Forward-smooth high-order uniformAharonov–Bohm asymptotics

M V Berry

H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

E-mail: asymptotico@bristol.ac.uk

Received 1 March 2016, revised 20 May 2016Accepted for publication 31 May 2016Published 24 June 2016

AbstractThe Aharonov–Bohm (AB) function, describing a plane wave scattered by aflux line, is expanded asymptotically in a Fresnel-integral based series whoseterms are smooth in the forward direction and uniformly valid in angle andflux. Successive approximations are valid for large distance r from the flux (orshort wavelength) but are accurate even within one wavelength of it. Coeffi-cients of all the terms are exhibited explicitly for the forward direction,enabling the high-order asymptotics to be understood in detail. The series isfactorally divergent, with optimal truncation error exponentially small in r.Systematic resummation gives further exponential improvement. Terms of theseries satisfy a resurgence relation: the high orders are related to the loworders. Discontinuities in the backward direction get smaller order by order,with systematic cancellation by successive terms. The relation to an earlierscheme based on the Cornu spiral is discussed.

Keywords: quantum, scattering, diffraction, uniform approximation

(Some figures may appear in colour only in the online journal)

1. Introduction

In the more than half-century since its derivation by Aharonov and Bohm [1], their wave-function, describing the scattering by a line of magnetic flux of a plane wave representingcharged quantum particles (e.g. electrons), has been much studied. Its physical significance,as a quantum effect on electrons moving in regions where there is a vector potential butno magnetic field, has provoked continuing discussion and controversy [2–5], and theAharonov–Bohm (AB) effect is an important physical example of a nonintegrable geometricphase [6, 7]. In addition, the mathematics underlying the AB wave has developed in severaldirections. It has a ‘many-whirls’ interpretation in terms of paths winding multiply round the

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 49 (2016) 305204 (16pp) doi:10.1088/1751-8113/49/30/305204

1751-8113/16/305204+16$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1

flux line [8–10]. It possesses phase singularities on the flux line, whose effect has beenobserved in an analogue experiment with water waves [11]. It has been generalised to includepenetration of the electrons into the region where the flux is concentrated [3].

Moreover, and importantly for the present paper, the AB wave has been approximated interms of Fresnel integrals (complex error functions) [3, 12–15], originating in an exactrepresentation of the AB wave for half-integer flux [1]. Such approximations are physicallyintuitive because there is a gauge in which the magnetic flux line is the edge of a phase-changing half-plane of vector potential, inspiring analogies with edge diffraction and theCornu spiral. The analogy leads to asymptotic formulas [14] for the AB wave far from theflux, the most recent of which [16] is extraordinarily accurate.

My aim here is to pursue the connection with the Fresnel integral, by showing how anapproximation scheme based on it can describe the mathematical structure of the AB wave indetail and with high accuracy. Several previous AB wave approximations are discontinuousin the forward direction, in the first [3, 14], or second [16] derivative. This is a disadvantagebecause it is in the forward direction that the AB wave possesses the distinctive and importantfeature that the scattered wave cannot be separated from the incident wave.

Here I derive (section 2) a systematic (large-distance, i.e. short-wavelength, or semi-classical) approximation scheme for which the AB wave is smooth (all derivatives con-tinuous) in the forward direction. The leading approximation involves a Fresnel integral, andthe correction terms are decreasing functions of distance from the flux. The scheme is uni-formly valid in direction (scattering angle), from forwards to backwards, and also in fluxstrength. Although it is a long-distance asymptotic expansion it represents the AB waveaccurately even within one wavelength of the flux line (see figures 2 and 3).

In the forward direction, all the terms of the approximation can be exhibited explicitly(section 3). This shows that this representation of AB is a factorially divergent series that canbe understood using familiar concepts from modern asymptotics (section 4), includingresummation giving further exponential accuracy beyond the least term, and a resurgencerelation for the coefficients. Although the scheme is forward-smooth, its terms are dis-continuous in the backward direction (section 5), with the discontinuities exactly calculableand cancelling in successive orders.

The scheme described here supersedes the earlier approximation based on fitting the ABwave to the Cornu spiral; the relation beween the two approaches is explored in the appendix.And although the representation is mathematically complete as a uniform asymptotic repre-sentation of AB, it does raise several questions, discussed in the concluding section 6.

