Non-Unitary evolution of quantum time-dependent non-Hermitian systems

TL;DR

This paper offers a new perspective on non-Hermitian quantum systems by emphasizing norm-preserving non-unitary evolution, providing a precise description and illustrating solutions with specific non-Hermitian Hamiltonians.

Contribution

It introduces a framework for understanding non-unitary evolution in time-dependent non-Hermitian quantum systems, emphasizing the importance of norm preservation and providing explicit solution methods.

Findings

Non-Hermitian evolution can be described similarly to Hermitian cases with proper norm preservation.

Solutions for specific non-Hermitian Hamiltonians involve time-dependent non-unitary transformations.

The approach clarifies the role of non-unitary transformations in solving non-Hermitian quantum systems.

Abstract

We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.