Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \subset \Omega \subset \mathbb R^3$ and the Dirichlet eigenvalues of $-\Delta_g$ on $\Omega$ are all simple (meaning they have multiplicity one)?
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1$\begingroup$ Yes, and such domains are generic. From perturbation theory, you know that given any vector field $V$ with associated flow $\phi_t$, and domain $\Omega$ with eigenvalue $\Lambda$, the spectrum of $\phi_t(\Omega)$ in a neighborhood of $\Lambda$ is either a smooth curve (if $\Lambda$ is simple) or a finite union of intersecting smooth curves (if not). You can show easily using domain variation formulas that for any non-simple $\Lambda$, there exists a vector field such that the spectrum of $\phi_t(\Omega)$ in a neighborhood of $\Lambda$ has at least two points a distance at least $ct$ apart. $\endgroup$– user378654Apr 17 at 16:13
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1$\begingroup$ Applying such flows iteratively will let you make the first $N$ eigenvalues simple while making an arbitrarily small deformation of the domain. You then take an appropriate limit that makes them all simple. $\endgroup$– user378654Apr 17 at 16:17
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$\begingroup$ Can you give some references for these claims? $\endgroup$– AliApr 21 at 14:08
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$\begingroup$ There's a book by Henroit on eigenvalues of elliptic operators on domains; he discusses the domain variation formulas and proves the claim I make. I think he also goes over the perturbation theory claim, or that can be found in Kato's book on the subject. It will probably be in Euclidean setting there, but everything works similarly on manifolds. Apologies for not having more precise references (I don't have the books in front of me), that's why I didn't write this as an answer. $\endgroup$– user378654Apr 22 at 16:56
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1$\begingroup$ One can ask a similar question for the Dirac operator on closed spin manifolds. Metrics with only simple Dirac eigenvalues were constructed by Mattias Dahl in dimension 3, but the problem is notoriously difficult and unknown in all higher dimensions. $\endgroup$– Bernd AmmannApr 22 at 17:14
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It does not answer the question, but provides an analogous statement for closed manifolds: On a closed (connected) manifold with a generic Riemannian metric all eigenvalues of the Laplace-Beltrami operator are simple.
This is proved in [K. Uhlenbeck, Generic Properties of Eigenfunctions, American Journal of Mathematics, Vol. 98, No. 4 (Winter, 1976), pp. 1059-1078 (20 pages)]
