论文标题
雅各比算子的光谱分析和正交多项式的渐近行为
Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials
论文作者
论文摘要
我们找到并讨论了与复发系数$ a_ {n}，b_ {n} $的正顺序多项式的渐近公式$ p_ {n}（z）$。我们的主要目标是考虑以下情况下的情况，即偏离元素$ a_ {n} \ to \ infty $作为$ n \ to \ infty $。对于相对较小和大的对角线元素$ b_ {n} $，获得的公式基本不同。 我们的分析与Jacobi运算符的光谱理论$ J $与系数$ a_ {n}，b_ {n} $以及对相应的第二阶差方程的研究。 我们通过$ n \ to \ insatz介绍了此类方程式的jost解决方案$ f_ {n}（z）$，$ n \ geq -1 $，并为他们为Schrödinger方程解决方案的半典型liouville -green ansatz扮演的角色建议。 这使我们能够通过针对微分方程开发的传统光谱理论方法来研究雅各比运算符及其特征功能的光谱结构$ p_ {n}（z）$。特别是，我们以$ p_ {n}（z）$为$ n \ to \ infty $的渐近公式中的所有系数在解决方案的wronskian $ p_ {n}（z）$和$ f_ {n}（z）$中。为$ p_ {n}（z）获得的公式$概括了经典的Hermite多项式的渐近公式，其中$ a_ {n} = \ sqrt {（n+1）/2} $和$ b_ {n} = 0 $。
We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\geq -1$, of such equations by a condition for $n\to\infty$ and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions $P_{n}(z)$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $P_{n}(z)$ as $n \to\infty$ in terms of the Wronskian of the solutions $ P_{n} (z) $ and $ f_{n} (z)$. The formulas obtained for $P_{n}(z)$ generalize the asymptotic formulas for the classical Hermite polynomials where $a_{n}=\sqrt{(n+1)/2}$ and $b_{n}=0$.
