional to

µξ2 + νξ2 = −κ√µ2 + ν2 = −κωT

2π,

whereκ is the curvature of the unit energy surface. Substituting these results in

the SPA of equation (A.2), we get

doscη (k) = k3/2∑

γ∈PT

∞∑

r=1

4√2πηγ

ωγ3/2√r|κγ|Tγ

cos(r(2kTγ −

π

2µγ ·α)− σγ

π

4

),

whereγ = (µ, ν) enumerates periodic tori on the unit energy surface,ηγ , ωγ , κγ ,

Tγ andσγ are the relevant parameters as defined in section 3.1. Finally, to get

the result, equation (3.1), we expressω in terms of the period and the topological

index:ωγ = |µγ|2π/Tγ.

Appendix B

Counting boundary intersections for

the right isosceles triangle

In this appendix, we show that the boundary intersections count for the eigenfunc-

tion ψl,m (with l > m > 0) of equation (3.15) is given by equation (3.16). Since

the boundary intersections are the points where the boundary function changes

sign, we start by calculating the boundary functionul,m = (1/kl,m)(n · ∇)ψl,m.

With proper choice of boundary parameters, we get

ulm =

√2πa[l sin(mx)−m sin(lx)] if s < a

√2πa(−1)l+m−1[l sin(my)−m sin(ly)] if a < s < 2a

2π/a[−l cos(lz) sin(mz)+m cos(mz) sin(lz)] if s > 2a,

(B.1)

wherex = πs/a, y = π(2 − s/a) andz = π(2 +√2 − s/a)/

√2 are linear

transformations ofs, normalized to vary through the range(0, π) on the respective

edges of the triangle.

The number of zeros ofu in each of the segments(0, a) and (a, 2a) is the

number of solutions forsin(lx)

l=

sin(mx)

m(B.2)

with 0 < x < π. We will show that this is exactlym − 1. Denotezi = πi/m

for i = 1, . . . , m − 1 (the zeros ofsin(mx)). Let δ = sin−1(m/l)/m < π/(2m).

93

94 APPENDIX B. COUNTING BI FOR THE TRIANGLE

Equation (B.2) can only have solutions in one of the (non overlapping) ranges

[zi−δ, zi+δ], because out of these ranges we have|f1(x)| ≡ | sin(mx)/m| > 1/l,

whereas|f2(x)| ≡ | sin(lx)/l| ≤ 1/l for all x. For oddi, we find that on the

higher edgex+ = zi+ δ we havef1(x+) = −1/l ≤ f2(x+) and on the lower edge

x− = zi− δ we havef1(x−) = 1/l ≥ f2(x−). Hence, the interval must contain at

least one solution of (B.2). For eveni, the argument repeats with reversed signs.

We will now show that each of these intervals can contain at most one solution.

Consider the interval aroundzi for some eveni (the proof for oddi is practically

the same, and will not be repeated here). Lety = x − zi (so on the interval we

have−δ < y < δ), thenf1(x) = sin(my)/m, andf2(x) = sin(ly + φ)/l, where

φ = lzi mod 2π. Assume that there are two solutions in the interval, located at

x1 = zi+y1 andx2 = zi+y2, with −δ ≤ y1 < y2 ≤ δ. Sincef1−f2 = 0 at these

two points, there must existy∗ ∈ (y1, y2) satisfyingf ′1(zi+ y∗)− f ′

2(zi+ y∗) = 0.

Hence we can write

cos(ly∗ + φ) = cos(my∗) ⇒ ly∗ + φ = ±(my∗). (B.3)

We will now assume thaty∗ > 0 and reach contradiction by showing thatf1(x2) >

f2(x2), contrary to the assumption thatx2 is a solution. If we assume thaty∗ < 0,

an equivalent argument would show thatf1(x1) < f2(x1) reaching contradic-

tion again (this part will not be described here). Since0 < y∗ < δ, we have

sin(my∗) > 0, and

f1(x∗) =

sin(my∗)

m>

∣∣∣∣sin(ly∗ + φ)

l

∣∣∣∣ = |f2(x∗)|.

We distinguish two cases:

1. If f2(x∗) > 0, we have from equation (B.3)ly∗ + φ = my∗. For points

larger thanx∗, we havef ′1(x

∗ + d) = cos(my∗ + md), andf ′2(x

∗ + d) =

cos(my∗+ld). Sincecos is decreasing in this region and sincef ′2 gains phase

faster, we find that ford ≪ 1 f ′1(x

∗ + d) > f ′2(x

∗ + d). As we increased,

the differencef1−f2 (which is positive atx∗) only increases. The functions

cannot intersect as long as the derivative difference remains positive. The

first point afterx∗ wheref ′1 − f ′

2 = 0 again is atd = 2π/(l+m). However,

95

since for any0 < x < π/2 the inequalitysin−1 x < πx/2 holds, we have

y2 < δ =1

msin−1 m

l<π

2l<

2π

l +m,

sof ′1 − f ′

2 is still positive atx2, andf1(x2) > f2(x2) as stated.

