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 JpJ_{p}, p>1p> -1, whose eigenfunctions are Laguerre polynomials. All operators JpJ_{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 JpJ_{p}, JqJ_{q} where pqp\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 (eJtf)n(e^{-J t} f)_{n} as t|t|\to\infty for Jacobi operators JJ whose eigenfunctions are different classical polynomials. For Laguerre polynomials, it turns out that the evolution (eJptf)n(e^{-J_{p} t} f)_{n} is concentrated in the region where nt2n\sim t^2 instead of ntn\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.