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Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are looking at Neumann eigenvalues).

A) What can be said in general about the geometry of $\Omega$?

B) Are there any pathological examples showing that the question is in some sense naive? e.g., examples of domains with discrete spectrum but strange geometry.

C) Are there examples of domains with a finite discrete spectrum followed by a continuous spectrum? i.e., something like $\{0\leq \lambda_1\leq\cdots\leq \lambda_k\} \cup [\lambda_{k+1},+\infty)$.

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This is more of a literature pointer than an answer. In the case of Dirichlet boundary conditions, the answer to the last question is yes: for example, the cross $$ C:=\{|x_1|\leq 1\}\cup \{x_2\leq 1\}\subset \mathbb{R}^2 $$ and the L-shaped domain $C\cap \{x_1\geq 0,x_2\geq 0\}$ both have exactly one eigenvalue $\lambda$ followed by a continuous spectrum $[\lambda',\infty)$ with $\lambda'>\lambda.$ See this talk (in Russian) and the references at the 19 minutes mark, the original one seem to be Schult, Ravenhall, Wyld' 89.

For Neumann boundary conditions, S. A. Nazarov indicates [2] that there are no such simple examples, but discussed multi-dimensional case.

[2]: Nazarov, S. A., Discrete spectrum of cranked, branching, and periodic waveguides, St. Petersbg. Math. J. 23, No. 2, 351-379 (2012); translation from Algebra Anal. 23, No. 2, 206-247 (2011). ZBL1238.35075, MR2841676.

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