A note on dimensions. Distances from the flux line will be represented in terms of thedimensionless variable r, measuring physical distances R in units of wavelength/2π. Withwavenumber k=2π/wavelength and introducing the particle mass m, energy E and Planckconstant �, r is defined as

( )�

= =r kRmE

R2

. 1.1

Magnetic flux Φ will be also be represented dimensionlessly, in quantum units for particleswith charge q, as

( )�

ap

=FF

=Fq

21.2

0

(Φ0 is different from the superconducting quantum flux unit for electrons, in which q=2e).

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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2. Uniform asymptotics

The exact formula derived in the original AB paper [1] described scattering of a plane waveincident from x=+∞ by a line of magnetic flux with quantum strength α. Here it will beconvenient to consider the plane wave incident from x=−∞, i.e. with θ=0 representingforwards in polar coordinates. Then the formula is the angular-momentum sum involvingBessel functions

( ) ( ) ( ( )) ( )

( { }) ( )

| | | |åy a q p

q q

= - +

= = =

aa

=-¥

¥-

-r

r

m J r

x r y r

, i exp i

cos , sin . 2.1m

mmAB

This is a smooth periodic function of θ and α. It is not difficult to approximate ψAB for smallr, by using the convergent series for the Bessel functions; that was the procedure used tounderstand the phase singularities on the flux line [11].

Here we are interested in large r. In the sum (2.1), the terms for |m|<r are comparable inmagnitude, after which (i.e. when the order of the Bessel functions exceeds its argument) theydecay rapidly. An expression suitable for deriving effective asymptotics is obtained byexpressing the Bessel functions as a standard contour integral (e.g. (8.412.6) in [17]) andevaluating the resulting trigonometric sum over m. This leads, after a sideways shift ofcontour and a little algebra, to a variant of formula (A.3) in [11], valid for 0�α�1 (whichrepresents no loss of generality because the AB wave is periodic in α):

( ) ( ) ( ) ( ) (( ) ) ( )òy ap

q a aq

= -- -+

r u r uu uu

,i

2d exp i cos

exp i sin sin 1cos cos

, 2.2ABC

in which C is the contour in figure 1(a).In the exponent, rcosu has relevant saddle points at u=−π and u=0, and there are

relevant poles at u=−π±|θ|. Reflecting this, it is convenient to deform and split thecontour into three as illustrated in figure 1(b). Thus

( )y y y y= + + . 2.3AB C1 C2 C3

C2 captures the pole at u=−π+|θ|, and its contribution is

( ) [ ( ( ))] ( )y q a q a q p q= + -r r, , exp i cos sgn . 2.4C2

Figure 1. Integration contours in (2.2) and (2.3).

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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The integrand corresponding to C3 is odd, so ψC3=0. For the remaining integral, C1 runsfrom the valley at u=−3π/2+i∞, through the saddle at u=−π, and down into the valleyat u=−π/2−i∞. As will become clear later, the discontinuity represented by sgnθ in (2.4)will be cancelled by the integral involving C1.

We seek a systematic approximation scheme for ψC1 for r? 1 that is uniformly valid for0�α�1 and all |θ|�π, including the forward direction where the two poles coincide withthe saddle at u=−π. This requires a simple variant of the standard ‘saddle-pole’ asymptotics(see e.g. [18]), involving two poles rather than one (it is possible to separate the poles usingpartial fractions, but the variant is simpler). Transform the C1 integral by mapping the variablefrom u to X as follows:

( )p= - + = - + =⎛⎝⎜

⎞⎠⎟u X u

XX ucos 1 , i.e. 2arcsin

2, 2 cos

12

. 2.52

Thus the Jacobian and denominator in (2.2) are

( )

q

q

= - =-

+ = -

=

uX u X

u X X

X

dd

2

sin

2

2, cos cos ,

where 2 sin 2.6

12

22

p2

p12

and the correspondences between saddles and poles in the map are

| | ( )pp q

= - = = == - =

u X u Xu X X

saddles: 0, 0 2 ,poles: . 2.7p

The map (2.5) is smooth for - < <X2 2 . The effect of the singularities (infiniteJacobian) at =X 2 will emerge in section 4.

