This question might already have been asked on this site and I have asked a similarish question before: Reference request to the theory and examples of the spectrum of the Laplace-Beltrami operator on different compact manifolds Unfortunately none of the posted questions and posted answers to my previous question have helped me with this: Which, if any, source contains a clear and rigorous treatment of the spectrums of unbounded linear operators? Moreover, precisely what is the argument for the discreteness of the spectrum of the Laplacian operator $\Delta = \sum_{i=1}^n\frac{d^2}{dx_i^2}$? Brief search on the quite dense book, "Methods of Modern Mathematical Physics, Analysis of Operators" by Reed and Simons produces at least one result which shows that the resolvent of the Dirichlet Laplacian on bounded sets is compact. This might be close to what I am looking for (or imply it immediately), but I am really looking for a concise reference which I can verify that shows the discreteness of $\sigma(\Delta)$ when $\Delta$ is restricted on compact sets. But while we are at it, I wouldn't mind learning more about the spectrum of said operator(s) in general. Is it so that the book by Reed and Simon contains all that I need, but it just takes work to find the right results? Or is there some other reference I should take a look at?
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1I'm not sure that the resolvent is compact when restricted to any compact set (probably somebody may clarify this). In some cases it is, for example when restricted to a compact domain (smooth manifold with smooth boundary) and considered as acting on Sobolev spaces $H_0$ (vanishing on the boundary). I think in general in the compact case you might want to choose the function spaces where the operator is acting carefully so that you end up with a Fredholm operator. One of the ways to prove that the spectrum is discrete is by showing that the operator is self-adjoint and is Fredholm. – Overflowian May 16 '23 at 17:05