The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs

Abstract

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux-across-surfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V(x) decays like |x|−γ at infinity with γ>n∈N then the generalized eigenfunctions of the corresponding Hamiltonian −1/2Δ+V are n−2 times continuously differentiable with respect to the momentum variable.