september 17, 14.20wscqs, crm montreal1 new types of solvability in pt_symmetric quantum mechanics

September 17, 14.20 WSCQS, CRM Montreal 1

New types of solvability in

PT_symmetric quantum mechanics


New types of solvability in

PT_symmetric quantum mechanics

(a review)[Workshop on Superintegrability in Classical and Quantum

Systems]

[September 16 - 21, 2002, CRM, Montreal]

M. Znojil (NPI, Rez near Prague)


a brief review of the recent developments

in an “extended” quantum theory where the spectra (of bound states) are required real but Hamiltonians

themselves need not remain Hermitian


TABLETABLE OFOF CONTENTSCONTENTS

I.I. THETHE CONCEPTCONCEPT OFOF PTPT SYMMETRYSYMMETRY

II. WHAT SHALL WE CALL “SOLVABLE“?II. WHAT SHALL WE CALL “SOLVABLE“?

III. PT- SYMMETRIC WORLDIII. PT- SYMMETRIC WORLD

IV. PSEUDO-HERMITICITYIV. PSEUDO-HERMITICITY

V. SUMMARYV. SUMMARY


I.

PT symmetric quantum mechanics

• THE EMERGENCE OF THE IDEA

• ITS EARLY APPLICATIONS



•real E

•boundary conditions

•isospectrality

• ITS EARLY APPLICATIONS

•WKB and numerical

•free motion

•expansions



real E for imaginary V

•(cubic anharmonic oscillator)

• [Caliceti et al ‘80, Bessis ‘92]

relevance of boundary conditions

•(complex contours)

•[Bender and Turbiner ‘93]

isospectrality

•(of ‘up’ and ‘down’ quartic oscilllators)

•[Buslaev and Grecchi ‘95]


• EARLY APPLICATIONS

WKB and numerical experiments

•with V(x) = i x^3

•[Bender and Boettcher ‘98]

a PT-sym. analogue of free motion

•(Bessel solutions)

•[Cannata, Junker, Trost ‘98]

strong-coupling expansions

• [Fernandez et al ‘98]


II.

Selected concepts of solvability


Sample menu

• ODE solvability

• symmetry reduction

• polynomial solvability

• SUSY partnership

• QES

• Hill determinants

• asymptotic series

• exceptional PDE


Details

•

• ODE solvability = one-dimensional [Morse’s V(x)]

• symmetry reduction = PDE -> ODE [central, D > 1]

• polynomial solvability = ch. of var. [Lévai’s method]

• SUSY partnership = new V’s [IST method]

• QES = algebraization [Hautot ‘72]

• Hill = non-Hermitian matrization [Znojil ‘94]

• asymptotic-series = artif. param’s [1/L]

• exceptional PDE = superintegrable etc


III.The emergence of less usual characteristics of solvability

for PT symmetric Hamiltonians




•

•ODE

• reduced symmetry

•polynomial solvability

•SUSY partnership

•QES

•Hill determinants

•Asymptotic series

•exceptional PDE




•

•ODE = solutions over contours

• reduced symmetry -> quasi-parity

•polynomial solvability = i p shift

•SUSY partnership (cf. IST method)

•QES (solving algebraic equations)

•Hill determinants (early non-Hermitian)

•Asymptotic series (artif. param’s)

•exceptional PDE (superintegrable, Calogero,…)


III. 1.Solutions over curved

complex contours

•Without PT symmetry (QES, sextic osc.) [BT ‘93]

• With PT symmetry



complex contours



(a) free-like

(b) WKB solvable

(c ) Laguerre solvable

(d) exact Jacobi

(d) QES



complex contours



(a) free-like (Bessel states)

(b) WKB solvable (V = (ix)^d)

(c ) Laguerrean: Morse and Coulomb

(d) exact Jacobi: Hulthén and CES

(d) QES (decadic)



complex contours



(a) free-like (Bessel states) [CJT ‘98]

(b) WKB solvable (V = (ix)^d) [BB ‘98, ‘99]

(c ) Laguerrean: Morse [Z’ 99] and Coulomb [LZ’00]

(d) exact Jacobi: Hulthén [Z’00] and CES [ZLRR’01]

(d) QES (decadic) [Z’00]


III. 2. D > 1 regularization recipe


III. 2.PT D > 1 regularization recipe

solutions over the straight complex lines of coordinates

•perturbative

•regularized:

•systematic:


solutions over the straight complex lines of coordinates:

• perturbative

(a) anharmonic oscillator [CGM ‘80]

• regularized:

