Scattering theory for Laguerre operators

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 $p\neq q$ 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.