A formula for eigenvalues of Jacobi matrices with a reflection symmetry

TL;DR

This paper derives a polynomial identity for eigenvalues of symmetric Jacobi matrices with reflection symmetry and explores its implications for infinite-dimensional operators, including the discrete Schrödinger operator with compact support.

Contribution

It introduces a new polynomial identity relating eigenvalues and matrix entries for symmetric Jacobi matrices, extending to infinite-dimensional operators with specific spectral properties.

Findings

Derived a polynomial identity for eigenvalues of symmetric Jacobi matrices.

Established spectral requirements for infinite-dimensional Jacobi operators.

Analyzed the case of the discrete Schrödinger operator with compact support.

Abstract

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix { entries} is obtained. In the limit $M\to\infty$ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, which super- and sub-diagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schr\"odinger operator acting in ${l}^2( \mathbb{N})$, which does not have bound and semi-bound states, and which potential has a compact support.