Constructing Exactly Solvable Pseudo-hermitian Many-particle Quantum Systems by Isospectral Deformation

TL;DR

This paper introduces a method to construct exactly solvable non-Hermitian many-particle quantum systems with real spectra, using isospectral deformation and pseudo-Hermiticity, exemplified by Calogero and XXZ models.

Contribution

It presents a general approach to generate pseudo-Hermitian quantum systems with real spectra that are isospectral to Hermitian counterparts, expanding solvable models in quantum mechanics.

Findings

Constructed pseudo-Hermitian Calogero model with real spectrum

Developed pseudo-Hermitian XXZ spin-chain model

Demonstrated unitary evolution in non-Hermitian systems

Abstract

A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.