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New types of solvability in PT_symmetric quantum mechanics

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New types of solvability in PT_symmetric quantum mechanics

WSCQS, CRM Montreal

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)

WSCQS, CRM Montreal

a brief review of the recent developments in an “extended” quantum theorywhere the spectra (of bound states) are required real but Hamiltonians themselves need not remain Hermitian

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I.THECONCEPTOFPTSYMMETRY

II. WHAT SHALL WE CALL “SOLVABLE“?

III. PT- SYMMETRIC WORLD

IV. PSEUDO-HERMITICITY

V. SUMMARY

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PT symmetric quantum mechanics

- THE EMERGENCE OF THE IDEA
- ITS EARLY APPLICATIONS

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- real E
- boundary conditions
- isospectrality
- ITS EARLY APPLICATIONS
- WKB and numerical
- free motion
- expansions

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- 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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- 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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- ODE solvability
- symmetry reduction
- polynomial solvability
- SUSY partnership
- QES
- Hill determinants
- asymptotic series
- exceptional PDE

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- 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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The emergence of less usual characteristics of solvability for PT symmetric Hamiltonians

WSCQS, CRM Montreal

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

WSCQS, CRM Montreal

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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Solutions over curved complex contours

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

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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

WSCQS, CRM Montreal

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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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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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:
- in quantum mechanics (AHO) [BG ‘95]
- in field theory (Schwinger Dyson eq.) [BM ‘ 97]

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- present context
- Calogero-Winternintz (at A=1) [Z’99]
- 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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Models solvable via classical OG polynomials:

- PT modified
- non-Hermitian
- systematic methods
- re-interpretations

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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

WSCQS, CRM Montreal

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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The methods of SUSY partnership

- starting from squre well:
- using alternative, PT specific SUSY schemes:
- referring to Lie algebras

WSCQS, CRM Montreal

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]

WSCQS, CRM Montreal

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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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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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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PDE cases

- the Winternitzian superintegrable V’s:
- the Calogerian three-body laboratory:

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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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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:
- pseudohermiticity as a source of new models
- constructions of the Hilbert-space metric
- superintegrability: a way towards asymmetry

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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:

WSCQS, CRM Montreal

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

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