Titchmarsh-Weyl formula for the spectral density of a class of Jacobi matrices in the critical case

TL;DR

This paper derives a formula linking the spectral density of certain unbounded Jacobi matrices in the critical case to the asymptotics of associated orthogonal polynomials, advancing spectral theory understanding.

Contribution

It establishes a Titchmarsh-Weyl type formula for spectral density in the critical case of unbounded Jacobi matrices, a case previously not well-understood.

Findings

Derived a spectral density formula for critical Jacobi matrices

Connected spectral density to orthogonal polynomial asymptotics

Enhanced understanding of spectral properties in critical regimes

Abstract

We consider a class of Jacobi matrices with unbounded entries in the so called critical (double root, Jordan box) case. We prove a formula for the spectral density of the matrix which relates its spectral density to the asymptotics of orthogonal polynomials associated with the matrix.