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We study Jacobi operators , , whose eigenfunctions are Laguerre polynomials. All operators 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 , where 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 as for Jacobi operators whose eigenfunctions are different classical polynomials. For Laguerre polynomials, it turns out that the evolution is concentrated in the region where instead of 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.