non-hermitian hamiltonians of lie algebraic type paulo eduardo goncalves de assis city university...

Non-Hermitian Hamiltonians of Non-Hermitian Hamiltonians of Lie algebraic typeLie algebraic type

Paulo Eduardo Goncalves de AssisPaulo Eduardo Goncalves de Assis

City University LondonCity University London

Non Hermitian Hamiltonians

Real spectra ?

Hermiticity: sufficient but not necessary

Non Hermitian Hamiltonians

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- 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

When is the spectrum real?

• PT-symmetry: Invariance under parity and time-reversal

Anti-unitarity:

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

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

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

Hermitian Hamiltonian real eigenvalues

complex eigenvalues Non-Hermitian Hamiltonian

Hermitian Hamiltonian real eigenvalues

complex eigenvalues Non-Hermitian Hamiltonian

Hermitian Hamiltonian real eigenvalues

complex eigenvalues Non-Hermitian Hamiltonian

IF

Hermitian Hamiltonian real eigenvalues

complex eigenvalues Non-Hermitian Hamiltonian

IF

Isospectral transformation:

non-Hermitian Hamiltonian Hermitian counterparts.

map

orthogonal

basis

bi-orthogonal

basis

Eigenstates of h and H are essentially different:

orthogonal

basis

bi-orthogonal

basis

Eigenstates of h and H are essentially different:

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

H is Hermitian with respect to the new metric.

H is Hermitian with respect to the new metric.

all observables transform

H is Hermitian with respect to the new metric.

all observables transform

non-Hermitian Hamiltonian X ambiguous physics

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.

What is being studied?

• Hamiltonians are formulated in terms of Lie algebras.

– General approach different 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.

sl2(R)-Hamiltonians

sl2(R)-Hamiltonians

sl2(R)-Hamiltonians

Representation:

invariant

Quasi-exactly solvable Turbiner et al

sl2(R)-Hamiltonians

Representation:

invariant

Quasi-exactly solvable Turbiner et al

sl2(R)-Hamiltonians

PT-symmetrize

Hermitian conjugates of J’s cannot be written in terms of them

Representation:

invariant

Quasi-exactly solvable Turbiner et al

su(1,1)-Hamiltonians

su(1,1)-Hamiltonians

su(1,1)-Hamiltonians

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

su(1,1)-Hamiltonians

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

su(1,1)-Hamiltonians

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

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

su(1,1)-Hamiltonians

Holstein-Primakoff Two-mode

su(1,1)-Hamiltonians

Hermitian partners

• Metric Ansatz:

Hermitian partners

• Metric Ansatz:

constraints

Hermitian partners

• Metric Ansatz:

constraints

exact action of the metric on the generators

Hermitian partners

• Metric Ansatz:

constraints

exact action of the metric on the generators

Recall:

Recall:

Recall:

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

Recall:

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

• Large variety of models may be mapped onto a Hermitian couterpart.

• 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.

Reducible Hamiltonian

Constraints

Reducible Hamiltonian

Constraints

Appropriate choices lead to the interesting sub cases:

Reducible Hamiltonian

Constraints

Appropriate choices lead to the interesting sub cases:

non-negative: µ- = µ-- = µ0- = 0

Reducible Hamiltonian

Constraints

Appropriate choices lead to the interesting sub cases:

non-positive: µ+ = µ++ = µ+0 = 0non-negative: µ- = µ-- = µ0- = 0

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-- = µ0- = 0

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 = µ0- = 0

non-negative: µ- = µ-- = µ0- = 0

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 = µ0- = 0

non-negative: µ- = µ-- = µ0- = 0

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 = µ0- = 0

eigenvalues and eigenstates

non-negative: µ- = µ-- = µ0- = 0

Non-reducible Hamiltonian

Constraints

Non-reducible Hamiltonian

Constraints

New solutions for limited sub cases:

non-positive: µ+ = µ++ = µ+0 = 0non-negative: µ- = µ-- = µ0- = 0

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 = µ0- = 0

non-negative: µ- = µ-- = µ0- = 0

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 = µ0- = 0

non-negative: µ- = µ-- = µ0- = 0

Eigenstates and Eigenvalues?

• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.

Diagonalization Hamiltonian in Harmonic Oscillator form

Eigenstates and Eigenvalues?

• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.

Diagonalization Hamiltonian in Harmonic Oscillator form

M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:

New Operators

Eigenstates and Eigenvalues?

• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.

Diagonalization Hamiltonian in Harmonic Oscillator form

M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:

New Operators

vacuum

eigenstates and eigenvalues of

Eigenstates and Eigenvalues?

• So far, we have only calculated metrics and discussed under which conditions non-Hermitian Hamiltonians possess real spectra.

Diagonalization Hamiltonian in Harmonic Oscillator form

M.S.Swanson, J.Math.Phys 45, (2004) 585.Generalized Bogoliubov:

New Operators

vacuum

eigenstates and eigenvalues of

More constraints for Ñ dependent Hamiltonian

More constraints for Ñ dependent Hamiltonian

metric-constraints and solvability-constraints combined ?

Exact: eigenvalues, eigenstates and suitable metric

More constraints for Ñ dependent Hamiltonian

metric-constraints and solvability-constraints combined ?

Exact: eigenvalues, eigenstates and suitable metric

Transition amplitudes

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.