  We consider the scattering theory for the Schrodinger equation with $-\Delta
-|x|^{\alpha}$ as a reference Hamiltonian, for $0< \alpha \leq 2$, 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
$-|x|^{\alpha}$ is replaced by a general second order polynomial.
