The reduced effect of a single scattering with a low-mass particle via a point interaction

TL;DR

This paper develops a second-order expansion for the effect of a single scattering event between a large quantum particle and a low-mass particle via a point interaction, providing precise error bounds in specific dimensions.

Contribution

It introduces a third-order approximation for the scattering effect with explicit error bounds, advancing the mathematical understanding of quantum scattering with point interactions.

Findings

Third-order error bounds in operator norm for dimensions one and three.

Explicit approximation formula for the scattering map.

Enhanced understanding of quantum scattering effects at low mass ratios.

Abstract

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in $\lambda$ to the map $G\to \Tr_{2}[(I\otimes \rho)S_{\lambda}^{*}(G\otimes I)S_{\lambda}]$, where $G\in \Bi(L^{2}(\R^{n}))$ is a heavy-particle observable, $\rho\in \Bi_{1}(\R^{n})$ is the density matrix corresponding to the state of the light particle, $\lambda=\frac{m}{M}$ is the mass ratio of the light particle to the heavy particle, $S_{\lambda}\in \Bi(L^{2}(\R^{n})\otimes L^{2}(\R^{n}))$ is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on $G$.