Periodic Jacobi Operators with Complex Coefficients

TL;DR

This paper explores the spectral theory of complex periodic Jacobi operators, establishing conditions for Hill discriminants and extending Borg's Theorem to non-self-adjoint cases, advancing understanding of their spectral properties.

Contribution

It provides new results on the inverse spectral problem and spectral characterization of complex periodic Jacobi operators, including polynomial discriminants and spectral interval analysis.

Findings

Any degree N polynomial with leading coefficient (-1)^N is a Hill discriminant for finitely many N-periodic Schrödinger operators.

Proved an analog of Borg's Theorem for non-self-adjoint Jacobi operators with spectrum as a closed interval.

Abstract

We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill discriminant of finitely many discrete $N$-periodic Schr\"{o}dinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem 5) which is the analog of Borg's Theorem for the non-self-adjoint Jacobi case.