Almost Sure Scattering of the Energy Critical NLS in $d>6$

Abstract

We study the energy-critical nonlinear Schr\"{o}dinger equation with randomised initial data in dimensions $d>6$. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $H^s(\mathbb{R}^d)$ whenever $s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\}$. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results of Spitz in dimension 4. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.