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Energy-dependent Hamiltonians and pseudo-Hermiticity

Energy-dependent Hamiltonians and pseudo-Hermiticity. Energy-dependent Hamiltonians and pseudo-Hermiticity. M. Znojil, H. B íla and, V. Jakubský (NPI, Rez near Prague). Energy-dependent Hamiltonians and pseudo-Hermiticity. M. Znojil, H. B íla and, V. Jakubský

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Energy-dependent Hamiltonians and pseudo-Hermiticity

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  1. Energy-dependent Hamiltonians and pseudo-Hermiticity Villa Lanna, Prague

  2. Energy-dependent Hamiltonians and pseudo-Hermiticity M. Znojil, H. Bíla and, V. Jakubský (NPI, Rez near Prague) Villa Lanna, Prague

  3. Energy-dependent Hamiltonians and pseudo-Hermiticity M. Znojil, H. Bíla and, V. Jakubský (NPI, Rez near Prague) A new application of quantum theorywhere non-Hermitian Hamiltonians are allowed [Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics] Villa Lanna, Prague

  4. TABLEOFCONTENTS I. ENERGY-DEPENDENCE II. MATHEMATICAL CHALLENGES III. ROLE OF PSEUDO-HERMITICITY IV. SUMMARY OF RESULTS Villa Lanna, Prague

  5. I. Energy dependence • WHAT IT IS • parametrized H = H(z) • equation H(z) |y(z)> = E(z) |y(z)> • nonlinearity z = E(z), solutions a-indexed • set = non-orthogonal, <a | b> = 1/R Villa Lanna, Prague

  6. I. Energy dependence • WHAT IT IS • parametrized H = H(z) • equation H(z) |y(z)> = E(z) |y(z)> • nonlinearity z = E(z), solutions a-indexed • set = non-orthogonal, <a | b> = 1/R • WHY IS IT USED/USEFUL • the Feshbach’s effective Hamiltonians • unifying physics below and above thresholds • QES kets: N-dependence = E-dependence Villa Lanna, Prague

  7. II. A complication or simplification? • THE IDEA OF SIMPLICITY • “effective” means less degrees of freedom • discrete freedom at every level Villa Lanna, Prague

  8. II. A complication or simplification? • THE IDEA OF SIMPLICITY • “effective” means less degrees of freedom • discrete freedom at every level • MODIFIED MATHEMATICS • I = S |a > R(a,b)<b| = S |a > <<a| = S|b>><b| Villa Lanna, Prague

  9. II. A complication or simplification? • THE IDEA OF SIMPLICITY • “effective” means less degrees of freedom • discrete freedom at every level • MODIFIED MATHEMATICS • I = S |a > R(a,b)<b| = S |a > <<a| = S|b>><b| • two possible “spectral decompositions” • left: L = S |a > R(a,b) E(b)<b| = S|b>> E(b)<b| • right: K = S |a > E(a) R(a,b)<b| = S|a> E(a)<<a| Villa Lanna, Prague

  10. III. Pseudo-Hermiticity of K and L • K = S|a> E(a)<<a| = h. c. of L = S|b>> E(b)<b| • One can find the new metric: • x L = K x Villa Lanna, Prague

  11. III. Pseudo-Hermiticity of K and L • K = S|a> E(a)<<a| = h. c. of L = S|b>> E(b)<b| • One can find the new metric: • x L = K x • and explicirt formulae: • x = S | a > r < a | = Hermitian • L /x = (1/x) K , 1/x = S | a >> 1/r << a |. Villa Lanna, Prague

  12. IV.(a) Pseudo-Hermitian H(z) allowed. • WHAT IT MEANS: • doubling of eigenstates: • right-action H(z) |y(z)> = E(z) |y(z)> • and left-action eq. <[y(z)| H(z) = E(z) <[y(z)| Villa Lanna, Prague

  13. IV.(a) Pseudo-Hermitian H(z) allowed. • WHAT IT MEANS: • doubling: right-action H(z) |y(z)> = E(z) |y(z)> • and left-action eq. <[y(z)| H(z) = E(z) <[y(z)| • solutions a- (=upper) and [a- (=lower-) indexed • sets = bi-orthogonal, with <[a | a> = 1 Villa Lanna, Prague

  14. IV.(a) Pseudo-Hermitian H(z) allowed. • WHAT IT MEANS: • doubling: right-action H(z) |y(z)> = E(z) |y(z)> • and left-action eq. <[y(z)| H(z) = E(z) <[y(z)| • solutions a- (=upper) and [a- (=lower-) indexed • sets = bi-orthogonal, with <[a | a> = 1 • and bi-complete, with I = S | a><[a | • with h.c. symmetry in I = S | a]><a | etc. Villa Lanna, Prague

  15. IV. (b) Pseudo-Hermitian H(z) assumed • H(z) = S|y(z)> E(z) <[y(z)| • h. c. = S |y(z) ]> E(z) <y(z)| = h H (1/h) Villa Lanna, Prague

