  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.