2. If f2(x∗) < 0, equation (B.3) givesly∗ + φ = −my∗, f ′2(x

∗ + d) =

cos(my∗ − ld) andf2(x∗ + d) = sin(ld − my∗)/l. For d < my∗/l (i.e.

y∗ < x − zi < y∗1 ≡ y∗(l + m)/l), this is negative, and certainly smaller

thanf1(x∗+ d). In this region we also havef ′2(x) > cos(my∗) > f ′

1(x). At

x = zi + y∗1 the derivativef ′2 is 1, but it decreases faster thanf ′

1 until they

become equal atx∗2 = zi + y∗2, wherey∗2 ≡ y∗(l +m)/(l −m). At x∗2, we

have

f1(x∗2) =

1

msin

(my∗

l +m

l −m

)>

1

lsin

(my∗

l +m

l −m

)= f2(x

∗2).

Hence, sincef ′1− f ′

2 is negative in the range(x∗, x∗2) andf1(x∗2)− f2(x∗2) >

0 we get thatf1 > f2 throughout this region. Now, ifx2 < x∗2 we get

f1(x2) > f2(x2) as required. Ifx2 > x∗2 then we have found a new pointx∗2,

with positivef2, wheref ′1 = f ′

2. This brings us back to case 1, and we can

proceed as described there, withx∗2 taking the place ofx∗.

This establishes the uniqueness of solutions inside each interval. Next, we will

show that thesem − 1 solutions are all of odd multiplicity (which implies thatu

changes sign in these points, and therefore they are boundary intersections). Let

f(x) ≡ f2(x)− f1(x) and suppose thatf(x0) = 0 for some0 < x0 < π. If x0 is

degenerate, then

0 = f ′(x0) = cos(lx0)− cos(mx0) ⇒ (l ∓m)x0 = 2πN

for some integerN . Substituting this inf , we get

0 = f(x0) ⇒ sin(lx0) = ±msin(mx0)

m= ±m

lsin(lx0).

But since1 > m/l > 0, this can only happen if bothlx0 andmx0 are multiples of

96 APPENDIX B. COUNTING BI FOR THE TRIANGLE

π. It is easy to verify that in this casef ′′(x0) = 0 andf (3)(x0) = ±(l2−m2) 6= 0.

Thus, all roots off in the interval are of multiplicity 1 or 3.

For the range2a < s < a(2 +√2), we defineµ = l +m, ν = l−m, and use

(B.1) to write

ulm = 2π

a[−l cos(lz) sin(mz) +m cos(mz) sin(lz)]

=π

a[µ sin(νz)− ν sin(µz)], (B.4)

so, repeating the previous argument, the number of boundaryintersections in this

range isν − 1 = l −m− 1.

The total number of boundary intersections in the 3 intervals is 2(m − 1) +

(l −m− 1) = m + l − 3. However, we still need to check the three corners. At

all the cornersu = 0, butu is not smooth there, so we need to check whether or

not it changes sign at these points. Returning to equation (B.1) and expanding for

values ofs approaching0,a,2a anda(√2+2) from both sides, we get that ats = 0

the sign is positive for both sides, fors = 2a the sign is(−1)l+m−1 on both sides,

so these points are never BI. Fors = a, the sign is(−1)m−1 if approaching from

below and(−1)l if approaching from above. Hence it is a boundary intersection

if and only if l+m = 0 (mod 2). Combining this result with the count inside the

intervals, we get the final result, equation (3.16).

Appendix C

Computing the trace formula for

Cη(k) of the right isosceles triangle

The computations in section 3.3 involved an expansion forCη(k) to higher orders

in k than are commonly used in such cases. This was mainly due to the fact

that we wanted to directly observe the contribution of the “round off term”ηR

to equation (3.19). This term does not appear in separable systems, and we saw

that it gives rise to contributions from “semi-periodic” orbits, but these are of

order√k. To get a meaningful expansion to such order, the integrals involved

in the “bulk term”CB had to be approximated beyond the usual stationary phase

approximation. In this appendix we provide some notes regarding the derivation

of the expansion (3.21) and the approximations that have been used in the process.

Starting from equation (3.19), we splitC into three parts as described in sec-

tion 3.3.1. In what follows we focus on the computation of thebulk termCB.

First, we note that for a functionFl,m that is invariant under swapping the roles of

l andm and also does not depend on their signs we can write

∑

l>m≥1

Fl,m =1

8

[∑

l,m

Fl,m − 2∑

l

Fl,0 − 2∑

l

Fll + 3F00

]

=1

8

∑

l,m

Fl,m − 1

4

∑

l

Fl,0 −1

4

∑

l

Fl,l +3

8F00,

where in the summations on the right side of the equationl,m take all possible

97

98 APPENDIX C. COMPUTINGCη(K) FOR THE TRIANGLE

integer values. If we writeηB = |l| + |m| then the expression forCB can be

written in this form, which is more convenient for applying the PSF. The first

term in the PSF expansion reads

18

∑

M,N

∫∫ ∞

−∞dl dm (|l|+ |m|)Θ(k − k|l|,|m|)e

2πi(|l|M+|m|N). (C.1)

Again, we focus on the contribution of a single term in this sum. We shall also

assumeM 6= N and that they are both positive. DenoteM = R cosφ andN =

R sin φ. Changing coordinates from(l, m) to (r, θ), the integral can be written as

1

8

∫ k/π

0

dr r2∫ 2π

0

dθ (σc cos(θ + φ) + σs sin(θ + φ))ei2πRr cos(θ) (C.2)

whereσc is the sign ofcos(θ + φ) andσs is the sign ofsin(θ + φ). The internal

integral in this expression can be written as

2 cosφ

∫ π

0

dθ σs sin(θ) cos(q cos θ) + 2 sinφ

∫ π

0

dθ σs cos(θ) cos(q cos θ))

+swap,

whereq = 2πRr, and “swap” stands for two terms which are equivalent to the

first ones after swapping the roles ofM andN . Continuing with the non-swapped

terms, we divide the integration range into 4 parts:(0, φ), (φ, π2), (π

2, π− φ), (π−

φ, φ). By inspecting the signs in each part and noting the cancellations, we get

4 cosφ

∫ π/2

φ

dθ sin(θ) cos(q cos θ) + 4 sinφ

∫ φ

0

dθ cos(θ) cos(q cos θ)).

