An infinite-temperature limit for a quantum scattering process

We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schrödinger 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 $λ=\frac{m}{M}$ tends to zero and the collisions become more frequent as $\frac{1}λ$, we show that our dynamics $Φ_{t,λ}$ approaches a limiting dynamics $Φ_{t,λ}^{\diamond}$ 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 $Φ_{t,λ}(G)$ and $Φ_{t,λ}^{\diamond}(G)$ in operator norm proportional to $λ^{2}$.

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