Reference request to the theory of the spectrum of unbounded linear operators: Proof for why the spectrum of the Laplacian is discrete on compact sets

Question

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?