Eigenvalue density for a class of Jacobi matrices

TL;DR

This paper derives the asymptotic eigenvalue distribution for a class of large real symmetric Jacobi matrices with elements that scale according to specific functions, extending understanding of their spectral behavior.

Contribution

It provides the first detailed asymptotic eigenvalue distribution for Jacobi matrices with elements varying according to a nondecreasing function, generalizing previous fixed-parameter results.

Findings

Eigenvalue distribution converges to a specific asymptotic form.

Results apply to matrices with elements scaling by a nondecreasing function.

Extends spectral analysis to broader classes of Jacobi matrices.

Abstract

We obtain the asymptotic distribution of eigenvalues of real symmetric tridiagonal matrices as their dimension increases to infinity and whose diagonal and off-diagonal elements asymptotically change with the index n as J_{nt+i nt+i}\sim a_i\phi(n), J_{nt+i nt+i+1}\sim b_i\phi(n), i=0,1,...,t-1, where a_i and b_i are finite, and \phi(n) belongs to a certain class of nondecreasing functions.