论文标题
拉瓜尔运营商的散射理论
Scattering theory for Laguerre operators
论文作者
论文摘要
我们研究雅各比运营商$ j_ {p} $，$ p> -1 $，其本征函数是laguerre的多项式。所有操作员$ j_ {p} $具有与正半轴一致的绝对连续的简单光谱。但是，这个事实绝不意味着Pairs $ J_ {p} $，$ j_ {q} $的波浪运算符，其中$ p \ neq q $存在。我们的目标是表明，尽管如此，这是正确的，并为这些波浪运营商找到明确的表达方式。我们还研究了$（e^{ - j t} f）_ {n} $的时间演变为$ | t | \ to \ to \ infty $，用于jacobi操作员$ j $，其特征性的经典多项式是不同的。对于Laguerre多项式，事实证明，Evolution $（e^{ - j_ {p} t} f）_ {n} $集中在$ n \ sim t^2 $而不是$ n \ sim | t | t | $的区域中。 作为我们考虑因素的副产品，我们获得了通用正交多项式的渐近公式中幅度和相之间的普遍关系。
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.
