Eur. Phys. J. Plus (2017) 132: 258
https://doi.org/10.1140/epjp/i2017-11524-7

Regular Article

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

¹ Theoretical Physics Laboratory, Department of Physics, University of Jijel, BP 98 Ouled Aissa, 18000, Jijel, Algeria
² Laboratoire de Physique Quantique et Systèmes Dynamiques, Faculté des Sciences, Université Ferhat Abbas Sétif 1, 19000, Sétif, Algeria

^* e-mail: bounames@univ-jijel.dz

Received: 24 March 2017
Accepted: 1 May 2017
Published online: 12 June 2017

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 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 generally not quasi-Hermitian and does not define an observable of the system but 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.