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.

These 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).

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).