The C1 integral now maps exactly to

( ) ( ) ( ) ( ) ( )òy ap

=--

r r XrX

X XH X,

i2

exp i dexp i

, 2.8C1L

2

2p2

in which L is the infinite straight contour with slope −π/4 through X=0, and H(X) is the X-even part of the numerator in (2.2), namely

( ) ( ( ) ( ))

( ) [ ( ) ( ) (( ) )] ( )

( )p

q a a

= + - -

= - - -

=H X h u h u

h uuX

u u

2 , where

dd

exp i sin sin 1 . 2.9

u u X12

Explicitly

( ) ( ) ( (( ) ( ))

( ) ( ( ))) ( )

ap a

q a

=-

´ - -

- -

H XX

X

X

2

2sin cos 1 arccos 1

exp i cos arccos 1 . 2.10

22

2

To get a uniform asymptotic scheme for large r, the expansion that incorporates the polesand saddles consistently is

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

4

( ) ( ) ( )å= + -=

¥

H X A X X B X . 2.11n

nn2

p2

0

2

Thus (2.8) becomes the (formally still exact) series

( ) [ ( ( ))]

( ) ( ) ( )ò å

y q a q a q p q

p

= + -

+ --

+=

¥⎡⎣⎢⎢

⎤⎦⎥⎥

r r

r X rXA

X XB X

, ; exp i cos sgn

i2

exp i d exp i . 2.12n

nn

AB

L

22

p2

0

2

The integrals involving the coefficients Bn are elementary (gaussian), and the integralinvolving A is

( )

( )| | ( )

| | ( ) | | ( )

| |ò òp

p

pp

--

= -

= - -

p

¥⎜ ⎟⎛⎝

⎞⎠

⎡⎣ ⎤⎦

XrX

X X XrX t t

XrX X r

dexp i 2i

exp i d exp12

i

iexp i erfc exp i , 2.13

X rL

2

2p2

pp2

22

pp2 1

4 p

p

in which for later convenience the second equality expresses the Fresnel integral in the firstline in terms of the complementary error function.

We need the coefficients in (2.11). The first two involve H(X) evaluated at the pole andthe saddle X=0:

( ) ( ) ( ) ( )= =-

A H X BH X H

X,

0. 2.14p 0

p

p2

The remaining coefficients Bn>0 involve only the derivatives of H(X) at the saddle X=0,namely the coefficients in

( ) ( )å==

¥

H X H X , 2.15n

nn

0

giving, by recursion

( ) ( )"=-

=-BB H

Xn 1, 2, 3, . 2.16n

n n1

p2

Now the scheme is complete. The term involving the coefficient A in (2.11), i.e. thecontribution from the Fresnel integral, is, after explicit evaluation of the coefficient A, usingerfc=1−erf, and a little algebra, is

( )( )( ) ( ) ( ( ))

( )

y a y a q aq

pa pa p q

» = +

´ - -⎡⎣⎢

⎤⎦⎥

r r r

r

, , exp i cos

cos isin erf exp i 2 sin . 2.17

AB erf

14

12

In the error function (Fresnel integral), the argument qr2 sin 12

represents the universalscaling law for the AB wave [15], which, as is well understood, it shares with edge diffraction[3]. Note that as a function of θ this is smooth (all derivatives continuous) in the forwarddirection θ=0: the discontinuity in the pole contribution (2.4) has been cancelled by thediscontinuity in (2.13). Note also that ψerf is the leading order in our earlier ‘cornufication’scheme [13], which will be discussed further in the appendix. In the backward direction, ψerf

is discontinuous; as will be discussed further in section 4, at this leading order of approx-imation the discontinuity, as well as the error, is O(1/r1/2).

Figure 2(a) shows how ψerf captures the main features of ψAB even for r=2π—only onewavelength from the flux line. This leading order does not however correctly capture the

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

5

wave scattered by the flux line; that will be accomplished by the terms involving the coef-ficients Bn in (2.12), to which we now turn.

