最终单调Jacobi和Verblunsky系数的谱边缘行为
Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients
摘要：
我们考虑雅可比矩阵的对角和非对角雅可比参数序列最终增加。我们描述了从属解在本质谱顶部的渐近行为，以及谱密度在本质谱顶部的渐近行为。特别是，允许对角和非对角Jacobi参数扰动的形式$- \sum_{j=1}^J c_j n^{-τ_j} + o(n^{-τ_1-1})$与$0 < τ_1 < τ_2 < \dots < τ_J$和$c_1>0$的自由情况下，我们发现的渐近行为的谱密度的$\log$订购$O(\log(2-x))$作为$x$方法$2$。除了它的本质意义外，上述结果还使我们能够描述单位圆上实值Verblunsky系数具有相同形式的正交多项式的谱密度的渐近性.
英文摘要：
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 $- \sum_{j=1}^J c_j n^{-τ_j} + o(n^{-τ_1-1})$ with $0 < τ_1 < τ_2 < \dots < τ_J$ 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.
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