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

Question

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?

Answer 1

The 2014 review article Existence, Uniqueness, and Construction of the Density-Potential Mapping in Time-Dependent Density-Functional Theory discusses the time-dependent Schrodinger equation and cites Existence of solutions for Schrödinger evolution equations.

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).

Answer 2

On bounded domains, for some nonlinear schrodinger equations, you might find papers by H. Brezis and T. Gallouet useful. The title of the paper is "Nonlinear Schrodinger evolution equations", and they're from around 1980.

There are also some remarks on Schrodinger with a potential on bounded domains in V. Sverak's PDE notes, which are available on his webpage. He has also mentioned several references.