T. Carleman's method on eigenvalues asymptotics

What is the best available less or more modern introduction to the subject?

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Can you be more specific on the operators whose eigenvalues you are interested in? –  Liviu Nicolaescu Nov 12 '12 at 18:56
@Liviu: say, Laplace-Beltrami operator on the closed compact Riemannian manifold, or Dirichlet Laplacian in a (smooth, if required) domain. –  Fedor Petrov Nov 12 '12 at 19:11

All approaches I am familiar with are based on some Tauberian theorem: one obtains informations about various weighted counts of eigenvalues and then one removes the weights via some Tauberian theorem.

There are three implementations of this strategy that I know of.

1. One based on the heat equation. For Laplace operators this can be implemented without relying on pseudo-differential operators. A good reference is the book of Berline-Getzler-Vergne: Heat kernels and Dirac operators. Another good reference for this approach, and probably more readable is J. Roe: Elliptic operators, topology and asymptotic methods.

2. One approach based on zeta functions. This uses the technique of pseudodifferential operators. A nice reference is M. Shubin: Pseudo-differential operators and Spectral theory.

3. Finally, there is an approach pioneered by L. Hormander based on the wave equation which gives the most accurate results. Shubin's book above contains a nice and very clear exposition of this method. This uses quite a bit of machinery from the theory of pseudo-differential operators. You can also find an elementary description of this method (free of pseudo-differential operators) in Chapter 17, vol. 3 of Hormander's The Analysis of Linear Partial Differential operators. The presentation there is a bit dense and I found it a bit challenging.

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Re Liviu's reference 3, go back to H\"ormander's original article The spectral function of an elliptic operator'', Acta Math 1968. Really lovely article and good exposition (easier than the book). –  Rafe Mazzeo Nov 13 '12 at 2:53
@ Rafe: Recently I needed explicit and very detailed info on the wave kernel on a Riemann manifold. The approach via Hadamard's parametrix construction described in vol. 3 of Hormander's book seems the only approach that could do this. –  Liviu Nicolaescu Nov 13 '12 at 10:51
I found that C.Sogge's [Lectures on eigenfunctions][1] is also a wonderful reference [1]: mathnt.mat.jhu.edu/sogge/zju/0LecturesOnEigenfunctions.pdf –  user23078 Nov 23 '12 at 9:41

For the case of an elliptic operator on a smooth bounded domain see

L. Garding, On the asymptotic distribution of eigenvalues and eigenfunctions of elliptic differential operators. Math. Scand. 1, 237-255 (1953).

The journal is available at http://www.digizeitschriften.de/dms/toc/?PPN=PPN35397434X_0001

More modern literature deals with more complicated cases treated by more sophisticated methods (there are books by Shubin, Ivrii, chapters in the treatise by Hormander, papers by Birman and Solomjak etc).

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