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THE METRIC OPERATORS FOR PSEUDO-HERMITIAN HAMILTONIAN

Published online by Cambridge University Press:  23 October 2023

WEN-HUA WANG
Affiliation:
School of Ethnic Education, Shaanxi Normal University, Xi’an 710062, China; e-mail: wenhua@snnu.edu.cn
ZHENG-LI CHEN*
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
WEI LI
Affiliation:
College of Science, Engineering University of PAP, Xi’an 710078, China; e-mail: 398979528@qq.com
*

Abstract

The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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