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

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 on general $\mathscr{L}$, whose Fourier symbol might be sign changing