1. Non-Hermitian Hamiltonians of Lie algebraic type Paulo Eduardo Goncalves de Assis City University London

2. Non Hermitian Hamiltonians Real spectra ? Hermiticity: sufficient but not necessary

3. Non Hermitian Hamiltonians -W.Heisenberg, Quantum theory of fields and elementary particles, Rev.Mod.Phys. 29 (1957) 269. -J.L.Cardy and R.L.Sugar, Reggeon field theory on a lattice, Phys.rev. D12 (1975) 2514. -F.G.Scholtz, H.B.Geyer and F. Hahne, Quasi-Hermitian operators in Quantum Mechanics and the variational principle, Ann. Phys. 213 (1992) 74. -T.Hollowood, Solitons in affine Toda field theory, Nucl.Phys. B384 (1992) 523. -D.I.Olive, N.Turok, and J.W.R.Underwood, Solitons and the energy momentum tensor for affine Toda theory, Nucl.Phys. B401 (1993) 663. -C.M.Bender and S.Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetries, Phys.Rev.Lett. 80 (1998) 5243. -C.Korff and R.A.Weston, PT Symmetry on the Lattice: The Quantum Group invariant XXZ spin-chain, J.Phys. A40 (2007) 8845. -A.K.Das, A.Melikyan and V.O.Rivelles, The S-Matrix of the Faddeev- Reshetikhin model, diagonalizability and PT-symmetry, J.H.E.P. 09 (2007) 104. Real spectra ? Hermiticity: sufficient but not necessary

4. When is the spectrum real? PT-symmetry: Invariance under parity and time-reversal Anti-unitarity:

5. When is the spectrum real? PT-symmetry: Invariance under parity and time-reversal Anti-unitarity: Unbroken PT : Not only the Hamiltonian but also the eigenstates are invariant under PT

6. When is the spectrum real? PT-symmetry: Invariance under parity and time-reversal Anti-unitarity: Unbroken PT : Not only the Hamiltonian but also the eigenstates are invariant under PT

7. When is the spectrum real? PT-symmetry: Invariance under parity and time-reversal Anti-unitarity: Unbroken PT : Not only the Hamiltonian but also the eigenstates are invariant under PT

8. Hermitian Hamiltonian real eigenvalues complex eigenvalues Non-Hermitian Hamiltonian

9. Hermitian Hamiltonian real eigenvalues complex eigenvalues Non-Hermitian Hamiltonian

10. Hermitian Hamiltonian real eigenvalues complex eigenvalues Non-Hermitian Hamiltonian IF

11. Hermitian Hamiltonian real eigenvalues complex eigenvalues Non-Hermitian Hamiltonian IF Isospectral transformation: non-Hermitian Hamiltonian Hermitian counterparts. map

12. orthogonal basis bi-orthogonal basis Eigenstates of h and H are essentially different:

13. orthogonal basis bi-orthogonal basis Eigenstates of h and H are essentially different:

14. orthogonal basis bi-orthogonal basis Eigenstates of h and H are essentially different: bi-orthogonality as non trivial metric Similarity transformation as a change in the metric Pseudo-Hermiticity

15. H is Hermitian with respect to the new metric.

16. all observables transform

17. H is Hermitian with respect to the new metric. all observables transform non-Hermitian Hamiltonian Xambiguous physics

18. What is being studied? Non-Hermitian Hamiltonians of Lie algebraic type, P.E.G.Assis and A.Fring, in preparation. non Hermitian Hamiltonian with real eigenvalues: constraints, metrics, Hermitian counterparts. eigenvalues and eigenfunctions when possible.

19. What is being studied? Hamiltonians are formulated in terms of Lie algebras. –General approachdifferent models –Successful framework for integrable or solvable models Non-Hermitian Hamiltonians of Lie algebraic type, P.E.G.Assis and A.Fring, in preparation. non Hermitian Hamiltonian with real eigenvalues: constraints, metrics, Hermitian counterparts. eigenvalues and eigenfunctions when possible.

20. sl 2 (R)-Hamiltonians

22. Representation: invariant Quasi-exactly solvable Turbiner et al

23. sl 2 (R)-Hamiltonians Representation: invariant Quasi-exactly solvable Turbiner et al

24. sl 2 (R)-Hamiltonians PT-symmetrize Hermitian conjugates of J’s cannot be written in terms of them Representation: invariant Quasi-exactly solvable Turbiner et al

25. su(1,1)-Hamiltonians

27. C.Quesne, J.Phys A40, (2007) F745.

28. su(1,1)-Hamiltonians C.Quesne, J.Phys A40, (2007) F745.

