# Scattering theory for Laguerre operators

### Dmitri Yafaev

Université de Rennes I, France; St. Petersburg State University, Russian Federation

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## Abstract

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 $p=q$ 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 $(e_{−Jt}f)_{n}$ as $∣t∣→∞$ 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 $n∼t_{2}$ instead of $n∼∣t∣$ 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.