(a) in quantum mechanics (AHO) [BG ‘95]

(b) in field theory (Schwinger Dyson eq.) [BM ‘ 97]


systematic approaches

• present context

(a) Calogero-Winternintz (at A=1) [Z’99]

(b) regularization by shift [LZ ‘00]

• SUSY context

(a) partners of a Hermitian V(x) [BR ‘00]

(b) shape invariant V(x) [Z’00]


III. 3. Models solvable via classical

OG polynomials:

•PT modified

•non-Hermitian

•systematic methods

•re-interpretations



OG polynomials:

•PT modified SI models: direct solutions

•non-Hermitian SUSY-generated V(x)

•Lévai’s systematic method with imaginary shift:

(a) unbroken PT symmetry

(b) PT symmetry spontaneously broken

•re-interpretations using Lie algebras

(a) ES context

(b) QES context



OG polynomials:

•PT modified SI models: direct solutions [Z’99]

•non-Hermitian SUSY-generated V(x) [A’99]

•Lévai’s systematic method with imaginary shift:

(a) unbroken PT symmetry [LZ’00]

(b) PT symmetry spontaneously broken [LZ’01]

•re-interpretations using Lie algebras

(a) ES context [BCQ’01,BQ’02]

(b) QES context [BB’98,Z’99,CLV’01]


III. 4.The methods of SUSY

partnership•starting from squre well:

• using alternative, PT specific SUSY schemes:

• referring to Lie algebras


III. 4.The methods of SUSY

partnership•starting from squre well:

(a) initial step [Z’01]

(b) non-standard PT SUSY hierarchy [BQ’02]

• using alternative, PT specific SUSY schemes:

(a) non-Hermitian SUSY repr’s [ZCBR’00]

(b) PSUSY and SSUSY schemes [BQ ‘02]

• referring to Lie algebras

(a) creation and annihilation anew [Z’00]

(b) PT scheme using sl(2,R) [Z’02]


III. 5.Quasi-exactly solvable PT

models

• initial breakthrough: quartic oscillators [BB’98]

• known QES revisited: Coul.+HO [Z’99] etc

• role of the centrifugal-like singularities:

(a) a few old sol’s revisited [Z’00,BQ’01]

(b) QES classes of V [Z‘00,Z’02]

(c) quasi-bases [Z’02]


III. 6.Constructions using the so

called Hill determinants

universal background:

(a) discretization via non-orthogonal bases

(b) proofs via oscillation theory [Z’94]

• PT sample with rigorous proof [Z’99]

• QES interpreted as a special case


III. 7.Perturbation expansions

using artificial parameters

• delta expansions as an initial motivation [BM’97]

• WKB [DP’98]

• 1/L expansions:

(a) challenge: ambiguity of the initial H(0) [ZGM’02]

(b) technique: feasibility of RS expansions [MZ’02]

(c) open problem: quasi-odd spectrum


III. 8.PDE cases

• the Winternitzian superintegrable V’s:

• the Calogerian three-body laboratory:


III. 8.PDE cases

• the Winternitzian superintegrable V’s:

(a) the problem of equivalence of the complexified separations of variables

[K,P,W,pc]

(b) the zoology of Hermitian limits V [JZ]

• the Calogerian three-body laboratory:

(a) PT symmetrized [ZT’01a]

(b) non-standard Hermitian limit [ZT’01b]

(c) next step: non-separable A > 3


IV.General formalism and

outlook• bi-orthogonal bases:

(a) diagonalizable and non-diagonalizable cases [Mostafazadeh ‘02]

(b) H = a real 2n x 2n matrix

(c) the Feshbach’s effective H(E): a nonlinearity

• outlook:

(a) pseudohermiticity as a source of new models

(b) constructions of the Hilbert-space metric

(c) superintegrability: a way towards asymmetry


V.Summary

• mathematics in interplay with physics

• immediate applicability


V.Summary


(from Hermitian to PT symmetric):

(a) unitarity

(b) Jordan blocks

(c ) quasi-parity


(a) Winternitzian models:

(b) Calogerian models:


V.Summary


(parallels between Hermitian and PT symmetric):

(a) unitarity <-> the metric in Hilbert space is not P

(b) Jordan blocks <-> unavoided crossings of levels

(c ) quasi-parity <-> PCT symmetry


(a) Winternitzian models:

non-equivalent Hermitian limits

(b) Calogerian models:

new types of tunnelling

september 17, 14.20wscqs, crm montreal1 new types of solvability in pt_symmetric quantum mechanics

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