The first of these terms can be integrated, and we leave the second as is for now.

Substituting the results back in (C.2), we get:

cosφ

4πR

∫ k/π

0

dr r sin(2πrR cosφ)

+1

2sin φ

∫ φ

0

dx cosx

∫ k/π

0

dr r2 cos(2πRr cosx),

99

which after integrating outr becomes

1

16π3R2Msin(2kM)− k

8π3R2cos(2kM)

+N

16π3R2

∫ φ

0

dx[k2 sin(2kR cosx) +

k

R cos xcos(2kR cosx)

− 1

2R2 cos2 xsin(2kR cosx)

]. (C.3)

we see that there we are left with 3 definite integrals to approximate. One edge of

the integration is on the stationary point 0, but the other one is on the pointφ. The

SPA leads to the main,O(k3/2), oscillatory term of (3.21), but to get theO(k1/2)

corrections one needs a more accurate scheme. To approximate theses integral we

have used the method of steepest descent [55]. Since it is known that

∫ π/2−i∞

0

eik cos zdz =π

2H0(k),

and in this integral, the path could be chosen to pass throughφ, we can write

∫ φ

0

eik cos zdz =π

2H0(k)−

∫ π/2−i∞

φ

eik cos zdz.

A path of constant phase was chosen fromφ to ǫ − i∞, transforming the inte-

gral into Laplace form (which allows asymptotic approximation to any required

order, via Watson’s lemma). The resulting series was subtracted from the known

asymptotic expansion forH0 [35], yielding the required result

∫ φ

0

eik cos zdz =

√π

2ei(k−

π4)k−1/2

+ei(k cosφ+π2)(k sinφ)−1 +

1

8

√π

2ei(k−

3π4)k−3/2

+1

tanφei(k cosφ+π)(k sin φ)−2.

The imaginary part of this allows appropriate approximation for the first integral

100 APPENDIX C. COMPUTINGCη(K) FOR THE TRIANGLE

of (C.3). The other two integrals

∫ φ

0

dxcos(k cos(x))

cosx, and

∫ φ

0

dxsin(k cos(x))

cos2 x

can approximated in a similar way.

Appendix D

Semi-classical expansion for the

boundary Green function

In this appendix, we explain some details which are relevantfor the derivation

of the boundary Green functiong and its expansion as a sum of orbits, equa-

tion (4.17). In particular, we discuss the operatorh of equation (4.9) and the

proper way to define its operation on hypersingular functions (such asg0 and

g). The work leading to this extension, as described in section D.1 was done in

collaboration with Klauss Hornberger at MPIPKS Dresden. Wethen move on to

approximatehg0 and show the connections between the diagonal and off-diagonal

partsg(N)0 andg(F )

0 . These connections enable the transformation of the hypersin-

gular expansion equation (4.15) into the regular form (4.17).

D.1 The integral equation

We begin with a derivation of the integral equation (4.9). This derivation closely

follows the derivation in [33]. However, the limiting process involved is discussed

here explicitly, in order to derive the proper extension ofh to the singular case.

The Dirichlet Green function is given by

G(r, r′; k) =

∞∑

n=1

ψn(r)ψn(r′)

kn2 − k2

. (D.1)

101

102 APPENDIX D. EXPANSION OF THE BOUNDARY GREEN FUNCTION

This is defined as 0 for any point wherer or r′ is outside ofΩ. It is continuous on

the boundary, but has a jump in the first derivative. For the case where bothr and

r′ are inΩ \ ∂Ω, we have(−∆− k2)G(r, r′) = δ2(r − r′). From this definition,

we see that

g(s, s′; k) =1

k2∂n′∂nG

(r(−)(s), r(−)(s′); k

),

where the notationr(−) is used to specify that the derivatives should be taken on

the insideof the boundary, i.e.f(r(−)(s)) ≡ limǫ→0+ f(r(s)− ǫn(s)). Similarly,

G0 is the free Green function as in section 4.2.1, satisfying(−∆−k2)G0(r, r′) =

δ2(r − r′) for r, r′ ∈ R2, and

g0(s, s′; k) =

2

k2∂n′∂nG0(r(s), r(s

′); k)

(in this case, since the derivatives ofG0 are continuous on∂Ω, the limit specifica-

tion is irrelevant—r could be replaced withr(+) orr(−), yielding the same result).