  16. IV. (b) Pseudo-Hermitian H(z) assumed • H(z) = S|y(z)> E(z) <[y(z)| • h. c. = S |y(z) ]> E(z) <y(z)| = h H (1/h) • postulating h = S |y(z) ]> q <[y(z)| • and getting (1/h) = S |y(z)> (1/q) <y(z)| . Villa Lanna, Prague

  17. IV. (b) Pseudo-Hermitian H(z) assumed • H = S|y(z)> E(z) <[y(z)| • h. c. = S |y(z) ]> E(z) <y(z)| = h H (1/h) • postulating h = S |y(z) ]> q <[y(z)| • and getting (1/h) = S |y(z)> (1/q) <y(z)| . • WHY IS IT SO EXCITING? • E-dependence = N-dependence (QES = example) • new representants = linear, E-independent Villa Lanna, Prague

  18. IV. (c) Pseudo-Hermitian K and L kept • SIMPLICITY (take just bras lowered: [ ): • K = S|a> E(a)<<[a| Villa Lanna, Prague

  19. IV. (c) Pseudo-Hermitian K and L kept • SIMPLICITY (take just bras lowered: [ ): • K = S|a> E(a)<<[a| • K(h. c.) = S|b ]>> E(b)<b| = m K (1/m) Villa Lanna, Prague

  20. IV. (c) Pseudo-Hermitian K and L kept • SIMPLICITY (take just bras lowered: [ ): • K = S|a> E(a)<<[a| • K(h. c.) = S|b ]>> E(b)<b| = m K (1/m) • assuming m = S |a]>> r <<[a| • and getting (1/m) = S |a> (1/r) <a| . Villa Lanna, Prague

  21. IV. (c) Pseudo-Hermitian K and L kept • SIMPLICITY (take just bras lowered: [ ): • K = S|a> E(a)<<[a| • K(h. c.) = S|b ]>> E(b)<b| = m K (1/m) • assuming m = S |a]>> r <<[a| • and getting (1/m) = S |a> (1/r) <a| . • L = S|a >> E(a)< [a| • L(h. c.) = S|b ]> E(b)<< b| = n L (1/n) Villa Lanna, Prague

  22. IV. (c) Pseudo-Hermitian K and L kept • SIMPLICITY (take just bras lowered: [ ): • K = S|a> E(a)<<[a| • K(h. c.) = S|b ]>> E(b)<b| = m K (1/m) • assuming m = S |a]>> r <<[a| • and getting (1/m) = S |a> (1/r) <a| . • L = S|a >> E(a)< [a| • L(h. c.) = S|b ]> E(b)<< b| = n L (1/n) • .n=S |a]> s <[a| and (1/n)=S |a>> (1/s) <<a| Villa Lanna, Prague

  23. IV. (d) Application: Klein Gordon • HAMILTONIANS? FESHBACH - VILLARS! • (two-by-two matrices) Villa Lanna, Prague

  24. IV. (d) Application: Klein Gordon • HAMILTONIANS? FESHBACH - VILLARS! • (two-by-two matrices) • POTENTIALS? SIMPLE MODELS m=m(x,E) • (exactly solvable examples) • interpretation: transitions to lower energies Villa Lanna, Prague

  25. IV. (d) Application: Klein Gordon • HAMILTONIANS? FESHBACH - VILLARS! • (two-by-two matrices) • POTENTIALS? SIMPLE MODELS m=m(x,E) • (exactly solvable examples) • interpretation: transitions to lower energies • FURTHER MERITS, E.G., SUPERSYMMETRY. • (Darboux factorization techniques) Villa Lanna, Prague

  26. IV. (e) Quasi-exactly solvable models • A BRIEF OUTLINE OF HISTORY: • unknown soldiers of sci [C ’68 etc] • ODE [H ’72, F ’79 etc] • Hill determinants and CF’s [SBD ’78 etc] • strongly singular models [Z ‘82 etc] • Lie algebras [e.g., T ’88] • Bethe ansatz [U ’93] • extensions to PT: quartic [BB ’98, Z ‘99 etc] Villa Lanna, Prague

  27. IV. (e) Quasi-exactly solvable PT models • A BRIEF OUTLINE OF THEIR NEAREST FUTURE: • (a) old sol’s revisited [new structures - wedges] • (b) new QES classes [decadic] • (c) quasi-bases [Z’02] Villa Lanna, Prague

  28. V. Summary • mathematics in interplay with physics • (parallels between pseudo- and Hermitian): • (a) unitarity <-> auxiliary metric P in Hilbert space • (b) Jordan blocks <-> unavoided crossings of levels • (c ) quasi-parity <-> PCT symmetry • immediate applicability • (a) Winternitzian models: • non-equivalent Hermitian limits • (b) Calogerian models: • open questions: new types of tunnelling Villa Lanna, Prague

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