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 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.

Universal relations in asymptotic formulas for orthogonal polynomials

“…Given unitarity of the operators (1.1), we deduce these relations from asymptotic formulas for time dependent evolution e −iΘ(J)t f as t → ∞ for suitable functions Θ(λ). These results were announced in [21]; their proofs are given in Sect. 4.…”