The Flux-Across-Surfaces Theorem for a Point Interaction Hamiltonian

TL;DR

This paper proves the flux-across-surfaces theorem for quantum Hamiltonians with point interactions, using explicit propagator expressions and covering cases with zero-energy resonance, advancing theoretical understanding of quantum scattering.

Contribution

It provides a rigorous proof of the flux-across-surfaces theorem for point interaction Hamiltonians, including zero-energy resonance cases, and suggests an alternative approach via generalized eigenfunctions.

Findings

Proved the theorem for Hamiltonians with point interactions.

Included cases with zero-energy resonance.

Outlined an alternative proof method using generalized eigenfunctions.

Abstract

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.