The characteristic function for complex doubly infinite Jacobi matrices

TL;DR

This paper introduces a characteristic function for complex doubly infinite Jacobi matrices, linking its zeros to the spectrum and eigenvalue multiplicities, with explicit formulas and examples.

Contribution

It defines a new analytic characteristic function for a class of complex Jacobi matrices and establishes its spectral properties, including eigenvalue correspondence and multiplicity.

Findings

Zeros of the characteristic function correspond to the spectrum.

Eigenvalue multiplicities match zero orders of the characteristic function.

Explicit formulas for eigenvectors and resolvent matrix elements are provided.

Abstract

We introduce a class of doubly infinite complex Jacobi matrices determined by a simple convergence condition imposed on the diagonal and off-diagonal sequences. For each Jacobi matrix belonging to this class, an analytic function, called a characteristic function, is associated with it. It is shown that the point spectrum of the corresponding Jacobi operator restricted to a suitable domain coincides with the zero set of the characteristic function. Also, coincidence regarding the order of a zero of the characteristic function and the algebraic multiplicity of the corresponding eigenvalue is proved. Further, formulas for the entries of eigenvectors, generalized eigenvectors, a summation identity for eigenvectors, and matrix elements of the resolvent operator are provided. The presented method is illustrated by several concrete examples.