On the {\eta} pseudo PT symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

TL;DR

This paper introduces a new approach to pseudo PT symmetry in time-dependent non-Hermitian quantum systems, deriving a novel metric that satisfies the Heisenberg evolution, and applies it to solve an SU(1,1) Hamiltonian.

Contribution

It proposes a new metric for time-dependent non-Hermitian Hamiltonians that satisfies the Heisenberg evolution, advancing the theoretical framework of pseudo PT symmetry.

Findings

Derived a new time-dependent metric for non-Hermitian systems.

Solved the SU(1,1) time-dependent non-Hermitian Hamiltonian.

Discussed physical applications of the new metric.

Abstract

In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{{\dag}} via the relation, H^{{\dag}}=PTHPT . We propose a derivation of pseudo PT symmetry and {\eta} -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric {\eta}(t)=PT{\eta}(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.