Diffusion bound for the nonlinear Anderson model

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.

While our Birkhoff normal form theory still works (from page 5 to page 17), we find the derivatives estimates on page 19 can not be obtained without any prior estimates of type H^a for small a>0

References in corpus (2)

• Logarithmic bounds on Sobolev norms for time-dependent linear Schrödinger equations
• Quasi-periodic solutions of nonlinear random Schrödinger equations

Cited by in corpus (1)

• An eigensystem approach to Anderson localization for multi-particle systems