Non-Hermitian Hamiltonian beyond PT symmetry for time-dependent SU(1,1) and SU(2) systems — Exact solution and geometric phase in pseudo-invariant theory

In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of $SU(1,1)$ and $SU\left(2\right)$ generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the $SU(1,1)$ and $SU\left(2\right)$ systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.