Evaluating the Bn integrals in (2.12), with their obvious r scaling, leads to the followingexpression for the AB wave, still formally exact:

( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ! ( ) ( )

å

y a y a

y a y a paq a

q ap pa

q a

= +

= +

=-

-

+

=+

⎜ ⎟⎛⎝

⎞⎠r r

r r

r

rd

r

d n B

, , O1

, where

, , exp i sin,

,

,i i

2 sin, . 2.18

N N

Nn

Nn

n

n

n

n

AB 3 2

erf0

12

12

/

Successive truncations ψN represent successive approximations to ψAB. Long calculationsusing (2.14), (2.15) and (2.16) lead to explicit expressions for the coefficient functions

( )q ad , ;n the first four are

( )

( )

( )( )

( )( )

( )( )

( )( )

( )

( )( ) ( )

( )( ) ( )

( )( ) ( )( )

( ) ( )( )( ) ( )

( ) ( )( ) ( )( )

( ) ( )( )( ) ( )( ) ( )( ) ( )

( ) ( )( )

( ) ( )

( ) ( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

q ap

p qq aq

q ap

p qq aq

q a a a q

q ap

p qq aq

q a a a q

q a a a a q

q ap

p qq aq

q a a a q

q a a a a q

q a a a

a a q

= -

= -

+ - - + +

=-

-

+ - - + +

- - - - + +

=-

-

+ - - + +

- - - - + +

+ - - -

´ - + +

⎡⎣⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

d

d

d

d

,exp i

2 sinexp i exp i

,exp i

4 2 sinexp i exp i

i sin exp i

,exp i

16 2 sin3 exp i exp i

3i sin exp i

i sin exp i

,exp i

192 2 sin45 exp i exp i

45i sin exp i

15i sin exp i

2i sin

exp i . 2.19

0

34

12

12

1

14

12

312

12

12

32

12

2

14

12

512

12

12

32

12

12

3 2 14

32

52

32

3

34

12

712

12

12

32

12

12

3 2 14

32

52

32

12

5 2 14

2 94

52

72

52

All the functions dn vanish for flux α=1/2, reflecting the fact, already recognised in theoriginal AB paper [1], that the leading order ψerf is exact for this case. And all the corrections,as well as the scattered wave in ψerf , vanish when α is integer and there is no AB effect.Notwithstanding the prefactors that diverge at θ=0, the dn are actually smooth functions ofθ. The forward direction will be discussed in detail in the next section.

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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Figure 2(b) shows the approximation N=0. This the first correction to ψεrf that correctlyincorporates the AB scattered wave, and corresponds to the lowest-order uniform approx-imation to ψAB. It is our counterpart of the lowest-order approximation in [16], whose secondderivative is forward-discontinuous. The agreement is dramatically improved: even forr=2π (one wavelength from the flux), the curves cannot be distinguished by eye. As a moreextended comparison, figure 3 shows the errors for the approximations ψerf, ψ0, ψ1, ψ2, ψ3.The improvement with increasing order is dramatic. And the fact that the errors are similarover the entire angular range confirms that the approximations are indeed uniformly validin θ.

3. Forwards

The first few forward coefficients (2.19) can be determined by taking the limit θ→0. Thisenables the form for general n to be identified as

( ) ( )( )! !

( ) ! ( )a p app

a a= - -

- + + -

+⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠d

n n

n n0, exp

14

i cos2 1

2i

2 1. 3.1n

n

3

12

12

Thus the approximants for the AB wave in the forward direction are (using erf(0)=0 in(2.17))

( ) ( ) ( )åy a pap

p pa= - ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r r

rT, exp i cos 1

2exp

14

i sin , 3.2N

N

n30

Figure 2. Comparison of exact AB wave |ψAB(r, θ, α)| (black curves) with (a) |ψerf(r, θ,α)| (equation (2.17); (b) N=0 uniform approximation in (2.18) (red dashed curves),for r=2π, α=1/4.

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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in which the terms in the sum are

( ) ( )! !( ) ! ( )

a a= -

- + + -

+⎜ ⎟⎛⎝

⎞⎠T

r

n n

n ni

2 2 1. 3.3n

n12

12

The increasing accuracy of the approximants as N increases is illustrated in figure 4, showingthat the scheme is uniformly accurate for all fluxes α, as well as all angles as we have seen.

4. High-order asymptotics: divergence and resurgence

It is obvious from (3.3) that the series in (3.2) is divergent in the manner familiar throughoutasymptotics [19, 20]: the terms grow factorially, with leading high-order behaviour(‘asymptotics of the asymptotics’)

( ) ! ( ) ( )�» - -⎜ ⎟⎛⎝

⎞⎠T n

rn

12

1i

21 . 4.1n

n

For large r, the terms decrease before starting to increase; the smallest term, corresponding tooptimal truncation, is of order approximately given by

Figure 3. Errors log10|ψAB−ψN| in AB approximations (equation (2.18) truncated atn=N), for r=2π, α=1/4 for −π�θ�π, for the indicated values of N.

