On the pseudo-Hermitian nondiagonalizable Hamiltonians

We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.

arXiv:quant-ph/0211161Journal of Mathematical PhysicsSep 25, 200344 citations

A non-Hermitian Hamiltonians: Pseudo 𝒫𝒯-symmetry

Mustapha Maamache, Nour El Houda Absi

International Journal of Modern Physics ASep 11, 20250 citations

Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians

Ali Mostafazadeh

We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.

arXiv:math-ph/0207009Journal of Mathematical PhysicsNov 25, 2002102 citations

Algebraic method for pseudo-Hermitian Hamiltonians

Junqing Li, Yan-Gang Miao, Zhao Xue

Jul 25, 20115 citations

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

Alberto Blasi, G. Scolarici, Luigi Solombrino

We extend the definition of generalized parity $P$ , charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$ -reflecting symmetry which generates the $P$ -pseudounitary and the $P$ -pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$ -unitary evolution of a physical system are also given.We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$-unitary evolution of a physical system are also given.

Oct 16, 200315 citations

We extend the definition of generalized parity

P

, charge-conjugation

C

and time-reversal

T

operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that

TP

and

CTP

are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian

H

to admit a

P

-reflecting symmetry which generates the

P

-pseudounitary and the

P

-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the

P

-unitary evolution of a physical system are also given.