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 $\{0\leq \lambda_1\leq\cdots\leq \lambda_k\} \cup [\lambda_{k+1},+\infty)$.