29. su(1,1)-Hamiltonians C.Quesne, J.Phys A40, (2007) F745.

30. su(1,1)-Hamiltonians C.Quesne, J.Phys A40, (2007) F745. D.P.Musumbu, H.B.Geyer, W.D.Heiss, J.Phys A39, (2007) F75. Swanson Hamiltonian

31. su(1,1)-Hamiltonians Holstein-PrimakoffTwo-mode

32. su(1,1)-Hamiltonians

33. Hermitian partners Metric Ansatz:

34. Hermitian partners Metric Ansatz: constraints

35. Hermitian partners Metric Ansatz: constraints exact action of the metric on the generators

36. Hermitian partners Metric Ansatz: constraints exact action of the metric on the generators

37. Recall:

39. Different possible subcases, e.g., purely linear or purely bilinear.

40. Recall: Different possible subcases, e.g., purely linear or purely bilinear. Large variety of models may be mapped onto a Hermitian couterpart.

41. Metric depends either only on momentum or coordinate operators. Recall: Different possible subcases, e.g., purely linear or purely bilinear. Large variety of models may be mapped onto a Hermitian couterpart.

42. Reducible Hamiltonian Constraints

43. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases:

44. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-negative: µ - = µ -- = µ 0- = 0

45. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-positive: µ + = µ ++ = µ +0 = 0 non-negative: µ - = µ -- = µ 0- = 0

46. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 non-negative: µ - = µ -- = µ 0- = 0

47. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 purely linear: µ ++ = µ -- = 0 and µ +0 = µ 0- = 0 non-negative: µ - = µ -- = µ 0- = 0

48. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 purely linear: µ ++ = µ -- = 0 and µ +0 = µ 0- = 0 non-negative: µ - = µ -- = µ 0- = 0

49. Reducible Hamiltonian Constraints Appropriate choices lead to the interesting sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 purely linear: µ ++ = µ -- = 0 and µ +0 = µ 0- = 0 eigenvalues and eigenstates non-negative: µ - = µ -- = µ 0- = 0

50. Non-reducible Hamiltonian Constraints

51. Non-reducible Hamiltonian Constraints New solutions for limited sub cases: non-positive: µ + = µ ++ = µ +0 = 0non-negative: µ - = µ -- = µ 0- = 0

52. Non-reducible Hamiltonian Constraints New solutions for limited sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 purely linear: µ ++ = µ -- = 0 and µ +0 = µ 0- = 0 non-negative: µ - = µ -- = µ 0- = 0

53. Non-reducible Hamiltonian Constraints New solutions for limited sub cases: non-positive: µ + = µ ++ = µ +0 = 0 purely bilinear: µ + = µ - = 0 purely linear: µ ++ = µ -- = 0 and µ +0 = µ 0- = 0 non-negative: µ - = µ -- = µ 0- = 0

54. Eigenstates and Eigenvalues? So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra. DiagonalizationHamiltonian in Harmonic Oscillator form

55. Eigenstates and Eigenvalues? So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra. DiagonalizationHamiltonian in Harmonic Oscillator form M.S.Swanson, J.Math.Phys 45, (2004) 585. Generalized Bogoliubov: New Operators

56. Eigenstates and Eigenvalues? So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra. DiagonalizationHamiltonian in Harmonic Oscillator form M.S.Swanson, J.Math.Phys 45, (2004) 585. Generalized Bogoliubov: New Operators vacuum eigenstates and eigenvalues of

57. Eigenstates and Eigenvalues? So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra. DiagonalizationHamiltonian in Harmonic Oscillator form M.S.Swanson, J.Math.Phys 45, (2004) 585. Generalized Bogoliubov: New Operators vacuum eigenstates and eigenvalues of

58. More constraints for Ñ dependent Hamiltonian

59. metric-constraints and solvability-constraints combined ? Exact: eigenvalues, eigenstates and suitable metric

60. More constraints for Ñ dependent Hamiltonian metric-constraints and solvability-constraints combined ? Exact: eigenvalues, eigenstates and suitable metric Transition amplitudes

61. Conclusions Calculated conditions and appropriate metrics with respect to which a large class of non Hermitian Hamiltonians bilinear in su(1,1) generators can be considered Hermitian. The same non Hermitian Hamiltonians could be diagonalized and it was shown, whithout metrics, that although being non Hermitian real eigenvalues do occur. Possibility to have complete knowledge of spectra, eigenstates (both of Hermitian and non Hermitian Hamiltonians) and meaningful metrics. Hamiltonians explored are very general, allowing interesting models as sub cases. Other algebras may be employed.