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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).

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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

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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

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TABLEOFCONTENTS

I.THECONCEPTOFPTSYMMETRY

II. WHAT SHALL WE CALL “SOLVABLE“?

III. PT- SYMMETRIC WORLD

IV. PSEUDO-HERMITICITY

V. SUMMARY

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slide5

I.

PT symmetric quantum mechanics

  • THE EMERGENCE OF THE IDEA
  • ITS EARLY APPLICATIONS

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slide6

THE EMERGENCE OF THE IDEA

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

WSCQS, CRM Montreal

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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]

WSCQS, CRM Montreal

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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]

WSCQS, CRM Montreal

slide9

II.

Selected concepts of solvability

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slide10

Sample menu

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

WSCQS, CRM Montreal

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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

WSCQS, CRM Montreal

slide12

III.

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

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slide13

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

WSCQS, CRM Montreal

slide14

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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slide15

III. 1.

Solutions over curved complex contours

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

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slide16

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

WSCQS, CRM Montreal

slide17

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)

WSCQS, CRM Montreal

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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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slide21

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]

WSCQS, CRM Montreal

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systematic approaches

  • 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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slide23

III. 3.

Models solvable via classical OG polynomials:

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

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slide24

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

WSCQS, CRM Montreal

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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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slide27

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]

WSCQS, CRM Montreal

slide28

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]

WSCQS, CRM Montreal

slide29

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

WSCQS, CRM Montreal

slide30

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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slide31

III. 8.

PDE cases

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

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slide32

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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slide33

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

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slide34

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:

WSCQS, CRM Montreal

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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

WSCQS, CRM Montreal

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