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 using 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.

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.