# Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

## TL;DR

This paper analyzes the asymptotic behavior of spectral density and subordinate solutions at the top of the essential spectrum for Jacobi matrices with eventually monotone parameters, extending results to orthogonal polynomials on the unit circle.

## Contribution

It provides new asymptotic descriptions for spectral density and subordinate solutions for Jacobi matrices with specific perturbations, including applications to Verblunsky coefficients.

## Key findings

- Asymptotic behavior of subordinate solutions at the spectrum edge characterized.
- Spectral density asymptotics derived for perturbations of the free case.
- Results applicable to orthogonal polynomials on the unit circle with real Verblunsky coefficients.

## Abstract

We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic behavior of the spectral density at the top of the essential spectrum. In particular, allowing on both diagonal and off-diagonal Jacobi parameters perturbations of the free case of the form $- \sum_{j=1}^J c_j n^{-\tau_j} + o(n^{-\tau_1-1})$ with $0 < \tau_1 < \tau_2 < \dots < \tau_J$ and $c_1>0$, we find the asymptotic behavior of the $\log$ of spectral density to order $O(\log(2-x))$ as $x$ approaches $2$. Apart from its intrinsic interest, the above results also allow us to describe the asymptotics of the spectral density for orthogonal polynomials on the unit circle with real-valued Verblunsky coefficients of the same form.