Figure 4. Errors log10|ψAB−ψN| in AB approximations (3.2) in forward directionθ=0, for r=2π, as functions of 0�α�1/2 for the indicated values oftruncation N.

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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( ) ⌊ ⌋ ( )=N r r2 , 4.2opt

in which ⌊ ⌋... denotes the integer part (floor function). This behaviour is illustrated by theblack dots in figure 5; the highest accuracy is achieved for truncation Nopt=6, as predictedfrom (4.2) for this radius r=π.

The accuracy corresponding to optimal truncation can be estimated by Borel summation[19] of the divergent post-truncation tail of (3.2), using (4.1). This gives

( ) ( )åp

»-

+

¥

+T Texp i

2. 4.3

Nn N

1

14

1opt

opt

As elsewhere in asymptotics [19], the optimal truncation error is proportional to the firstomitted term; in this case, the precise proportionality factor arises from the factors (−i)n in(4.1). In the series (3.2) for ψAB, the corresponding truncation error, obtained by usingStirling’s approximation for the factorial, is exponentially small:

( ) ( ) ( ) ( ) ( ) ( )( )( )y a y a

pap

- » - -r rr

rr, 0, ,

i exp i sin 24

i exp 2 . 4.4N rN r

AB optopt

Figure 6 illustrates the accuracy of this estimate.Although the divergence of the series has been calculated from the explicit form (3.3) of

the coefficients, it was anticipated from the singularities of the map (2.5) at X2=2. ByDarboux’s principle [19], these singularities, closest to the expansion point X=0, determinethe high-order behaviour (4.3) of the coefficients. This can also be derived by expanding inpowers of X the form of H(X) near X2∼2, which from (2.10) (for θ=0) is

( ) ( )pa» -

-H X

X

2 sin 2

2. 4.5

2

The formula (4.4) gives the leading-order correction after optimal truncation of the series(3.2). It is possible to go much further. One of several effective resummation schemes isAirey’s way [21] (see also section 21.2 of [19] ) of calculating the ‘terminant’ or ‘convergingfactor’ R(r,α), defined (see (3.2) by writing the exact AB wave as

Figure 5. Errors in forward-direction asymptotics θ=0, for r=π, i.e.half a wavelngthfrom the flux, and α=1/4. Black dots: errors log10|ψAB−ψN| in AB approximations(3.2) for increasing truncations N. Red dots: errors log10|ψAB−ψ(K)| in exponentiallyimproved AB approximations (4.6) for increasing truncations K of the terminant sum(4.11), with the red dots horizontally shifted to Nopt(r)+1+K=7+K, to indicateimprovement beyond optimal truncation of the bare asymptotic series (3.2).

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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( )( ) ( )

( ( ) ) ! ( ) ( )( ) ( )

å

y a pap

p pa

a

= -

´ + - -⎜ ⎟

⎡⎣⎢

⎛⎝⎜⎜

⎛⎝

⎞⎠

⎞⎠⎟⎟

⎤⎦⎥⎥

r rr

T N rr

R r

, 0, exp i cos 12

exp i sin

1i

2, . 4.6

N r

n

N r

AB 314

0

12 opt

opt opt

Using (3.3), conveniently defining

( )m aº º -F r2i ,12

, 4.7

and abbreviating Nopt(r) by N, the terminant is given by a formal sum representing thedivergent tail of the series:

( ) ( )!( ) ! ( )!

! ( ) ! ( ) ( )

å

å

a

m m

=-

=+ + - + + +

+ + + +

=

¥

+ +

=

¥⎜ ⎟⎛⎝

⎞⎠

R rF

NT

NF

NF

N m N mN N N m N m

,2

12 1 1

1 2 2 3. 4.8

m

N

N m

m

m

m

01

0

The technique is based on expanding the summand in powers of 1/N, i.e.

