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What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?

In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \subseteq \mathbb{R}^3$ without boundary.

Let's say for instance that $f$ is an eigenfunction of a certain self-adjoint operator $A$. And let's say that $f$ has two nodal domains. Is there any general result saying about the boundedness of the domains? Are they both unbounded or one of them could be bounded?

Any suggestion would be very much appreciated!

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  • $\begingroup$ Check out arxiv.org/abs/1808.03342 $\endgroup$
    – Neal
    Commented Sep 18, 2018 at 18:18
  • $\begingroup$ @Neal Thanks! I will take a look! :) $\endgroup$
    – Onil90
    Commented Sep 18, 2018 at 18:26

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