Non-Hermitian Hamiltonians of Lie algebraic type

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Non-Hermitian Hamiltonians of Lie algebraic type

• Paulo Eduardo Goncalves de Assis
• City University London

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Non Hermitian Hamiltonians
Real spectra ?
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Non Hermitian Hamiltonians
Real spectra ?

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  theory, Nucl.Phys. B384 (1992) 523.
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  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.
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  S-Matrix of the Faddeev-Reshetikhin model,
  diagonalizability and PT-symmetry, J.H.E.P. 09
  (2007) 104.

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When is the spectrum real?

• PT-symmetry Invariance under parity and
  time-reversal

Anti-unitarity
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When is the spectrum real?

• PT-symmetry Invariance under parity and
  time-reversal

Anti-unitarity
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When is the spectrum real?

• PT-symmetry Invariance under parity and
  time-reversal

Anti-unitarity
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When is the spectrum real?

• PT-symmetry Invariance under parity and
  time-reversal

Anti-unitarity
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real eigenvalues

• Hermitian Hamiltonian

Non-Hermitian Hamiltonian
complex eigenvalues
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real eigenvalues

• Hermitian Hamiltonian

Non-Hermitian Hamiltonian
complex eigenvalues
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real eigenvalues

• Hermitian Hamiltonian

Non-Hermitian Hamiltonian
complex eigenvalues
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real eigenvalues

• Hermitian Hamiltonian

Non-Hermitian Hamiltonian
complex eigenvalues
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Eigenstates of h and H are essentially different
orthogonal basis
bi-orthogonal basis
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Eigenstates of h and H are essentially different
orthogonal basis
bi-orthogonal basis
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Eigenstates of h and H are essentially different
orthogonal basis
bi-orthogonal basis
bi-orthogonality as non trivial metric
Similarity transformation as a change in the
metric
Pseudo-Hermiticity
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H is Hermitian with respect to the new metric.
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H is Hermitian with respect to the new metric.
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H is Hermitian with respect to the new metric.
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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.

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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.

• Hamiltonians are formulated in terms of Lie
  algebras.
• General approach different models
• Successful framework for integrable or solvable
  models

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sl2(R)-Hamiltonians
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sl2(R)-Hamiltonians
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sl2(R)-Hamiltonians
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sl2(R)-Hamiltonians
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sl2(R)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
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su(1,1)-Hamiltonians
Holstein-Primakoff
Two-mode
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su(1,1)-Hamiltonians
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Hermitian partners

• Metric Ansatz

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Hermitian partners

• Metric Ansatz

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Hermitian partners

• Metric Ansatz

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Hermitian partners

• Metric Ansatz

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Recall
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Recall
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Recall
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Recall
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Recall

• Metric depends either only on momentum or
  coordinate operators.

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• Reducible Hamiltonian

Constraints
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• Reducible Hamiltonian

Constraints
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
non-positive µ µ µ0 0
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
non-positive µ µ µ0 0
purely bilinear µ µ- 0
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
non-positive µ µ µ0 0
purely bilinear µ µ- 0
purely linear µ µ-- 0 and µ0 µ0-
0
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
non-positive µ µ µ0 0
purely bilinear µ µ- 0
purely linear µ µ-- 0 and µ0 µ0-
0
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• Reducible Hamiltonian

Constraints
non-negative µ- µ-- µ0- 0
non-positive µ µ µ0 0
purely bilinear µ µ- 0
purely linear µ µ-- 0 and µ0 µ0-
0
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• Non-reducible Hamiltonian

Constraints
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• Non-reducible Hamiltonian

Constraints
New solutions for limited sub cases
non-positive µ µ µ0 0
non-negative µ- µ-- µ0- 0
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• Non-reducible Hamiltonian

Constraints
New solutions for limited sub cases
non-positive µ µ µ0 0
non-negative µ- µ-- µ0- 0
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• Non-reducible Hamiltonian

Constraints
New solutions for limited sub cases
non-positive µ µ µ0 0
non-negative µ- µ-- µ0- 0
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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
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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
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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
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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
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More constraints for Ñ dependent Hamiltonian
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More constraints for Ñ dependent Hamiltonian
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More constraints for Ñ dependent Hamiltonian
Transition amplitudes
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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.