A possible method for non-Hermitian and non-$PT$-symmetric Hamiltonian systems

TL;DR

This paper proposes a method to analyze non-Hermitian, non-$PT$-symmetric Hamiltonians by constructing a suitable $ ext{eta}_+$ operator, ensuring real spectra and consistent quantum properties, even in noncommutative space extensions.

Contribution

It introduces a novel approach using $ ext{eta}_+$-pseudo-Hermiticity to study non-Hermitian, non-$PT$-symmetric Hamiltonians, including their noncommutative space extensions.

Findings

Real spectra and positive-definite inner products are achieved.

The method applies to coupled Hamiltonians in noncommutative space.

Energy level corrections are calculated without losing physical properties.

Abstract

A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator $\eta_+$ and defining the annihilation and creation operators to be $\eta_+$-pseudo-Hermitian adjoint to each other. The operator $\eta_+$ represents the $\eta_+$-pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-$PT$-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator $\eta_+$ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-$PT$-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.