Stability in Non-Normal Periodic Jacobi Operators: Advancing B\"org's Theorem

TL;DR

This paper extends stability results for non-normal periodic Jacobi operators, linking matrix entry oscillations to spectral properties, and applies findings to finite difference approximations of differential equations.

Contribution

It generalizes B"org's theorem to non-normal cases, connecting oscillations of matrix entries with pseudospectrum connectedness, and broadens the scope of stability analysis.

Findings

Extended stability results to non-normal operators

Linked matrix oscillations to pseudospectrum connectedness

Applied results to finite difference approximations

Abstract

Periodic Jacobi operators naturally arise in numerous applications, forming a cornerstone in various fields. The spectral theory associated with these operators boasts an extensive body of literature. Considered as discretized counterparts of Schr\"odinger operators, widely employed in quantum mechanics, Jacobi operators play a crucial role in mathematical formulations. The classical uniqueness result by G. B\"org in $1946$ occupies a significant place in the literature of inverse spectral theory and its applications. This result is closely intertwined with M. Kac's renowned article, 'Can one hear the shape of a drum?' published in $1966$. Since $1975,$ discrete versions of B\"org's theorem have been available in the literature. In this article, we concentrate on the non-normal periodic Jacobi operator and the discrete versions of B\"org's Theorem. We extend recently obtained stability results to encompass non-normal cases. The existing stability findings establish a correlation between the oscillations of the matrix entries and the size of the spectral gap. Our result encompasses the current self-adjoint versions of B\"org's theorem, including recent quantitative variations. Here, the oscillations of the matrix entries are linked to the path-connectedness of the pseudospectrum. Additionally, we explore finite difference approximations of various linear differential equations as specific applications.