# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.07496