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 $p\ne q$ 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 $|t|\to \infty$ 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.