Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures

Abstract

We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectra, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola–Kanai model as a particular case. Explicit solutions are given in both the classical and quantum pictures.

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

“…In addition, the new indices are not required to be PTsymmetric to allow all-real eigenvalues in their point spectrum. The transition from these results to the time-dependent case is straightforward [54], where PT-symmetric structures find interesting applications [55,56].…”

Exact solution for the time dependent non-Hermitian generalized Swanson oscillator

“…The starting point is the non-Hermitian time-independent Hamiltonian of the generalized Swanson oscillator defined by [20,21] 1…”

“…Exact solutions have been found for the time dependent case by using Lewis and Riesenfeld time dependent invariants and time-dependent non-unitary transformation [18,19]. Among recent works, Zelaya et al [20] have presented an extension of the Swanson model with arbitrary time dependent real parameters; there, the Schrödinger equation for a special case of this model produces a generalization of the Caldirola-Kanai oscillator. In particular, in the concrete generalization of the Swanson oscillator provided in Reference [20], it is missing a study of the non-Hermitian case defined by arbitrary time dependent complex-valued functions.…”

“…Among recent works, Zelaya et al [20] have presented an extension of the Swanson model with arbitrary time dependent real parameters; there, the Schrödinger equation for a special case of this model produces a generalization of the Caldirola-Kanai oscillator. In particular, in the concrete generalization of the Swanson oscillator provided in Reference [20], it is missing a study of the non-Hermitian case defined by arbitrary time dependent complex-valued functions. Therefore, influenced by Zelaya's et al seminal landmark article, we proceed in this work to expand the above analysis to study time dependent non-unitary transformations that allow to obtain exact solutions of the generalized Swanson oscillator.…”

Quasienergy operators and generalized squeezed states for systems of trapped ions

“…The quantum Hamiltonian and the evolution operator can be expressed in terms of the Lie algebra generators, which allows one to obtain the propagator [31]. The approach can be applied to investigate quantum dynamics for charged particles in electromagnetic fields [32], with examples that span linear electrodynamic traps, the Kanai-Caldirola forced harmonic oscillator [33], a charged particle in either oscillating magnetic field or constant magnetic field and oscillating electric field (combined Paul and Penning trap). Late investigations demonstrate that by using the Lewis-Riesenfeld invariant theory and Fock states, the quantum dynamics of a particle with time-varying mass in a Paul trap can be investigated.…”