S. Teufel et al., The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs, J MATH PHYS, 40(4), 1999, pp. 1901-1922
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].