Exactly solvable time-dependent non-Hermitian quantum systems from point transformations

TL;DR

This paper introduces a method using complex point transformations to construct exactly solvable, time-dependent non-Hermitian quantum systems, including invariants, Dyson maps, and metrics, enabling explicit solutions.

Contribution

It presents a novel approach employing point transformations to generate and solve time-dependent non-Hermitian quantum systems with explicit invariants and mappings.

Findings

Constructed non-Hermitian invariants for time-dependent systems

Derived Dyson maps and metric operators explicitly

Provided solutions to time-dependent Schrödinger equations

Abstract

We demonstrate that complex point transformations can be used to construct non-Hermitian first integrals, time-dependent Dyson maps and metric operators for non-Hermitian quantum systems. Initially we identify a point transformation as a map from an exactly solvable time-independent system to an explicitly time-dependent non-Hermitian Hamiltonian system. Subsequently we employ the point transformation to construct the non-Hermitian time-dependent invariant for the latter system. Exploiting the fact that this invariant is pseudo-Hermitian, we construct a corresponding Dyson map as the adjoint action from a non-Hermitian to a Hermitian invariant, thus obtaining solutions to the time-dependent Dyson and time-dependent quasi-Hermiticity equation together with solutions to the corresponding time-dependent Schr\"odinger equation.