On the pseudo-Hermitian nondiagonalizable Hamiltonians

Abstract

We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, showing that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a semidefinite inner product, and that the pseudo-Hermiticity property is equivalent to the existence of an antilinear involutory symmetry. Moreover, 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.

22 pages

References in corpus (7)

• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• Pseudo-Hermiticity, weak pseudo-Hermiticity and eta-orthogonality condition
• Erratum: Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians [J. Math. Phys. 43, 6343 (2002); math-ph/0207009]
• Pseudohermitian Hamiltonians, time-reversal invariance and Kramers degeneracy
• The challenge of non-hermitian structures in physics

Cited by in corpus (26)

• Pseudo-Hermitian Representation of Quantum Mechanics
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
• Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics
• Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
• Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
• A Physical Realization of the Generalized PT-, C-, and CPT-Symmetries and the Position Operator for Klein-Gordon Fields
• Non-anti-hermitian quaternionic quantum mechanics
• Erratum: Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians [J. Math. Phys. 43, 6343 (2002); math-ph/0207009]
• Virial theorem and generalized momentum in quaternic quantum mechanics
• $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum
• Quantum centrality testing on directed graphs via PT-symmetric quantum walks
• Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries
• QT-Symmetry and Weak Pseudo-Hermiticity
• Irreducible second order SUSY transformations between real and complex potentials
• Alternative descriptions and bipartite compound quantum systems
• Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Alternative Descriptions in Quaternionic Quantum Mechanics
• Exact propagators for complex SUSY partners of real potentials
• Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
• PT symmetry as a necessary and sufficient condition for unitary time evolution
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Surprising spectra of PT-symmetric point interactions
• Some Remarks on Quantum Brachistochrone