Abstract

Orthogonal polynomials \(P_{n}(\lambda)\) are oscillating functions of \(n\) as \(n\to\infty\) for \(\lambda\) in the absolutely continuous spectrum of the corresponding Jacobi operator \(J\). We show that, irrespective of any specific assumptions on the coefficients of the operator \(J\), the amplitude and phase factors in asymptotic formulas for \(P_{n}(\lambda)\) are linked by certain universal relations found in the paper. Our proofs rely on the study of a time-dependent evolution generated by suitable functions of the operator \(J\).

中文翻译：

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正交多项式渐近公式的普遍关系

摘要

正交多项式\(P_{n}(\lambda)\)是\(n\) as \(n\to\infty\) for \(\lambda\)在相应雅可比算子的绝对连续谱中的振荡函数\(J\)。我们表明，无论对算子\(J\)的系数有任何特定假设，\(P_{n}(\lambda)\) 的渐近公式中的幅度和相位因子都通过在纸。我们的证明依赖于对由算子\(J\) 的合适函数产生的时间相关演化的研究。