Time-dependent [formula omitted]-operators as Lewis-Riesenfeld invariants in non-Hermitian theories

Abstract

C-operators were introduced as involution operators in non-Hermitian theories that commute with the time-independent Hamiltonians and the parity/time-reversal operator. Here we propose a definition for time-dependent C(t)-operators and demonstrate that for a particular signature they may be expanded in terms of time-dependent biorthonormal left and right eigenvectors of Lewis-Riesenfeld invariants. The vanishing commutation relation between the C-operator and the Hamiltonian in the time-independent case is replaced by the Lewis-Riesenfeld equation in the time-dependent scenario. Thus, C(t)-operators are always Lewis-Riesenfeld invariants, whereas the inverse is only true in certain circumstances. We demonstrate the working of the generalities for a non-Hermitian two-level matrix Hamiltonian. We show that solutions for C(t) and the time-dependent metric operator may be found that hold in all three PT-regimes, i.e., the PT-regime, the spontaneously broken PT-regime and at the exceptional point.