Investigation of non-Hermitian Hamiltonians in the Heisenberg picture

TL;DR

This paper explores the Heisenberg picture for non-Hermitian, pseudo-Hermitian Hamiltonians, establishing a method to find real eigenvalues and constructing new Hamiltonians with different symmetry properties.

Contribution

It introduces a Heisenberg picture framework for pseudo-Hermitian systems and demonstrates how to derive real eigenvalues via similarity transformations, including the construction of new non-Hermitian Hamiltonians.

Findings

Hermitian counterparts yield the same second order equations of motion.

Verification of iso-spectral property through similarity transformations.

Construction of two novel non-Hermitian Hamiltonians with different symmetry properties.

Abstract

The Heisenberg picture for non-Hermitian but $\eta$-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but $\eta$-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two new non-Hermitian Hamiltonians are constructed and investigated, where one is non-Hermitian and non-PT-symmetric and the other is non-Hermitian but PT-symmetric. Moreover, the complementarity and compatibility between our treatment and the PT symmetry are discussed.