For each of these Green functions, we introduce a “mixed boundary-derivative”

function, defined fors ∈ ∂Ω andr′ ∈ Ω \ ∂Ω

µ(s, r′) = (∂(+)n + ∂(−)

n )G(r(s), r′) = ∂nG(r(−)(s), r′)

µ0(s, r′) = (∂(+)

n + ∂(−)n )G0(r(s), r

′) = 2∂nG0(r(−)(s), r′),

where∂(±)n f(r(s)) ≡ limǫ→0+n·f(r(s)±ǫn(s)). Note thath(s, s′) = µ0(s, r(s

′)),

andg(s, s′) = ∂n′µ(s, r(−)(s′))/k2.

We now fix an arbitraryr′ ∈ Ω \ ∂Ω, and consider the functionFr′(r) ≡G(r, r′)−G0(r, r

′). It is easy to verify that it is continuous at∂Ω, and that for all

r /∈ ∂Ω it satisfies the Helmholtz equation(−∆− k2)Fr′(r) = 0. From potential

theory [56], it follows1 thatF can be expressed as a single layer potential, with

some “surface charge distribution”ϕ(s)

Fr′(r) =

∮G0(r, r(s1))ϕr′(s1) ds1. (D.2)

1The discussion in [56] considers solutions of Laplace’s equation. However, the relevantderivations and results can be generalized to Helmholtz solutions, with only minor modifications.

D.1. THE INTEGRAL EQUATION 103

The surface distribution can be found using the single layerjump condition:

ϕr′(s1) = (∂(−)n − ∂(+)

n )Fr′(r(s1))

= (∂(−)n − ∂(+)

n )G(r1, r′) = µ(s1, r

′).

On the other hand, applying the mixed partial derivative(∂(−)n + ∂

(+)n ) (atr(s)) to

both sides of equation (D.2), we get

µ− µ0 = hµ =

∮

∂Ω

h(s, s1)µ(s1, r′) ds1

for all r′ ∈ Ω\∂Ω. Finally, to get (4.9), we apply∂(−)n′ on each side of this

equation. The conclusion from this discussion is that in order to get the correct

form of the integral operator, we need to putr′, the starting point of the path,

inside the billiard and slightly off the boundary:r′(−ǫ) = r(s′)− ǫn(s′), then take

the limit ǫ → 0 at thefinal stage. We now need to take a closer look at this limit,

for the specific case whereh is applied toµ0(s1, r(−)(s′)).

To simplify the discussion and enable focusing on explicit integrals, the dis-

cussion in the remainder of this appendix shall be limited tothe specific case of

the circle billiard (the general case does not differ in a significant way). In the case

of the circle (of radiusa), and using the notation depicted in figure 4.4, we always

haveψi+1 = ψi =∣∣ si+1−si

2a

∣∣, anddi = 2a sin∣∣ si+1−si

2a

∣∣. For classical paths (which

satisfy the SPA conditionψi = ψi) all angles are equal. The operators defined in

equations (4.11)–(4.12) become (withψ(s, s′) ≡∣∣ s−s′

2a

∣∣):

h(s, s′; k) = −ik2sinψH+

1 (2ka sinψ) (D.3)

g0(s, s′; k) =

i

4

[H+

2 (2ka sinψ) + (1− 2 sin2 ψ)H+0 (2ka sinψ)

]

=i

2

[H+

1 (2ka sinψ)

2ka sinψ− sin2 ψ ·H+

0 (2ka sinψ)

](D.4)

To make use of these equations in the limiting process described above, we

must chooser′ − ǫn′ as the starting point (instead ofr′), and express the relevant


angles and lengths in terms ofǫ. After some trigonometric manipulations we get

d2 = 4 sin2s− s′

2a(1− ǫ) + ǫ2

cos(ψ + ψ′) = 1− 2 sin2 s− s′

2a

cos(ψ − ψ′) =sin2(s−s′2a

)(1− ǫ+ 1

2ǫ2)− 1

4ǫ2

sin2(s−s′2a

)(1− ǫ) + 1

4ǫ2

. (D.5)

This should be inserted into the expression forg0. Denotingt(s1) = sin s1−s′2a

, we

get

hg(−ǫ)0 =

k

8

∫ πa

−πads1 sin(

s− s12

)H+1

(2ka sin

s− s12

)(D.6)

·[t2(1− ǫ+ 1

2ǫ2)− 1

4ǫ2

t2(1− ǫ) + 14ǫ2

H+2 (kd

′)

+ (1− 2t2) ·H+0 (kd

′)

],

with d′ = 2a√t2(1− ǫ) + ǫ2/4. We see that for any finiteǫ there is no singularity

at t = 0.

Chooseδ small enough so that2kaδ ≪ 1. Assumingǫ ≪ δ, split the inte-

gration rangeΓ = [−πa, πa] into two parts:Γout = [−πa,−2aδ] ∪ [2aδ, πa], and

Γin = [−2aδ, 2aδ].

For the outer regionΓout, |t| & δ ≫ ǫ, so ǫ can be safely neglected in the

integrand:∫Γout

h(s, s1)g(−ǫ)0 (s1, s

′)ds1 ∼∫Γout

h(s, s1)g0(s1, s′)ds1.

