Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

TL;DR

This paper establishes almost sure local well-posedness for the cubic nonlinear Schrödinger equation with higher order operators, extending previous results to more general operators and dimensions using directional space estimates.

Contribution

It introduces new estimates in directional spaces to prove well-posedness for Schrödinger equations with operators of degree  , generalizing known results for the standard Laplacian.

Findings

Proves almost sure local well-posedness for higher order Schrödinger operators.

Extends results to any spatial dimension under natural assumptions.

Utilizes directional space estimates to achieve these extensions.

Abstract

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $\sigma \geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schr\"odinger equation (i.e. $\mathscr{L} = \Delta$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.