An infinite-temperature limit for a quantum scattering process

Abstract

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 ${\mathbb{R}}^n$, $n=1,2,3$, 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 ${\lambda }^2$.

18 pages

References in corpus (2)

• On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles
• Master-equations for the study of decoherence

Cited by in corpus (1)

• Quantum linear Boltzmann equation