Scattering theory for the Schrodinger equation with repulsive potential

We consider the scattering theory for the Schrodinger equation with -Delta - vertical bar x vertical bar(alpha) as a reference Hamiltonian, for 0 < a <= 2, in any space dimension. We prove that, when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of alpha, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where - vertical bar x vertical bar(alpha) is replaced by a general second order polynomial. (c) 2004 Elsevier SAS. All rights reserved.