Transcript Non-Hermitian Hamiltonians of Lie algebraic type

Non-Hermitian Hamiltonians of
Lie algebraic type
Paulo Eduardo Goncalves de Assis
City University London
Non Hermitian Hamiltonians
Real spectra ?
Hermiticity: sufficient but not necessary
Non Hermitian Hamiltonians
Real spectra ?
Hermiticity: sufficient but not necessary
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tensor for affine Toda theory, Nucl.Phys. B401 (1993) 663.
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having PT symmetries, Phys.Rev.Lett. 80 (1998) 5243.
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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
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:
orthogonal
basis
bi-orthogonal
basis
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?
• 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
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
Representation:
invariant
Quasi-exactly solvable
Turbiner et al
PT-symmetrize
Hermitian conjugates of J’s cannot be written in terms of them
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
Swanson Hamiltonian
C.Quesne, J.Phys A40, (2007) F745.
D.P.Musumbu, H.B.Geyer, W.D.Heiss, J.Phys A39, (2007) F75.
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.
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.
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-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
purely bilinear: µ+ = µ- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Reducible Hamiltonian
Constraints
Appropriate choices lead to the interesting sub cases:
non-negative:
µ- = µ-- = µ0- = 0
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-reducible Hamiltonian
Constraints
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = µ0- = 0
Non-reducible Hamiltonian
Constraints
New solutions for limited sub cases:
non-negative:
µ- = µ-- = µ0- = 0
non-positive: µ+ = µ++ = µ+0 = 0
purely bilinear: µ+ = µ- = 0
purely linear: µ++ = µ-- = 0 and
µ+0 = 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
Generalized Bogoliubov:
New Operators
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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
Generalized Bogoliubov:
New Operators
eigenstates and eigenvalues of
vacuum
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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
Generalized Bogoliubov:
New Operators
eigenstates and eigenvalues of
vacuum
M.S.Swanson, J.Math.Phys 45, (2004) 585.
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.