An infinite-temperature limit for a quantum scattering process

Abstract

We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schr\"odinger part and a noisy Poission-timed scattering part. The dynamics describes the evolution of a test particle in $\R^{n}$, $n=1,2,3$, immersed in a gas, and the noisy scattering part is defined by the reduced effect of an individual interaction, where the interaction between the test particle and a single gas particle is via a repulsive point potential. In the limit that the mass ratio $\lambda=\frac{m}{M}$ tends to zero and the collisions become more frequent as $\frac{1}{\lambda}$, we show that our dynamics $\Phi_{t,\lambda}$ approaches a limiting dynamics $\Phi_{t,\lambda}^{\diamond}$ with second order error. Working in the Heisenberg representation, for $G\in \Bi(L^{2}(\R^{n}))$ $n=1,3$ we bound the difference between $\Phi_{t,\lambda}(G)$ and $\Phi_{t,\lambda}^{\diamond}(G)$ in operator norm proportional to $\lambda^{2}$.