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In virtually all books, lecture notes, and articles I've encountered, it is taken as a fact that the Laplace-Beltrami operator $\Delta$ has a discrete spectrum on compact Riemannian manifolds. Some texts attempt to reference supporting material, but the cited resources are often so extensive that a beginner would need to read hundreds of pages to verify whether the reference indeed supports the claim. Consequently, I am seeking helpful guides that can walk me through the necessary theory of eigenfunctions and eigenvalues of the Laplace operator: i) with simple examples in a Euclidean setting (no need to go beyond $\mathbb{R}^2$), and ii) on Riemannian manifolds (involving the Laplace-Beltrami operator).

Essentially, I am looking for a text which covers discussions such as 1.) What is spectrum for Laplacian in $\mathbb{R}^n$? and 2.) Spectrum of the Laplace operator with Neumann condition on intervals by introducing/referring to necessary theory and shows some examples of computing the said spectrum with some example boundary-initial value problems.

Additionally, I would like to comment that the post: What is spectrum for Laplacian in $\mathbb{R}^n$? offers useful insights into what I am interested in: understanding when and why the spectrum of $\Delta$ is what it is, and how we interpret $\Delta$ as an operator. However, I would prefer to refer to established sources, and the comments in the post do not delve into the depth I desire.

Edit: Rosenberg's book, The Laplacian on Riemannian manifold, seems to have some components which I am seeking: The book has a strong start with the basic examples, but then jumps, understandably, to heavier machinery with Hodge theory. I have no need for Hodge theory as of now. I am satisfied with a example rich text which takes the time to look at the eigenvalue-function problem with different domains and different boundary conditions.

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  • $\begingroup$ I haven’t checked out the book yet so I can’t promise it’s what you’re after, but have you checked out Rosenberg’s book about the Laplacian? $\endgroup$
    – Mr. Brown
    Apr 27, 2023 at 8:25
  • $\begingroup$ @Mr.Brown Rosenberg seems to have some components which I am after. But the book jumps almost immediately to Hodge theory, which might be a bit too advanced for my current purposes. But it will take some time before I am be certain. $\endgroup$ Apr 27, 2023 at 19:30

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You might be interested in Chapter 2 of Bump's Automorphic Forms and Representations. One main goal is studying the spectrum of the Beltrami-Laplace operator on compact quotients of the upper half plane, emphasizing their relationship to representations of $\mathrm{GL}_2(\mathbb{R})$. See especially Section 2.3, which develops a lot of the necessary theory and ends by proving the $\ell^2$-convergence of the eigenvalues of $\Delta_k$.

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    $\begingroup$ Thanks! The section 2.3 seems to be a good start for what I am looking for. Do you happen to know any source which is more "low level" example heavy, e.g. computation of the spectrum (with discussion regarding the theory) of $\Delta$ on $S^1, S^2, S^1\times S^2$ and some rectangular/partially rectangular domains? $\endgroup$ Apr 26, 2023 at 16:26
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Notes:

  • Yaiza Cazani's Notes. Contains many computations of the Weyl law in simple domains.

  • Grebenkov-Nguyen Notes. Section 3 of Grebenkov-Nguyen contains explicit computations of eigenfunctions for simple domains with a lot of symmetry.

  • Zelditch Notes. Section 2 gives explicit computations.

Books:

  • Eigenvalues in Riemannian Geometry by Chaval
  • The Laplacian on a Riemannian manifold by Rosenberg
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