Two dimensional nonlinear Schrödinger equation with spatial white noise potential and fourth order nonlinearity

We consider NLS on 𝕋^2 with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and renormalized approximating equations. In particular we extend a previous result by A. Debussche and H. Weber available in the cubic and sub-cubic setting.