Lewis–Riesenfeld invariants for PT-symmetrically coupled oscillators from two-dimensional point transformations and Lie algebraic expansions

Abstract

We construct Lewis–Riesenfeld invariants from two-dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic sp(4) Lie algebra. This allows for an alternative systematic approach to find Lewis–Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time-dependent Dyson maps from their respective time-independent counterparts in the reference system.