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Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to infinity. What is the easiest/simplest way to estblish this fact? By simplest, I mean an abstract, formal, approach to the problem, not too heavily based on technical calculation. In the same line of thought, is there a philosophical/intuitive reason to see why the spectrum should behave like this?

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    $\begingroup$ I guess the route I think of is like this. You show that the resolvent (or the semigroup) of $\Delta_g$ maps $L^2$ into an appropriate Sobolev space. The latter is compactly embedded in $L^2$ by Rellich's theorem, so the resolvent is a compact operator. So the eigenvalues of the resolvent (with multiplicity) converge to zero, and hence the eigenvalues of the Laplacian converge to infinity. $\endgroup$ – Nate Eldredge May 18 at 14:04
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    $\begingroup$ To Nate Eldredge's route, which is a standard one, I would add: $\Delta_g$ is self-adjoint and, by elliptic regularity, its domain is the Sobolev space $H^2$. So the resolvent factors through $H^2$, hence is compact by Rellich's Theorem. $\endgroup$ – Sönke Hansen May 18 at 14:29
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    $\begingroup$ @Nate: So I guess this changes the question into a request for an abstract/formal approach to the Rellich embedding theorem :) I've never actually seen a proof for a general Riemannian manifold . . . might an easy proof exist for spaces with enough symmetries. For example, what does Rellich look like for a compact homogeneous space? $\endgroup$ – Max Schattman May 18 at 14:30
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    $\begingroup$ @SönkeHansen: I was thinking about that, but I seem to recall that the proof of elliptic regularity goes through the Sobolev embedding argument itself, so I wasn't sure off the top of my head whether that is circular. $\endgroup$ – Nate Eldredge May 18 at 14:57
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    $\begingroup$ @MaxSchattman: I think you just use a partition of unity. When $M$ is compact, can find a finite number of smooth functions $\psi_k$, each compactly supported in a coordinate chart, whose sum is 1. Now if I have a sequence of functions $f_n$ in my Sobolev space, the functions $f_n \psi_1$ are in the corresponding Sobolev space of the coordinate chart. Apply the Euclidean version of Rellich to pass to an $L^2$ convergent subsequence, and repeat iteratively for $\psi_2, \psi_3, \dots, \psi_k$. Adding them up, you have a subsequence of the original $f_n$ converging in $L^2$. $\endgroup$ – Nate Eldredge May 18 at 15:09
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A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to those of the Laplacian. Then two properties that you stated follow from the general properties of compact operators.

This approach is due to Hilbert. He wrote a long series of papers on the subject in 1904-1912. (see also his book with Courant, Methods of mathematical physics). Modern expositions are usually based on Hilbert's ideas. The notions of Hilbert space and compact operator were essentially distilled from these works.

An earlier philosophy of Poincaré interprets these eigenvalues as poles of certain meromorphic function in the plane (in modern language it is essentially the resolvent), and the poles of a meromorphic function are isolated and tend to infinity. Poincaré was the first to prove under general conditions the existence of an infinite sequence of eigenvalues tending to infinity. (Sur les équations de la physique mathématique, Rend. Circ. mat. Palermo, 1894 8, 57-155.)

Three remarks should be made:

a) At the time of Poincaré and Hilbert, the modern formal notion of compact Riemannian manifold did not exist. (The notion of compact was introduced by Aleksandrov and Urysohn in 1924, and the notion of manifold by Weyl 1913, and only for dimension 2). Even in the classical book on the subject by Titchmarsh, Eigenfunction expansions..., 1958, the words "manifold" and "compact" are not mentioned!

b) There was a very large number of problems about vibrations which were solved "explicitly" in 18th and 19th century. So Hilbert and Poincaré had a lot of "empirical material" to generalize. Fourier should be mentioned: his work inspired Hilbert and Poincare. He solved many concrete eigenvalue problems but had no tools to attack the general problem.

c) As a physical fact, existence of infinite discrete spectrum was first discovered (for the case of a string) by a music theorist Marin Mersenne in 1637. This started a long story of research about these eigenvalues.

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What follows is only an answer to the philosophical/intuitive question and follows only one specific point of view (there are probably many others).

Eigenvalue problem for the Laplace operator on a Riemannian manifold $(M,g)$ is a quantization of the problem of the classical motion of a particle "freely moving", i.e. following geodesics, on $(M,g)$. More precisely, the phase space of this classical mechanics problem is the symplectic manifold $T^{*}M$ (cotangent bundle, with its standard symplectic form) and the Hamiltonian is the function on $T^{*}M$ given by $H=|p|_g^2/2$, where $p$ is linear form on cotangent fibers.

Eigenspaces of the Laplace operator with eigenvalues less than E are quantization of the part of the phase space with $H <E$. If M is compact, then $H<E$ is a subset of $T^{*}M$ of finite volume and so its quantization will produce a finite dimensional vector space (of dimension roughly the volume in units of Planck constant $\hbar$): in particular, there will be finitely many eigenvalues below $E$, all with finitely many multiplicities. When $E$ goes to infinity, the volume of $H<E$ goes to infinity and so eigenvalues go to infinity.

In fact, this picture tells you that the number of eigenvalues less than $E$ should be of the order of the volume of the set $H<E$, which is of order $vol(M,g) E^{dim(M)/2}$, where $vol(M,g)$ is the volume of $(M,g)$, and $dim(M)$ is the dimension of $n$. This is indeed true (Weyl law).

(Maybe a trivial comment, but one never knows: it is probably helpful to think about the case of the circle, and more generally flat tori, where everything is trivial Fourier analysis, before thinking about more general situations).

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    $\begingroup$ Even in the "trivial" flat $m$-torus case, where the eigenvalues are given by an explicit formula, nontrivial analytic number theory is needed to understand the asymptotics the spectral function for $m\geq 2$. The principal term is of course controlled by the (co)volume, but already in the simplest case of square torus, estimating the error term is the Gauss circle problem. $\endgroup$ – Victor Protsak May 18 at 18:00

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