Quantum Diffusion via an Approximate Semigroup Property

TL;DR

This paper introduces a novel wavepacket decomposition method to derive an effective diffusion equation from the weakly random Schrödinger equation, valid in dimensions d≥2, by exploiting a semigroup property of the main term.

Contribution

It presents a new approach based on wavepacket decomposition and geometric analysis to establish diffusive limits for the random Schrödinger equation in higher dimensions.

Findings

Main term exhibits a semigroup property enabling iterative timescale extension.

Error terms are controlled via geometric bounds on exceptional paths.

First derivation of effective diffusion from the random Schrödinger equation in dimensions d≥2.

Abstract

In this paper we introduce a new approach to the diffusive limit of the weakly random Schrodinger equation, first studied by L. Erdos, M. Salmhofer, and H.T. Yau. Our approach is based on a wavepacket decomposition of the evolution operator, which allows us to interpret the Duhamel series as an integral over piecewise linear paths. We relate the geometry of these paths to combinatorial features of a diagrammatic expansion which allows us to express the error terms in the expansion as an integral over paths that are exceptional in some way. These error terms are bounded using geometric arguments. The main term is then shown to have a semigroup property, which allows us to iteratively increase the timescale of validity of an effective diffusion. This is the first derivation of an effective diffusion equation from the random Schrodinger equation that is valid in dimensions $d\geq 2$.