Infinite series of time-dependent Dyson maps

TL;DR

This paper introduces a novel scheme generating an infinite series of time-dependent Dyson maps linking non-Hermitian and Hermitian Hamiltonians, utilizing symmetries and invariants, exemplified by a PT-symmetric oscillator system.

Contribution

It presents a new method for constructing multiple Dyson maps for non-Hermitian Hamiltonians using symmetry principles and invariants, with detailed application to a coupled oscillator system.

Findings

Infinite series of Dyson maps can be systematically constructed.

Symmetries and Lewis-Riesenfeld invariants facilitate explicit Dyson map construction.

Application to a PT-symmetric oscillator system demonstrates the scheme's effectiveness.

Abstract

We propose and explore a scheme that leads to an infinite series of time- dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying sym- metries responsible for this feature respected by various Lewis-Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric oper- ators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion