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New types of solvability in PT_symmetric quantum mechanicsPowerPoint Presentation

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

WSCQS, CRM Montreal

TABLE “extended” quantum theoryOFCONTENTS

I.THECONCEPTOFPTSYMMETRY

II. WHAT SHALL WE CALL “SOLVABLE“?

III. PT- SYMMETRIC WORLD

IV. PSEUDO-HERMITICITY

V. SUMMARY

WSCQS, CRM Montreal

I. “extended” quantum theory

PT symmetric quantum mechanics

- THE EMERGENCE OF THE IDEA
- ITS EARLY APPLICATIONS

WSCQS, CRM Montreal

- THE EMERGENCE OF THE IDEA “extended” quantum theory
- real E
- boundary conditions
- isospectrality
- ITS EARLY APPLICATIONS
- WKB and numerical
- free motion
- expansions

WSCQS, CRM Montreal

- THE EMERGENCE OF THE IDEA “extended” quantum theory
- 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]

WSCQS, CRM Montreal

- EARLY APPLICATIONS “extended” quantum theory
- 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]

WSCQS, CRM Montreal

Sample menu “extended” quantum theory

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

WSCQS, CRM Montreal

Details “extended” quantum theory

- 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

WSCQS, CRM Montreal

III “extended” quantum theory.

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

WSCQS, CRM Montreal

III “extended” quantum theory.

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

III “extended” quantum theory.

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,…)

WSCQS, CRM Montreal

III. 1 “extended” quantum theory.

Solutions over curved complex contours

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

WSCQS, CRM Montreal

III. 1 “extended” quantum theory.

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

III. 1 “extended” quantum theory.

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)

WSCQS, CRM Montreal

III. 1 “extended” quantum theory.

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 “extended” quantum theory.

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: “extended” quantum theory

- perturbative
- (a) anharmonic oscillator [CGM ‘80]
- regularized:
- in quantum mechanics (AHO) [BG ‘95]
- in field theory (Schwinger Dyson eq.) [BM ‘ 97]

WSCQS, CRM Montreal

systematic approaches “extended” quantum theory

- 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]

WSCQS, CRM Montreal

III. 3 “extended” quantum theory.

Models solvable via classical OG polynomials:

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

WSCQS, CRM Montreal

III. 3 “extended” quantum theory.

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

III. 3 “extended” quantum theory.

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]

WSCQS, CRM Montreal

III. 4 “extended” quantum theory.

The methods of SUSY partnership

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

WSCQS, CRM Montreal

III. 4 “extended” quantum theory.

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

III. 5 “extended” quantum theory.

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]

WSCQS, CRM Montreal

III. 6 “extended” quantum theory.

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

WSCQS, CRM Montreal

III. 7 “extended” quantum theory.

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 “extended” quantum theory.

PDE cases

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

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III. 8 “extended” quantum theory.

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

WSCQS, CRM Montreal

IV “extended” quantum theory.

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

WSCQS, CRM Montreal

V “extended” quantum theory.

Summary

- mathematics in interplay with physics
- immediate applicability

WSCQS, CRM Montreal

V “extended” quantum theory.

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

V “extended” quantum theory.

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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