On spectrum of Jacobi operator with exponentially increasing matrix elements

TL;DR

This paper investigates the spectral properties of a class of Jacobi operators with exponentially increasing matrix elements, establishing their connection to Sturm–Liouville problems with self-similar weights and deriving eigenvalue asymptotics.

Contribution

It introduces a novel analysis of Jacobi matrices with exponential growth, linking their eigenvalue problems to Sturm–Liouville operators with self-similar weights and deriving asymptotic formulas.

Findings

Eigenvalue asymptotics differ for definite and indefinite metrics.

Eigenvalue problem is equivalent to a Sturm–Liouville problem with self-similar weight.

Conditions under which the operator is self-adjoint are established.

Abstract

The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the matrix and in some cases can arise indefinite metric. We proved that eigenvalue problem for this operator is equivalent to the eigenvalue problem of Sturm--Liouville operator with discrete self-similar weight. The asymptotic formulas for eigenvalues are obtained. These formulas differ for cases of definite and indefinite metrics.