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New types of solvability in PT_symmetric quantum mechanics

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

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  1. New types of solvability in PT_symmetric quantum mechanics 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) WSCQS, CRM Montreal

  3. 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

  4. TABLEOFCONTENTS I.THECONCEPTOFPTSYMMETRY II. WHAT SHALL WE CALL “SOLVABLE“? III. PT- SYMMETRIC WORLD IV. PSEUDO-HERMITICITY V. SUMMARY WSCQS, CRM Montreal

  5. I. PT symmetric quantum mechanics • THE EMERGENCE OF THE IDEA • ITS EARLY APPLICATIONS WSCQS, CRM Montreal

  6. THE EMERGENCE OF THE IDEA • real E • boundary conditions • isospectrality • ITS EARLY APPLICATIONS • WKB and numerical • free motion • expansions WSCQS, CRM Montreal

  7. 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

  8. 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

  9. II. Selected concepts of solvability WSCQS, CRM Montreal

  10. Sample menu • ODE solvability • symmetry reduction • polynomial solvability • SUSY partnership • QES • Hill determinants • asymptotic series • exceptional PDE WSCQS, CRM Montreal

  11. 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

  12. III. The emergence of less usual characteristics of solvability for PT symmetric Hamiltonians WSCQS, CRM Montreal

  13. 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

  14. 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,…) WSCQS, CRM Montreal

  15. III. 1. Solutions over curved complex contours • Without PT symmetry (QES, sextic osc.) [BT ‘93] • With PT symmetry WSCQS, CRM Montreal

  16. 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

  17. 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

  18. 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] WSCQS, CRM Montreal

  19. III. 2. D > 1 regularization recipe WSCQS, CRM Montreal

  20. III. 2. PT D > 1 regularization recipe solutions over the straight complex lines of coordinates • perturbative • regularized: • systematic: WSCQS, CRM Montreal

  21. 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

  22. 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] WSCQS, CRM Montreal

  23. III. 3. Models solvable via classical OG polynomials: • PT modified • non-Hermitian • systematic methods • re-interpretations WSCQS, CRM Montreal

  24. 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

  25. 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] WSCQS, CRM Montreal

  26. III. 4. The methods of SUSY partnership • starting from squre well: • using alternative, PT specific SUSY schemes: • referring to Lie algebras WSCQS, CRM Montreal

  27. 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

  28. 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

  29. 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

  30. 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 WSCQS, CRM Montreal

  31. III. 8. PDE cases • the Winternitzian superintegrable V’s: • the Calogerian three-body laboratory: WSCQS, CRM Montreal

  32. 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 WSCQS, CRM Montreal

  33. 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 WSCQS, CRM Montreal

  34. V. Summary • mathematics in interplay with physics • immediate applicability WSCQS, CRM Montreal

  35. 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

  36. 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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