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

The quantum probability flux of a particle integrated over time and a dista nt 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-sur faces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phy s. Rev. D 11, 366-372 (1975)]. We prove the flux-across-surfaces theorem fo r short range potentials and wave functions without energy cutoffs. The pro of is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103-116 (1996)], and on smoothness properties of generaliz ed eigenfunctions: It is shown that if the potential V(chi) decays like \ch i\(-gamma) g at infinity with gamma>n epsilon N then the generalized eigenf unctions of the corresponding Hamiltonian -1/2 Delta+V are n-2 times contin uously differentiable with respect to the momentum variable. (C) 1999 Ameri can Institute of Physics. [S0022-2488(99)00604-0].