Scattering theory for the Schrodinger equation with repulsive potential

Details Scattering theory for the Schrodinger equation with repulsive potential Scattering theory for the Schrodinger equation with repulsive potential

2004-02-11

arxiv.org/abs/math/0402170v1

J. Math. Pures Appl. 84 (2005), no. 5, 509-579.

Mathematics

Analysis of PDEs

47 pages, a4wide, no figure

Scientific paper

10.1016/j.matpur.2004.10.007

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.

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