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I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature.

I'd appreciate multiple suggestions that explore the topic using different approaches too.

I'm particularly interested in the variational characterization of the eigenvalues and eigenfunctions.

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    $\begingroup$ In what context? Domains in Euclidean space (what kind of boundary conditions)? If you're working on manifolds, there is Chavel's Eigenvalues in Riemannian Geometry. $\endgroup$ – Nate Eldredge Jun 14 '17 at 14:29
  • $\begingroup$ @NateEldredge Thanks for your comment and the reference, I'll check it out. I was deliberately vague in phrasing the question, because I'd like to cast a wide net here. However, if I need to be more specific, I was actually thinking of Dirichlet and Neumann eigenvalues and eigenfunctions on $\Omega \subset \mathbb{R}^n$ bounded. $\endgroup$ – user89890 Jun 14 '17 at 15:48
  • $\begingroup$ variational characterization is given in Evans PDE book. $\endgroup$ – Piyush Grover Jun 14 '17 at 16:35
  • $\begingroup$ On $\mathbb R^n$, there's not much to discuss, the Fourier transform gives you an explicit spectral representation (and in particular $-\Delta$ doesn't have eigenvalues). $\endgroup$ – Christian Remling Jun 14 '17 at 23:26
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Actually, it is dangerous to answer this question, since there are a lot of good resources in this direction and there are many famous professors here that know this field much better than me. But I want to introduce a nice book to you which I believe it is a nice one in your direction:

"Spectral Theory in Riemannian Geometry", which is written by "Olivier Lablée" here.

It is very readable book, specially chapter $6$: "Can one hear the holes of a drum?"

Also, you can see the book "Functional Analysis, Spectral Theory, and Applications" which is written by "Einsiedler" and "Ward" here.

I hope these books be useful for beginning (as for study and also for typical answer), specially case one.

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I took a reading course based on Sogge's book "Hangzhou Lectures on Eigenfunctions of the Laplacian" a long time ago. This may serve as a standard reference because most of the results mentioned in this book are well-known among researchers.

The subject of spectral geometry of elliptic operators is very deep, and you should be able to read more in-depth articles afterwards suitable for your interest. It is really easy to ask open questions yourself and finding them difficult to answer, and then you may learn something by reading through literature. This may sound horribly generic, but the subject is connected to many other fields of mathematics (abstract harmonic analysis, microlocal analysis, geometric analysis, Hodge theory, to name a few). So I think listing an exhaustive list may be impossible.

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A good recent survey is Geometrical Structure of Laplacian Eigenfunctions by Grebenkov and Nguyen, with lots of nice pictures and over 500 references.

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