ForΓin, kd′ . 2kaδ ≪ 1 and we can use the small argument approximation

D.2. COMPUTATION OFG1 105

for the Hankel functions. Changing the integration variable tot, we get

∫

Γin

h(s, s1)g(−ǫ)0 (s1, s

′)ds1

∼ q

8SH+

1 (qS)(−i)4

πq2

∫ δ

−δdtt2(1− ǫ+ 1

2ǫ2)− 1

4ǫ2

(t2(1− ǫ) + 1

4ǫ2)2

=−i

πqSH+

1 (qS)1

1 − ǫ

[ǫ

2√1− ǫ

tan−1

(2√1− ǫ

ǫδ

)− (1− ǫ

2)2δ

ǫ2

4+ δ2(1− ǫ)

]

∼ i

πq

SH+1 (qS)δ

,

where we have introduced the notationsq = 2ka andS = sin s−s′2a

. Finally, we

get

limǫ→0

hg(−ǫ)0 = lim

δ→0

[∫

Γout(δ)

h(s, s1)g0(s1, s′)ds1 +

i

πq

SH+1 (qS)δ

],

which according to equation (4.14) is exactly=∫Γh(s, s1)g0(s1, s

′)ds1.

D.2 Computation of g1

As we have seen in section 4.2, the specific form of the estimates forhg(N)0 = g1

allowed us to transform equation (4.15) into a form which is non-singular and

easier to approximate (equation (4.17)). We will show the needed results for the

case of the circle. Assuming, without lost of generality,s′ = 0 anda = 1, the

relevant integral is written as

(hg0)(s, 0) =k

4=

∫ π

−πds1 sin(

s− s12

)H+1

(2k sin

s− s12

)(D.7)

·[H+

1 (2k sin(12s1))

2k sin(12s1)

− sin2(1

2s1) ·H+

0 (2k sin(1

2s1))

].

We change the integration variable froms1 to x = 2ka sin s1−s′2a

, and also intro-

duce the following notations:q = 2ka, S = sin s−s′2a

, C =√1− S2 = cos s−s

′

2a,


y = S√q2 − x2 − Cx. With these, the integral becomes

hg0 =1

8=

∫ q

−q

2dx√q2 − x2

yH+1 (y)

[H+

1 (x)

x−(x

q

)2

H+0 (x)

]. (D.8)

In the diagonal strip|s1| < xC/k, we have| s12a| < xC

q≪ 1. Hencex ∼

q| s12a| < xC, so for computingg1 = hg

(N)0 , the limits of integration in (D.8) should

be changed from±q to ±xC . In this range we can use the fact that|x| < xC ≪ q

and get, to highest order inxC/q

g(N)1 ∼ 1

4q=

∫ xC

−xCdx

y

xH+

1 (y)H+1 (x) ∼

1

4q=

∫ ∞

−∞dx

y

xH+

1 (y)H+1 (x) (D.9)

(we have usedxC ≫ 1), andy ∼ qS − Cx.

For further estimation we need to Analise thex dependence ofy. As men-

tioned in section 4.2, we splitg1 into two parts: the “off diagonal part”g(F )1 , sup-

ported on|s− s′| > xC/k, and the “diagonal part”,g(N)1 supported on|s− s′| ≤

xC/k.

The off-diagonal part

if |s− s′| > xC/k, we haveqS > q sin(xCq) ∼ xC ≫ 1. Choose a smaller cutoff

1 ≪ xC1≪ xC, and use it to replace infinity in the integration range of (D.9).

Now for all x in this range, we haveCx ≤ xC1≪ xC ∼ qS. Using these to

approximateyH1(y), we get

yH+1 (y) ∼ qSH+

1 (qS)e−iCx ∼√

2qSπ

ei(qS−3

4π)e−iCx,

and

g(F )1 ∼

√S8πq

ei(qS−3

4π)=

∫ ∞

−∞dx e−iCxH

+1 (x)

x. (D.10)

Now we can use the definition (4.13) to express the integral ina regular form:

=

∫ ∞

−∞dx e−iCxH

+1 (x)

x=

∫ ∞

0

dx

[cos(Cx)

(H+

0 (x) +H+2 (x)

)+

4i

πx2

]. (D.11)

D.2. COMPUTATION OFG1 107

The real part of this integral gives

∫ ∞

0

dx cos(Cx) (J0(x) + J2(x)) =1− (2C2 − 1)√

1− C2= 2S.

For the imaginary part, we will use

∫ ∞

0

cos(Cx)Y0(x)dx =

∫ ∞

0

sin(Cx)Y1(x)dx = 0. (D.12)

Inserting the following identity:

Y2(x) = Y0(x)− 2d

dx(Y1(x) +

2

πx)− 4

πx2

into the imaginary part of (D.11), integrating by parts and using (D.12), we get

∫ ∞

0

dx

[cos(Cx) (Y0(x) + Y2(x)) +

4

πx2

]

= −2C∫ ∞

0

sin(Cx) 2

πxdx+

∫ ∞

0

4

πx2(1− cos(Cx)) dx

= −2C +4Cπ

∫ ∞

0

sin2 x

x2dx = 0.

Finally inserting this into (D.10), we get

g(F )1 ∼

√S3

2πqei(qS−

3

4π) ∼ g

(F )0 (s, s′; xC, k).

The diagonal part

For g(N)1 we will only calculate the behaviour near the diagonal|s − s′| ≪ 1/k.

Sinceg(N)1 must merge smoothly withg(F )

1 when|s−s′| approachesxC, this should

suffice to show thatg(N)1 is bounded.

