Radiative Transport Limit for the Random Schr\"{o}dinger Equation with Long-Range Correlations

TL;DR

This paper investigates the asymptotic behavior of the phase space energy distribution of solutions to the Schr"{o}dinger equation with long-range correlated random potentials, showing convergence to a radiative transfer equation with regularizing effects and fractional Laplacian approximation.

Contribution

It establishes the convergence of the Wigner transform to a radiative transfer equation for long-range correlated potentials and introduces a fractional Laplacian approximation.

Findings

Wigner transform converges to a radiative transfer equation

Long-range correlations induce regularizing effects

Radiative transfer equation approximated by a fractional Laplacian

Abstract

In this paper we study the asymptotic phase space energy distribution of solution of the Schr\"{o}dinger equation with a time-dependent random potential. The random potential is assumed to be with slowly decaying correlations. We show that the Wigner transform of a solution of the random Schr\"{o}dinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The derivations of these results are based on an asymptotic analysis using perturbed-test-functions, martingale techniques, and probabilistic representations.