Diffusion bound for the nonlinear Anderson model

TL;DR

This paper establishes a power-law upper bound on diffusion in a 1D nonlinear Anderson model, removing previous restrictions and resolving a longstanding problem in nonlinear disordered systems.

Contribution

It proves a diffusion bound without decay restrictions on nonlinearity, advancing understanding of nonlinear disordered quantum systems.

Findings

Power-law diffusion upper bound proved

Decaying condition on nonlinearity removed

Resolution of Bourgain's diffusion problem

Abstract

In this paper, we prove the power-law in time upper bound for the diffusion of a 1D discrete nonlinear Anderson model. We remove completely the decaying condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud. 163: 21--42, 2007.). This gives a resolution to the problem of Bourgain (Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear disordered systems. The proof uses a novel ``norm'' based on tame property of the Hamiltonian.