If |s− s′| ≪ 1/k, we haveqS ≪ 1, andCx ∼ x becomes the major contribu-


tion toy for most of the integration range.

g(N)1 ∼ 1

4q=

∫ ∞

−∞dx

|x− qS|x

H+1 (|x− qS|)H+

1 (x)

=1

4q

∫ ∞

−∞dx

( |x− qS|x

H+1 (|x− qS|)H+

1 (x) + qSH+1 (qS)

2i

πx2

).

Based on a result from appendix B of [33], the last integral evaluates to2ei(qS−π2),

so the result is (for|s− s′| ≪ 1/k):

g(N)1 ∼ − i

2qeiqS .

Comparing to the result of section D.2, we see that this is of lower order ink

( 1k= o( 1√

k)).

List of Abbreviations and Symbols

Abbreviations

BI Boundary intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5

BIC Boundary intersections count . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .5

BGS The Bohigas Giannoni and Schmit conjecture . . . . . . . . . . . . . .. . . . . . . . . . . 2

BRW Boundary adjusted random waves . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 13

CRW Curved boundary adjusted random waves . . . . . . . . . . . . . . . . . .. . . . . . . . . 14

GPT Generalized periodic tori . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 32

NI Nodal intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .5

NRW Normally derived random waves . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 14

PPO Primitive periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 73

PSF Poisson summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .29

PT Periodic tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 31

RMT Random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 2

RW Random waves (Berry’s random wave model) . . . . . . . . . . . . . . . .. . . . . . . . .4

SRF Short range field (Gaussian model from [26]) . . . . . . . . . . . . . .. . . . . . . . . . 13

SPA Stationary phase approximation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 68

SPO Special periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 38

Symbols—Latin

A Area of the billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2

b(s) Local density of boundary intersections . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 9

c0–c2 Elements of the single point covariance matrix of (u,u) . . . . . . . . . . . . . . . 15

109

110 ABBREVIATIONS AND SYMBOLS

Cη Accumulated BIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 47

d Displacement vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 63

d Spectral density (of states) . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 3

dη Density of BIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 29

dσX Smoothed spectral density ofX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

f Numerically computed oscillating part of a trace formula . .. . . . . . . . . . . 41

g Boundary Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 62

g0 Free boundary Green function . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 63

g Theoretical estimate for a (truncated and smoothed) oscillating part . . . . 41

G0 Free 2D Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 63

h Kernel of the boundary integral equation . . . . . . . . . . . . . . . .. . . . . . . . . . . . 63

h Integral operator with kernelh(s, s′) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

H The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 32

Ii Action corresponding to coordinatei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

k Wavenumber, square root of the eigenvalue . . . . . . . . . . . . . . .. . . . . . . . . . . . 2

kn Thenth wavenumber (ordered by value) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .2

Lγ Length of a classical orbitγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

L Length of the billiard’s boundary . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2

l, m Quantum numbers (of an integrable 2D system) . . . . . . . . . . . . .. . . . . . . . . 29

M,N Indices enumerating periodic orbits . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 31

Mγ Monodromy matrix of periodic orbitγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

n(s) Outwards pointing normal to∂Ω atr(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

pi Canonical momentum of coordinatei ∈ 1, 2 . . . . . . . . . . . . . . . . . . . . . . . . . 31

qi Canonical coordinate,i ∈ 1, 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

q(n) Estimate for thenth wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

r Number of repetitions of a primitive orbit . . . . . . . . . . . . . . .. . . . . . . . . . . . 31

r Point (inR2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 2

r(s) Natural (arc-length) parametrization of a curve . . . . . . . . .. . . . . . . . . . . . . . . 5

R Normalized correlation function . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 18

s Natural (arc-length) parameter . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5

t Direction of the tangent to∂Ω atr(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

T Tangent (stability) matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 69

Tγ The period of orbitγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

111

un(s) Thenth boundary function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 5

W∆ Spectral window (cutoff) function, with width∆ . . . . . . . . . . . . . . . . . . . . . 41

Symbols—Greek

αi Maslov index for coordinatei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

γ Index enumerating (periodic) orbits or tori . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3

Γ A partial curve to∂Ω, or general curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

δ(x) The Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 3

η Total number of boundary intersections . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4

Θ(x) The Heaviside step fuction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 32

κ(s) Curvature at the pointr(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

µ, ν Co-prime indices, enumerating primitive orbits . . . . . . . . .. . . . . . . . . . . . . 31

νn Number of nodal domains of thenth eigenfunction . . . . . . . . . . . . . . . . . . . . 4

ρσ Smoothing kernel of widthσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

σ Width of a spectral interval or smoothing kernel . . . . . . . . . .. . . . . . . . . . . .38

ψ Wavefunction, Helmholz eigenfunction . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 2

ψn Thenth eigenfunction, by increasing energy . . . . . . . . . . . . . . . . .. . . . . . . . . 2

Ω The domain of the billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 1

∂Ω The boundary of the billiard . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .1

112 ABBREVIATIONS AND SYMBOLS

Bibliography

List of related publications

[Rel1] A. Aronovitch, U. Smilansky, The statistics of the points where nodal lines

intersect a reference curve, Journal of Physics A Mathematical General 40

(2007) 9743–9770.

[Rel2] A. Aronovitch, U. Smilansky, Trace formulae for counting

nodal domains at the boundaries of 2D quantum billiards,

PreprintarXiv:arXiv:1006.5656.

Bibliography

[1] O. Bohigas, M. J. Giannoni, C. Schmit, Characterizationof chaotic quan-

tum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1)

(1984) 1–4.

