# Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions. Unfortunately, I am a beginner in the theory of this equation.

Does anybody have a good reference for the time-dependent case on bounded domains? So in particular, I am interested in questions concerning the potential. How can it be chosen, such that we have existence/uniqueness.

Moreover, I would also be interested in time-dependent Schrödinger equations on two-dimensional closed manifolds like the sphere for example.

So equations like $i \partial_t \psi(x,t) = (-\Delta_{\mathbb{S}^2}+V(x,t) ) \psi(x,t)$

either in domains $(x,t) \in (a,b) \times (0,T)$ (for $(a,b)$ bounded) or $(x,t) \in M \times(0,T)$ where $M$ is a two-dimensional manifold?

They discuss $\mathbb R^n,$ but they seem to be mainly concerned with smoothness conditions on $V$, so the methods should apply to bounded manifolds (at least "flat" ones).