Stability in non-normal periodic Jacobi operators: advancing Börg’s theorem

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 Schrödinger operators, widely employed in quantum mechanics, Jacobi operators play a crucial role in mathematical formulations. The classical uniqueness result by G. Börg 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 Börg’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 Börg’s Theorem. We extend recently obtained stability results to cover 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 covers the current self-adjoint versions of Börg’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.