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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

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up vote 5 down vote accepted

If the first (lowest) eigenfunction $f_0$ is smooth, then $V$ is smooth. Indeed, assuming $M$ connected, it is a classical fact that $f_0$ doesn't vanish (it is the first case of Courant's nodal theorem for instance), and obviously $V=\lambda_0 +\Delta f_0/f_0$.

With boundary and Neumann condition, the same argument applies, and with Dirichlet condition, $V$ is at least smooth in the interior.

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