07-281 Yury Kreimer, Yoram Last, Barry Simon

Monotone Jacobi parameters and non-Szego weights (310K, PDF) Nov 16, 07

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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 \[ Q(x)=\beta^{-1} C^{\frac{1}{\beta}} \frac{\Gamma(\frac32)\Gamma(\frac{1}{\beta}-\frac12)(2-x)^{\frac12 -\frac{1}{\beta}}}{\Gamma(\frac{1}{\beta}+1)}\, (1+O((2-x))) \]

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