( ) ! ( )!! ( ) ! ( )

( ) ( )åm m m+ + - + + +

+ + + +=

=

¥N m N mN N N m N m

c mN

2 1 11 2 2 3

,. 4.9

mk

kk

0

The first few coefficients are

( ) ( ) ( )

( ) [ ( ) ( )

( ) ] ( )

m m m

m m m m

m

= = - + +

= + - + + -

+ - + + +

c m c m m m

c m m

m m m

, 1, , 2 1 ,

,124

18 12 4 4 1 3

3 3 4 2 3 . 4.10

0 112

2 3

22 2 2

2 2 3 4

The method relies on the fact that all coefficients involve positive powers of m, enabling thesum over m in (4.9)—the sum over terms in the divergent tail of the series—to be evaluated asgeometric series.

Figure 6. Error log10|ψAB−ψNopt| for optimal truncation ⌊ ⌋= =N N r2 ,opt in forwarddirection, for α=1/4, as a function of r. Black curve: exact error; red dashed curve:estimate (4.4).

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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Thus the terminant becomes

( ) ( ) ( )åaa

==

¥

R rA r

N,

,. 4.11

k

kk

0

The first few coefficients are

( ) ( ) ( )( ) ( )( )

( ) ( ) [ ( )

( ) ( )( ) ] ( )

a am m

a m m m m

m m m mm m

=-

=- + + -

-

=-

- + + - + -

+ - + + - + -+ - +

A rg

A rg g

g

A rg

g

g g

g

,1

1, ,

2 1 1 4 12 1

,

,1

4 11 4 2 4 1 6 2

4 6 11 3 4 3 8 2

3 8 2 , 4.12

0 1

2 2 2

3

2 52 4 2 4

2 4 2 2 4 3

2 4 4

in which g is the following quantity, whose modulus is close to unity:

⌊ ⌋ ( )º =gFN

rr

i22

. 4.13

The approximants ψ(K)(r,α) in this exponentially improved representation are obtainedby truncating the series (4.11) at the term k=K and substituting this into (4.6). The lowestorder K=0, together with Stirling’s formula, reproduces the estimate (4.4). The higherorders dramatically increase the accuracy, as illustrated by the red dots in figure 5. This seriesalso has a smallest term, whose astonishing accuracy, for positions only half a wavelengthfrom the flux, is O(10−8). Further improvement would require further resummation.

The coefficients in (3.3), namely

( ) ( ) ! ( )!( ) ! ( )mm m

=- +

+K

n nn n2 1

, 4.14n

satisfy a remarkable resurgence relation, of the type familiar elsewhere in asymptotics [20], inwhich the high orders n?1 are expressed in terms of the low orders n=0, 1, 2K This is areinterpretation of a known asymptotic series for a product of gamma functions (equation(5.11.19) of [22, 23] (see also p15 of [19]):

( ) ( ) ( ) ( ) ( ) ( )( )! ( )

!( ) ( ) ( )

( )( )( )

( )( )( ) ( )"

åm mm

m mm

mpmp

mm mm

m mm

» =-

+-

+ --

=+

´ -- -

+- - -

-

=

⎛⎝⎜

⎞⎠⎟

K Kn K

m n mm

K

nn

KK

n

Kn n

2 11

2 1

sin2 1

31 1

34 1 2

. 4.15

n n Mm

Mm

m,

2

0 02 2

0

21

2

22

2

The simplest truncation M=0 reproduces the leading-order large n asymptotics (4.1).Increasing M generates more accurate approximations, with optimal truncation at M∼n/2,after which the terms start to increase. For optimal truncation, the error is

( ) ( )( ) ( )m mm

-~

K K

K n1

2. 4.16n n n

nn

, 23 2

/

/

In (4.15), the maximum allowed value of M is n.It would be possible to use the resurgence relation, in conjunction with repeated Borel

summation, in a more sophisticated ‘hyperasymptotic’ resummation scheme [24–26]. But forthe present problem this is less accurate than the scheme based on (4.11) (its advantages

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

11

appear when approximating functions on or near their Stokes lines, which is not the case herebecause of the ‘anti-Stokes’ factors (−i)n in (3.3)).

In this section, we have considered θ=0. But similar arguments, albeit algebraicallymore complicated, would yield analogous exponentially improved approximations away fromforwards.