[2] E. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian sys-

tems: scars of periodic orbits, Physical Review Letters 53 (16) (1984) 1515–

1518.

[3] H. Weyl, The laws of asymptotic distribution of the eigenvalues of linear par-

tial differential equations, Math. Ann 71 (1912) 441–479.

[4] C. Gordon, D. Webb, S. Wolpert, One cannot hear the shape of a drum, Bul-

letin of the American Mathematical Society 27 (1992) 134–138.

113

http://arxiv.org/abs/arXiv:1006.5656

114 BIBLIOGRAPHY

[5] S. Zelditch, The inverse spectral problem, Surveys in Differential Geometry

IX (2004) 401–467.

[6] M. Berry, M. Tabor, Level clustering in the regular spectrum, Proceedings of

the Royal Society of London. Series A, Mathematical and Physical Sciences

356 (1686) (1977) 375–394.

[7] E. Wigner, Random matrices in physics, Siam Review 9 (1) (1967) 1–23.

[8] M. Berry, M. Robnik, Semiclassical level spacings when regular and chaotic

orbits coexist, Journal of Physics A: Mathematical and General 17 (1984)

2413.

[9] R. Balian, C. Bloch, Distribution of eigenfrequencies for the wave equation in

a finite domain. I. Three-dimensional problem with smooth boundary surface,

Ann. Physics 60 (1970) 401–447.

[10] R. Balian, C. Bloch, Distribution of eigenfrequenciesfor the wave equation

in a finite domain. III. Eigenfrequency density oscillations, Ann. Physics 69

(1972) 76–160.

[11] M. V. Berry, M. Tabor, Closed orbits and the regular bound spectrum, Pro-

ceedings of the Royal Society of London. Series A, Mathematical and Physi-

cal Sciences (1934-1990) 349 (1976) 101–123, 10.1098/rspa.1976.0062.

[12] M. C. Gutzwiller, Periodic orbits and classical quantization conditions, Jour-

nal of Mathematical Physics 12 (3) (1971) 343–358.

[13] M. V. Berry, Semiclassical theory of spectral rigidity, Proceedings of the

Royal Society of London. Series A, Mathematical and Physical Sciences

(1934-1990) 400 (1985) 229–251, 10.1098/rspa.1985.0078.

[14] E. B. Bogomolny, J. P. Keating, Gutzwiller’s Trace Formula and Spectral

Statistics: Beyond the Diagonal Approximation, Physical Review Letters 77

(1996) 1472–1475.

[15] M. Sieber, K. Richter, Correlations between Periodic Orbits and their Role

in Spectral Statistics, Physica Scripta Volume T 90 (2001) 128–133.

BIBLIOGRAPHY 115

[16] Nodal patterns in physics and mathematics, The European Physical Journal

- Special Topics 145, 10.1140/epjst/e2007-00142-7.

[17] A. Pleijel, Remarks on courant’s nodal line theorem, Communications on

Pure and Applied Mathematics 9 (3) (1956) 543–550.

[18] G. Blum, S. Gnutzmann, U. Smilansky, Nodal domains statistics: A criterion

for quantum chaos, Physical Review Letters 88 (11) (2002) 114101.

[19] S. Gnutzmann, U. Smilansky, N. Sondergaard, Resolvingisospectral

’drums’ by counting nodal domains, Journal of Physics A Mathematical Gen-

eral 38 (2005) 8921–8933.

[20] R. Band, T. Shapira, U. Smilansky, Nodal domains on isospectral quantum

graphs: the resolution of isospectrality?, Journal of Physics A Mathematical

General 39 (2006) 13999–14014.

[21] P. D. Karageorge, U. Smilansky, Counting nodal domainson surfaces of rev-

olution, Journal of Physics A Mathematical General 41 (20) (2008) 205102–

+.

[22] D. Klawonn, Inverse nodal problems, Journal of PhysicsA: Mathematical

and Theoretical 42 (2009) 175209.

[23] M. V. Berry, Regular and irregular semiclassical wave functions, Journal of

Physics A Mathematical General 10 (1977) 2083–2091.

[24] E. Bogomolny, C. Schmit, Percolation Model for Nodal Domains of Chaotic

Wave Functions, Physical Review Letters 88 (11) (2002) 114102.

[25] E. Bogomolny, R. Dubertrand, C. Schmit, SLE description of the nodal lines

of random wavefunctions, Journal of Physics A: Mathematical and Theoreti-

cal 40 (3) (2007) 381.

[26] G. Foltin, S. Gnutzmann, U. Smilansky, The morphology of nodal lines ran-

dom waves versus percolation, Journal of Physics A Mathematical General

37 (2004) 11363–11371.

116 BIBLIOGRAPHY

[27] A. Aronovitch, U. Smilansky, The statistics of the points where nodal lines

intersect a reference curve, Journal of Physics A Mathematical General 40

(2007) 9743–9770.

[28] J. A. Toth, S. Zelditch, Counting nodal lines which touch the boundary of an

analytic domain, J. Differential Geom. 81 (3) (2009) 649–686.

[29] I. Polterovich, Pleijel’s nodal domain theorem for free membranes, Proc.

Amer. Math. Soc. 137 (3) (2009) 1021–1024.

[30] J. A. Toth, I. Wigman, Counting open nodal lines of random waves on planar

domains, PreprintarXiv:arXiv:0810.1276.

