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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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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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If you know that the eigenfunction is nowhere vanishing -- which holds e.g. for the first eigenfunction -- then your question is a local question, i.e. you can write it in coordinates. Then it follows from regularity theory for elliptic equations as explained e.g. in the book by Gilbarg and Trudinger that the eigenfunction is smooth if and only if the potential is smooth.

Even if the potential is non-smooth, there are some weaker forms of regularity available. I will be happy to provide details if somebody requests it.

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