Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
arXiv:2405.15326 · doi:10.1088/1751-8121/ad8a2b
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 Schrödinger-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.
arXiv comment24 pages, 16 figures, typos fixed and references added. Final version appeared in J. Phys. A
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