##### Author

Hislop, Peter D.

5737 Department of Mathematics

Kirkpatrick, Kay

Schenker, Jeffrey

Olla, Stefano

60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]

##### Type

Document de travail / Working paper

##### Item number of pages

18

##### Abstract (EN)

We show that a quantum particle in R d , for d 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement x 2 ρ(x, x, t) dx grows like t 3 in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension [7, 15]. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice Z d , for d 1, the mean square displacement is diffusive growing like t 1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice Z d with a time-dependent Markovian potential [17]. 1. Statement of the Problem and Result A quantum particle in a random potential can move diffusively, ballistically, or superballistically depending on the circumstances. In this note, we derive some results about the mean square displacement of a quantum particle subject to a time-dependent white noise Gaussian potential V ω (x, t) that is correlated in space and uncorrelated in time. We prove that the mean square displacement is superballistic for models on R d and diffusive for models on the lattice Z d. We consider the Schrödinger equation with a time-dependent potential given by i∂ t ψ(x, t) = − 2 2m ∆ψ(x, t) + V ω (x, t)ψ(x, t), (1) where x is in R d (resp. Z d) and the operator −∆ is the Laplacian on R d (resp. discrete Laplacian on Z d). The potential V ω (x, t) is a mean zero Gaussian stochastic process with covariance V ω (x, t)V ω (x ′ , t ′) = V 2 0 g(x − x ′)δ(t − t ′), (2) where the strength of the disorder is V 0 > 0, and the spatial correlation function g ∈ C 2 (R d ; R) is a real, even function with sufficiently rapid decay at infinity. We assume the physically reasonable convexity condition that |(∇g)(0)| = 0 and that the Hessian matrix Hess(g)(0) is negative definite. The angular brackets in (2) denote averaging with respect the Gaussian probability measure. Generalizing from the single particle wave function ψ, the evolution of a density matrix ρ ∈ I, where I is the ideal of trace class operators, is governed