The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

Abstract

We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schrödinger group satisfies a natural $L^1-L^{\infty }$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.

32 pages, 4 figures