5. Backwards

The exact AB wave (2.1) is a smooth periodic function of θ, including the backward direction:( ) ( )y p a y p a= -r r, , , , .AB AB But the terms in the series (2.18) are discontinuous in the

backward direction. As a first step in understanding this, consider ψerf. From (2.17), and usingthe standard asymptotic expansion of erf, the discontinuity is

( )( )

( )( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) !! ( )å

y a y p a y p a

pa p

p pa

p

D º - -

= -

=-- -

-=

¥

r r r

r r

r

r rn

, , , , ,

i exp i sin 2 erfc exp i 2

exp i sin 2

2i

42 1 5.1

n

n

n

erf erf erf

14

14

0

Each term in this expansion is cancelled by the discontinuity in the correction terms, labelledn in (2.18). This follows from the discontinuities in the coefficients dn, given exactly by

( )( ) ( ) ( )

( )( )

( ) !! ( )

a p a p a

pap

p

D º - -

=- -

-

d d d

n

, ,

2cos

exp i i

22 1 5.2

n n n

n

n

14

2

As a consequence, the discontinuities in the leading-order approximation ψerf and alsothe approximants ψN, are of the same order as the errors in the approximations at θ=π andθ=–π, namely r−1/2 for ψerf , and r−(N+3/2) for ψN:

( )

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

y a y p a y p a

y a y p a y p ay a y p a y p a

y a y p a y p a

D º - -

» D º - =D º - -

» D º - =

-

- +

r r r

r r r O rr r r

r r r O r

, , , , ,

, , , , , ,, , , , ,

, , , , , . 5.3

AB

N N N

N AB NN

erf erf erf

erf erf12

32

The large r behaviour of these three functions is illustrated in figure 7. It might seemparadoxical that the two errors |Δ±ψ| appear close together and the discontinuities |Δψ| areapproximately twice as big. But this an artefact of plotting the moduli of these complexquantties. In the complex plane, the values of the approximants ψN(r,π,α) and ψN(r,−π,α) lieon opposite sides and roughly equidistant from the exact ψAB(r,π,α), as figure 8 shows.

6. Concluding remarks

The scheme developed here is a systematic sequence of approximations, formally valid forlarge dimensionless distance r but surprisingly accurate even within one wavelength of theflux. Restoring dimensions, the asymptotic variable is

�=kR RmE2 (see (1.1)), so the

approximation is semiclassical as well as short-wave and large-distance. (Strictly, the semi-classical asymptotics includes the rapid �1/ oscillations embodied in the periodic dependence

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

12

on the quantum flux (1.2).) The approximation scheme is uniformly valid in direction θ andflux α, with the leading order ψerf (equation (2.17) involving the Fresnel integral, and suc-cessive truncations (N in (2.18)) representing the exact ψAB with error O(1/rN+3/2).

The derivation presented here was based on the integral ψC1 in the decomposition (2.3),given by (2.2) with the contour C2 in figure 1(b), with ψC2 given by (2.4) and ψC3 zero bysymmetry. The integral was evaluated directly, using saddle-pole asymptotics. An alternativederivation is possible, based on the following representation of ψC1 in terms of Hankelfunctions (derived in [3], based on expressions in [1], and here written with θ=0 repre-senting forwards):

( )( )( ) ( )

[ ( ) ( ) ( )] ( ) ( )( ) ( ) - -òy a pa q pa r r q

r q r a

=- - ´ -

´ +a a

¥

- -

r r

H H

,12i

sin exp i cos d exp i cos

i exp i , 0 1 . 6.1r

C112

11 1

Figure 7. Backward discontiuities |Δψ| (full black curves) and backward errors |Δ±ψ|(red curves and black dashed curves), defined by (5.3) for ψerf (multiplied by r1/2) andψN (multiplied by rN+3/2). Full black curves: π; rN+1/2 × backward discontinuity forθ = π and θ = −π; red curves: rN+1/2 × error for θ = π; black dashed curves: rN+1/2 ×error for θ = −π, for α=1/4 and approximations (a) ψεrf, (b) ψ0, (c) ψ1, (d) ψ2.

Figure 8. Complex backward amplitudes for r=2π, α=1/4. Red dot: exact ψAB(2π,π, 1/4); black dots: approximations ψ3(2π, +π, 1/4) (upper right) and ψ3(2π, –π, 1/4)(lower left).

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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The derivation would proceed using known asymptotics for the Hankel functions, but wouldlead to the same result (2.17) and (2.18) as here derived bare-handedly directly from thecontour integral representation of ψC1.

