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

September 17, 14.20 WSCQS, CRM Montreal 2

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)

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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

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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

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I.

PT symmetric quantum mechanics

• THE EMERGENCE OF THE IDEA

• ITS EARLY APPLICATIONS

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• THE EMERGENCE OF THE IDEA

•real E

•boundary conditions

•isospectrality

• ITS EARLY APPLICATIONS

•WKB and numerical

•free motion

•expansions

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• THE EMERGENCE OF THE IDEA

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]

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• 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]

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II.

Selected concepts of solvability

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Sample menu

• ODE solvability

• symmetry reduction

• polynomial solvability

• SUSY partnership

• QES

• Hill determinants

• asymptotic series

• exceptional PDE

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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

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III.The emergence of less usual characteristics of solvability

for PT symmetric Hamiltonians

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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

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III.The emergence of less usual characteristics of solvability

for PT symmetric Hamiltonians

•

•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,…)

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III. 1.Solutions over curved

complex contours

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

• With PT symmetry

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III. 1.Solutions over curved

complex contours

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

• With PT symmetry

(a) free-like

(b) WKB solvable

(c ) Laguerre solvable

(d) exact Jacobi

(d) QES

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III. 1.Solutions over curved

complex contours

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

• With PT symmetry

(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)

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III. 1.Solutions over curved

complex contours

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

• With PT symmetry

(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]

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III. 2. D > 1 regularization recipe

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III. 2.PT D > 1 regularization recipe

solutions over the straight complex lines of coordinates

•perturbative

•regularized:

•systematic:

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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]

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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]

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III. 3. Models solvable via classical

OG polynomials:

•PT modified

•non-Hermitian

•systematic methods

•re-interpretations

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III. 3. Models solvable via classical

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

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III. 3. Models solvable via classical

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]

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III. 4.The methods of SUSY

partnership•starting from squre well:

• using alternative, PT specific SUSY schemes:

• referring to Lie algebras

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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]

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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]

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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

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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

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III. 8.PDE cases

• the Winternitzian superintegrable V’s:

• the Calogerian three-body laboratory:

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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

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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

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V.Summary

• mathematics in interplay with physics

• immediate applicability

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V.Summary

• mathematics in interplay with physics

(from Hermitian to PT symmetric):

(a) unitarity

(b) Jordan blocks

(c ) quasi-parity

• immediate applicability

(a) Winternitzian models:

(b) Calogerian models:

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V.Summary

• mathematics in interplay with physics

(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

• immediate applicability

(a) Winternitzian models:

non-equivalent Hermitian limits

(b) Calogerian models:

new types of tunnelling