The Linear Boltzmann Equation as the Low Density Limit of a Random Schrödinger Equation

We study the long time evolution of a quantum particle interacting with a random potential in the Boltzmann-Grad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collision kernel is given by the full quantum scattering cross section of the obstacle potential. 1 The Model and the Result The Schrödinger equation with a random potential describes the propagation of quantum particles in an environment with random impurities. In the first approximation one neglects the interaction between the particles and the problem reduces to a onebody Schrödinger equation. With high concentration of impurities the particle is localized, in particular no conduction occurs [1, 2, 3, 8, 11, 12]. In the low concentration regime conduction is expected to occur but there are no rigorous mathematical proof of the existence of the extended states except for the Bethe lattice [16, 17]. In this paper we study the long time evolution in the low concentration regime in a specific scaling limit, called the low density or Boltzmann-Grad limit. Our model is the quantum analogue of the low density Lorentz gas. As the time increases, the concentration will be scaled down in such a way that the total interaction between the particle and the obstacles remains bounded for a typical configuration. Therefore our result is far from the extended states regime which requires to understand the behavior of the Schrödinger evolution for arbitrary long time, independently of the fixed (low) concentration of impurities. We start by defining our model and stating the main result. Supported by MacCraken Fellowship. Courant Institute, New York University. 251 Mercer Street, New York, NY 10012, [email protected] Partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027. Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich Germany, [email protected]. On leave from School of Mathematics, GeorgiaTech, USA