This scheme is different from the ‘many-whirls’ representation [8, 10], in which the ABwave is decomposed into contributions describing different numbers of windings around theflux line. Whirling waves express the topology of AB: noncontractible loops in the planepunctured by removal of the point where the flux is located. And, related to this, they enablethe Dirac magnetic phase factor to be incorporated concisely and exactly. Mathematically, thewhirling waves are terms in a series obtained by Poisson-transforming the sum over angularmomenta m in the original AB wave (2.1). By contrast, the integral (2.2) which is our starting-point is obtained by summing the m series exactly in closed form.

One feature that our approximation scheme shares with the decomposition into whirlingwaves, although the details are different, is that the individual terms are discontinuous—forour scheme, the discontinuities are in the backward direction θ=±π. This raises the questionof whether it is possible to construct an approximation scheme whose contributions are allsinglevalued and respect the θ periodicity of AB. I am not sure, but suspect that no seriesbased on Fresnel integrals (error functions) will have this property.

Acknowledgments

This work was stimulated by Professor John Hannay’s complementary leading-orderapproximation, and I thank him for showing it to me before publication, as well asfor discussions. I also thank Professor Ovidiu Costin for discussions about expone-ntially improved approximations like that in section 4, Dr Martin Sieber and Dr StefanFischer for helpful correspondence and conversations, and Dr Eliahu Cohen for carefullyreading the paper, leading to several corrections. My research is supported by the Leverh-ulme Trust.

Appendix. Relation with cornufication

It is interesting to discuss the relation between the erf-based scheme developed here and therepresentation considered earlier [13], in which the AB wave was fitted to a Fresnel integral(erf with complex argument). This was based on representing the AB wave exactly, in a formresembling (2.17), with (now using a slightly different notation) erf replaced by a complexfunction K(r,α), and noting that in its complex plane K clings very close to the Cornu spiral(figure A1) for all values of α and as r varies in the plane, except when r is very close to theorigin. Explicitly, the ‘cornufied’ map from r to K is

( )( ) ( ( ) ( ( ))) ( )ap

papa y q a q aq=

-- - +rK r r,

exp i

2 sincos , , exp i cos . A.1

14

AB

This led to the suggestion that K can be represented in the form of a complex Fresnelintegral F

( ) ( ( )) ( )a a=r rK F w, , , A.2

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

14

in the convenient form (see (2.13))

( ) ( )ò p p p= = -p ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎛

⎝⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟F w u u wd exp

12

i12

exp14

i erf exp14

i , A.3w

0

22

with the argument w determined as the solution of (A.1) (and slightly complex toaccommodate the small departures from the spiral). A series was found for w(r, α), startingwith qr2 sin .1

2But the procedure implicitly required the inverse function of erf, and

complications associated with different branches were not examined, leading to a still-unresolved (and unrecognised in [13]) lack of smoothness in the representation for θ slightlynegative.

It could be that this anomaly can be resolved by a more careful analysis of w(r, α). If so,a possible connection with the scheme presented here might be be analogous to the two waysof representing small � asymptotic solutions y(x) of the Schrödinger equation in WKB theory,i.e.

( ) ( ( ) ( ) ) ( )

( ) ( ( ) ( ) ( ) ( ) ) ( ) ( )

"

"

��

�� �

= + +

= = + + +⎜ ⎟

⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠

y xw

a x a x a

y x w x w x w x w x b

exp i

expi

. A.4

00 1

0 12

2

The analogy is that the scheme elaborated in this paper corresponds to the linearrepresentation (a), with erf replacing exp with the zero order argument w0(x), and thecornufication scheme corresponds to the nonlinear representation (b), again with erf replacingexp but with the more complicated argument w(x).

Alternatively, forcing the AB wave into the form (A.1) and (A.4) might be too strong aconstraint—a Procrustean fit that cannot be smooth, in which case the analogy would fail.

Figure A1. Black dots: cornufication map (A.1), from r to complex K, for positionswith r=π, 2π, 4π, fluxes α=1/10, 1/4, 1/2 and angles 0�|θ|�π. Red curve:Cornu spiral F(w) (equation (A.3)), plotted parametrically for real w.

J. Phys. A: Math. Theor. 49 (2016) 305204 M V Berry

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