Monotone Jacobi parameters and non-Szego weights

Abstract

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