A remark on the discrete spectrum of non-self-adjoint Jacobi operators

TL;DR

This paper investigates the spectral properties of non-self-adjoint Jacobi operators, providing new bounds on perturbation determinants, refining existing inequalities, and discussing spectral enclosures for these operators.

Contribution

It introduces a new bound for the perturbation determinant of non-self-adjoint Jacobi operators and refines the Lieb--Thirring inequality for these cases.

Findings

Derived a new bound for the perturbation determinant

Refined the Lieb--Thirring inequality for non-self-adjoint Jacobi operators

Discussed spectral enclosures for such operators

Abstract

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for such operators is also discussed.