📝 SummarXiv

Abstract

Here we first present an alternative formulation of the Lewis \& Riesenfeld theorem for solving the Schr\"odinger equation with nonautonomous Hermitian and pseudo-Hermitian Hamiltonians. We then employ this framework to characterize the spontaneous breaking of time-dependent antilinear symmetries of these Hamiltonians. We demonstrate that, under unbroken antilinear symmetries, the Lewis \& Riesenfeld phases are real and odd functions of time, which allows us to recover the well-known real spectra of time-independent pseudo-Hermitian Hamiltonians. However, in the spontaneously broken regime, imaginary components of the Lewis \& Riesenfeld phases arise, leading to coalescence effects analogous to those in the time-independent scenario. Finally, we present an illustrative example of unbroken and broken $\mathcal{PT}$-symmetry for a time-dependent Hamiltonian modeling the non-Hermitian dynamical Casimir effect.