arXiv 2202.020872022-02-04math.CAmath.FAmath.SP

Jacobi算子的谱分析与正交多项式的渐近性

Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

D. R. Yafaev

备注：: 86 pages

摘要：

我们发现并讨论了具有递推系数$a_{n}, b_{n}$的正交多项式$P_{n}(z)$的渐近公式。我们的主要目标是考虑非对角元素$a_{n}\to\infty$为$n\to\infty$的情况。对于相对较小和较大的对角元素$b_{n}$，得到的公式有本质上的不同。我们的分析是密切联系在一起的Jacobi算子$J$与系数$a_{n}, b_{n}$和相应的二阶差分方程的研究的谱理论。通过对$n\to\infty$的一个条件，我们引入了这类方程的Jost解$f_{n}(z)$，$n\geq -1$，并提出了一个类似于薛定谔方程解的半经典Liouville-Green近似的近似.这使我们能够研究Jacobi算子的谱结构和它们的本征函数$P_{n}(z)$的传统方法的谱理论发展的微分方程。特别地，我们将$P_{n}(z)$的渐近公式中的所有系数表示为根据解$ P_{n} (z) $和$ f_{n} (z)$的Wronskian的$n \to\infty$。对$P_{n}(z)$得到的公式推广了经典Hermite多项式的渐近公式，其中$a_{n}=\sqrt{(n+1)/2}$和$b_{n}=0$ .

英文摘要：

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 $b_{n}=0$.

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