Monotone Jacobi parameters and non-Szego weights

TL;DR

This paper explores the relationship between the asymptotic behavior of Jacobi parameters and spectral weights near the edges of the spectrum, revealing explicit formulas for weights in specific cases.

Contribution

It establishes a connection between Jacobi parameter asymptotics and spectral weight asymptotics, providing explicit edge behavior formulas for certain parameter regimes.

Findings

Spectral weights near spectrum edges follow specific asymptotic forms.

Explicit formula for weight function near x=2 for given Jacobi parameters.

Demonstrates the link between Jacobi parameters and spectral measure asymptotics.

Abstract

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\equiv 1$, $b_n =-C n^{-\beta}$ ($0<\beta< \frac23)$, one has $d\mu(x)= w(x) dx$ on $(-2,2)$, and near $x=2$, $w(x)=e^{-2Q(x)}$ where \[ Q(x)=\beta^{-1} C^{\frac{1}{\beta}} \frac{\Gamma(\frac32)\Gamma(\frac{1}\beta}-\frac12)(2-x)^{\frac12 -\frac{1}{\beta}}}{\Gamma(\frac{1}{\beta}+1)}(1+O((2-x))) \]