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arXiv 2007.08418math.CAmath.FAmath.SP
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Laguerre算子的散射理论

Scattering theory for Laguerre operators

D. R. Yafaev

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摘要：

研究了Jacobi算子$J_{p}$，$p> -1$，它们的特征函数是Laguerre多项式.所有算子$J_{p}$都有与正半轴一致的绝对连续的简单谱。然而，这一事实并不意味着存在$p\neq q$的$J_{p}$、$J_{q}$对的波算符。我们的目标是表明，尽管如此，这是真的，并找到明确的表达式，这些波算子。对于特征函数为不同经典多项式的Jacobi算子$J$，我们还研究了$(e^{-J t} f)_{n}$作为$|t|\to\infty$的时间演化.对于Laguerre多项式，事实证明，演化$(e^{-J_{p} t} f)_{n}$集中在$n\sim t^2$而不是$n\sim |t |$的区域中，如在标准情况下发生的。作为我们的考虑的副产品，我们得到了一般正交多项式的渐近公式的振幅和相位之间的普遍关系。

英文摘要：

We study Jacobi operators $J_{p}$, $p> -1$, whose eigenfunctions are Laguerre polynomials. All operators $J_{p}$ have absolutely continuous simple spectra coinciding with the positive half-axis. This fact, however, by no means imply that the wave operators for the pairs $J_{p}$, $J_{q}$ where $p\neq q$ exist. Our goal is to show that, nevertheless, this is true and to find explicit expressions for these wave operators. We also study the time evolution of $(e^{-J t} f)_{n}$ as $|t|\to\infty$ for Jacobi operators $J$ whose eigenfunctions are different classical polynomials. For Laguerre polynomials, it turns out that the evolution $(e^{-J_{p} t} f)_{n}$ is concentrated in the region where $n\sim t^2$ instead of $n\sim |t |$ as happens in standard situations. As a by-product of our considerations, we obtain universal relations between amplitudes and phases in asymptotic formulas for general orthogonal polynomials.

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