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Periodic Jacobi operators naturally arise in numerous applications, forming acornerstone in various fields. The spectral theory associated with theseoperators boasts an extensive body of literature. Considered as discretizedcounterparts 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 significantplace in the literature of inverse spectral theory and its applications. Thisresult is closely intertwined with M. Kac's renowned article, 'Can one hear theshape of a drum?' published in $1966$. Since $1975,$ discrete versions ofB\"org's theorem have been available in the literature.  In this article, we concentrate on the non-normal periodic Jacobi operatorand the discrete versions of B\"org's Theorem. We extend recently obtainedstability results to encompass non-normal cases. The existing stabilityfindings establish a correlation between the oscillations of the matrix entriesand 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 matrixentries are linked to the path-connectedness of the pseudospectrum.Additionally, we explore finite difference approximations of various lineardifferential equations as specific applications.

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