On the invariant method for the time-dependent non-Hermitian Hamiltonians

Abstract

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator, which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

arXiv comment11 pages

References in corpus (21)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
• Time evolution of non-Hermitian Hamiltonian systems
• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems
• The non-Hermitian Swanson model with a time-dependent metric
• Geometric Phase in PT-Symmetric Quantum Mechanics
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Two-Point Green's Function in PT-Symmetric Theories
• Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''
• Which operator generates time evolution in Quantum Mechanics?
• Reply to Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''
• Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

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