We treat some classes of linear and semilinear stochastic partial differential equations of Schrödinger type on $\mathbb{R}^d$, involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-Itô chaos expansions and pseudodifferential operator methods. The initial data and potential term of the Schrödinger operator are assumed to be generalized stochastic processes that have spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate (intersections of weighted) Sobolev-Kato-Kondratiev spaces.