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Journal of Physics A Mathematical and General

Flux and scattering into cones for long range and singular potentials

Werner O. Amrein, David B. Pearson

Published Aug 7, 1997• 19 citations
Publisher

Abstract

We prove that the probability of finding a scattered quantum-mechanical particle at large times in a truncated cone is identical with the scattered flux, integrated over time, across a distant spherical surface subtending this cone. The theory applies to potentials with arbitrary local singularities and decaying faster than at large distances.

DOI
10.1088/0305-4470/30/15/022

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We prove that the probability of finding a scattered quantum-mechanical particle at large times in a truncated cone is identical with the scattered flux, integrated over time, across a distant spherical surface subtending this cone. The theory applies to potentials with arbitrary local singularities and decaying faster than at large distances.
The relation between the physical meaning of a nonrelativistic NNN-particle flux and its mathematical representation in quantum mechanics is discussed. We prove a theorem that equates the probability of finding an NNN-fragment system in the distant future in a given cone with the total probability that the fragments cross a distant surface subtended by the cone. Together with the scattering-into-cones theorem this result proves that the usually calculated number of fragments whose momenta in the distant future lie in a given cone is equal to the total counted number of fragments that, at any time, cross a subtended distant surface. It thus adds to both physically and mathematically cleaner underpinnings of scattering theory.
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.
We consider a non-relativistic quantum particle in Rd\mathbb{R}^dRd, d=2d=2d=2 or d=3d = 3d=3, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of the singular potentials and the distances between the points simultaneously go to zero as their number grows, while the total interaction strength remains finite. Assuming that the singular potentials have negative scattering lengths and are uniformly distributed, we prove the strong resolvent convergence as N→∞N \to \inftyN→∞ of the family of operators to a Schrödinger operator with a regular electrostatic potential. The result is obtained via ΓΓΓ-converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense.