Hello,

I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the Schrödinger Operator $L=-\Delta+ V$ acting on (smooth) functions. Do you know for what kind of potentials $V:M\rightarrow \mathbb{R}$, the eigenfunctions will be smooth?

Explicitly:

Are the eigenfunctions smooth if V is bounded? Or is it necessary that the potential is smooth?

What happens, if the manifold M has a boundary $\partial M$ with Dirichlet/Neumann boundary conditions assumed. Does the regularity of the eigenfunctions depend on these boundary conditions?

It would be also helpful, if you tell me good textbooks where I can read about the above problems.

Regards