MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


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?


  • 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.


share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.