T. Carleman's method on eigenvalues asymptotics What is the best available less or more modern introduction to the subject?
 A: 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.


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*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.

*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.

*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.
A: 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).
