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The Flux-Across-Surfaces Theorem under conditions on the scattering state

D. Dürr, T. Moser, Peter Pickl

Published Sep 13, 2013• 8 citations
OpenAlex

Abstract

Abstract. The flux-across-surfaces theorem (FAST) which describes the outgoing asymptotics of the quantum flux of the scattering state is basic to scattering theory. The FAST has been proven for potential scattering under conditions on the outgoing asymptote ψout (and of course under suitable conditions on the scattering potential). In this article we prove the FAST under conditions on the scattering state itself. 1

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The flux-across-surfaces theorem under conditions on the scattering state

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The flux-across-surfaces theorem (FAST) describes the outgoing asymptotics of the quantum flux density of a scattering state. The FAST has been proven for potential scattering under conditions on the outgoing asymptote ψ out (and of course under suitable conditions on the scattering potential). In this paper, we prove the FAST under conditions on the scattering state itself. In the proof, we will also establish new mapping properties of the wave operators.

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 15, 1975•43 citations
Abstract. The flux-across-surfaces theorem (FAST) which describes the outgoing asymptotics of the quantum flux of the scattering state is basic to scattering theory. The FAST has been proven for potential scattering under conditions on the outgoing asymptote ψout (and of course under suitable conditions on the scattering potential). In this article we prove the FAST under conditions on the scattering state itself. 1
The flux-across-surfaces theorem establishes a fundamental relation in quantum scattering theory between the asymptotic outgoing state and a quantity which is directly measured in experiments. We prove it for a hamiltonian with a point interaction, using the explicit expression for the propagator. The proof requires only assuptions on the initial state and it covers also the case of zero-energy resonance. We also outline a different approach based on generalized eigenfunctions, in view of a possible extension of the result.
We review the foundations of the scattering formalism for one-particle potential scattering and discuss the generalisation to the simplest case of many non-interacting particles. We point out that the “straight path motion” of the particles, which is achieved in the scattering regime, is at the heart of the crossing statistics of surfaces, which should be thought of as detector surfaces. We prove the relevant version of the many-particle flux across surfaces theorem and discuss what needs to be proven for the foundations of scattering theory in this context.
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 review the foundations of the scattering formalism for one particle potential scattering and discuss the generalization to the simplest case of many non interacting particles. We point out that the "straight path motion" of the particles, which is achieved in the scattering regime, is at the heart of the crossing statistics of surfaces, which should be thought of as detector surfaces. We sketch a proof of the relevant version of the many particle flux across surfaces theorem and discuss what needs to be proven for the foundations of scattering theory in this context.
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.