[31] S. Gnutzmann, P. D. Karageorge, U. Smilansky, Can one count the shape of

a drum?, Physical Review Letters 97 (9) (2006) 090201.

[32] S. O. Rice, Mathematical Analysis of Random Noise, BellSystems Tech. J.,

Volume 23, p. 282-332 23 (1944) 282–332.

S. O. Rice, Mathematical Analysis of Random Noise-Conclusion, Bell Sys-

tems Tech. J., Volume 24, p. 46-156 24 (1945) 46–156.

[33] A. Backer, S. Furstberger, R. Schubert, F. Steiner, Behaviour of boundary

functions for quantum billiards, Journal of Physics A Mathematical General

35 (2002) 10293–10310.

[34] Y. Elon, S. Gnutzmann, C. Joas, U. Smilansky, Geometriccharacterization

of nodal domains: the area-to-perimeter ratio, Journal of Physics A: Mathe-

matical and Theoretical 40 (11) (2007) 2689.

[35] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables, Dover Publications, New York,

1964.

[36] A. G. Monastra, U. Smilansky, S. Gnutzmann, Avoided intersections of

nodal lines , Journal of Physics A Mathematical General 36 (2003) 1845–

1853.

http://arxiv.org/abs/arXiv:0810.1276

BIBLIOGRAPHY 117

[37] M. V. Berry, Statistics of nodal lines and points in chaotic quantum billiards:

perimeter corrections, fluctuations, curvature, Journal of Physics A Mathe-

matical General 35 (2002) 3025–3038.

[38] C. T. Wheeler, Curved boundary corrections to nodal line statistics in chaotic

billiards, Journal of Physics A Mathematical General 38 (2005) 1491–1504.

[39] M. S. Longuet-Higgins, The statistical analysis of a random, moving sur-

face, Philosophical Transactions of the Royal Society of London. Series A,

Mathematical and Physical Sciences 249 (966) (1957) 321–387.

[40] M. Kac, On the distribution of values of trigonometric sums with linearly

independent frequencies, Amer. J. Math. 65 (1943) 609–615.

[41] M. L. Mehta, Random Matrices, Second edition, AcademicPress, Inc.,

Boston, MA, 1991.

[42] E. B. Bogomolny, U. Gerland, C. Schmit, Models of intermediate spectral

statistics, Phys. Rev. E 59 (2) (1999) R1315–R1318.

[43] O. Bohigas, Random matrix theories and chaotic dynamics, in: M. Giannoni,

A. Voros, J. Zinn-Justin (Eds.), Chaos et physique quantique, no. LII (1989)

in Les Houches, North-Holland, 1991, pp. 87–199.

[44] I. Percival, Semiclassical theory of bound states, Adv. Chem. Phys 36 (1).

[45] H.-J. Stockmann, Quantum Chaos: An Introduction, Cambridge University

Press, Cambridge, UK, 1999.

[46] D. Klawonn, R. Band, A. Aronovitch, Counting nodal domains in the right

isosceles triangle, Work in progress.

[47] M. V. Berry, M. Wilkinson, Diabolical Points in the Spectra of Triangles,

Royal Society of London Proceedings Series A 392 (1984) 15–43.

[48] F. James, Statistical methods in experimental physics, World Scientific Pub

Co Inc, 2006, Ch. 11, pp. 313–318.

118 BIBLIOGRAPHY

[49] J. Hadamard, Lectures on Cauchy’s problem in linear partial differential

equations, Vol. III, Dover Publications, New York, 1952, Ch. 1, pp. 133–158.

[50] J. de Klerk, Hypersingular integral equations–past, present, future, Nonlin-

ear Analysis 63 (5-7) (2005) e533 – e540.

[51] U. Smilansky, Semiclassical quantization of chaotic billiards: A scattering

approach, in: E. Akkermans, G. Montambaux, J.-L. Pichard, J. Zinn-Justin

(Eds.), Mesoscopic Quantum Physics, no. LXI (1994) in Les Houches, North-

Holland, 1996, pp. 377–433.

[52] M. V. Berry, M. Robnik, Statistics of energy levels without time-reversal

symmetry: Aharonov-Bohm chaotic billiards , Journal of Physics A Mathe-

matical General 19 (1986) 649–668.

[53] H. Primack, H. Schanz, U. Smilansky, I. Ussishkin, Penumbra diffraction in

the quantization of dispersing billiards, Phys. Rev. Lett.76 (10) (1996) 1615–

1618.

[54] J. B. Keller, Corrected bohr-sommerfeld quantum conditions for nonsepara-

ble systems, Annals of Physics 4 (2) (1958) 180 – 188.

[55] P. Miller, Applied asymptotic analysis, Amer Mathematical Society, 2006,

Ch. 6.

[56] R. Kress, Linear integral equations, Applied mathematical sciences,

Springer-Verlag, 1989, Ch. 6, pp. 57–81.

Author’s Declaration

I declare that the work in this thesis was carried out in accordance with the regu-

lations of the Weizmann Institute of Science. The work is original except where

indicated by special reference in the text and no part of the dissertation has been

submitted for any other degree.

Amit AronovitchDecember 2010

119

ae b ˘ e af e c doctor of philosophy ˇb ba › home › feamit › pdfs › thesis.pdf · i thank...

Documents