Time-Dependent and Time-Independent Potential Scattering for Asymptotically Coulomb Potentials

Abstract

We consider the three-dimensional quantum mechanical problem of nonrelativistic potential scattering, where the Hamiltonian is H2 = H1 + V = H0 + Vc + V;H0 is the free particle Hamiltonian, Vc is the Coulomb potential, and V(x) is a real-valued potential function defined for x ∈ R3. We show the existence of the wave operators W±(H2,HD) = s-lim eiH2t e−iHD(t) on L2(R3) for V = V2 + V′, where V2∈L2(R3),V′∈L∞(R3)∩Lp(R3)(p<3),and e−iHD(t) is the family of unitary operators used by Dollard to show the existence of the wave operators W±(H1,HD) appropriate for the pure Coulomb case. If V is spherically symmetric then W±(H2,HD) are shown to be absolutely continuous complete. If in addition V(r) is continuous in (0,∞), then W±(H2,HD) are continuous complete. In both cases S=W+*W− is unitary. The connection between the more physical, time-dependent wave operator approach and the traditional time-independent method is made, and phase shift formulas are obtained for the wave and scattering operators.