Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics

TL;DR

This paper investigates the pseudo-Hermitian properties of Chebyshev differential matrices in spectral collocation methods and their implications for non-Hermitian Liouville quantum mechanics, highlighting spectral stability issues and model extensions.

Contribution

It introduces the pseudo-Hermitian perspective of Chebyshev differential matrices and explores their impact on spectral properties in non-Hermitian quantum mechanics models.

Findings

Pseudo-Hermitian Chebyshev matrices affect eigenstate completeness.

Spectral instability can be controlled by the compactification parameter.

Expanded models of non-Hermitian Liouville quantum mechanics are proposed.

Abstract

The spectral collocation method (SCM) exhibits a clear superiority in solving ordinary and partial differential equations compared to conventional techniques, such as finite difference and finite element methods. This makes SCM a powerful tool for addressing the Schr\"odinger-like equations with boundary conditions in physics. However, the Chebyshev differential matrix (CDM), commonly used in SCM to replace the differential operator, is not Hermitian but pseudo-Hermitian. This non-Hermiticity subtly affects the pseudospectra and leads to a loss of completeness in the eigenstates. Consequently, several issues arise with these eigenstates. In this paper, we revisit the non-Hermitian Liouville quantum mechanics by emphasizing the pseudo-Hermiticity of the CDM and explore its expanded models. Furthermore, we demonstrate that the spectral instability can be influenced by the compactification parameter.