Weyl law

In 1911 year, when Weyl is a young mathematician specticlizing in integrable system and PDE, He proved the important result about the asystomztion of eigenvalues of Dirichelet problem in $latex \Omega\subset R^n$ is a compact domain;i.e.

$latex N(\lambda)=(2\pi)^d Vol(\Omega)\lambda^{\frac{d}{2}}(1+o(1))$

Which in fact is a conjecture of *** in *** in published in 1910.

This is definitely a very amazing achievement of mathematician, The realist meaning we can actually charge with the spectrum asyspesion.

In fact, we know, the only thing we know is that the eigenfunction with different spectrum is orthogonal and we have are a cretition named maximum-minmum principle for the k eigenvalue. but how could we charge with the asymotum of them? It seems not to be chargeable, though we have a $latex L^2$ isometry, the spectrum expansion, ut it still not seems to be chargeable the main difficult come from the compacness this just mean a divide of the whole space, and we consider the X-ray tansigation from every point to the whole space, it need to be passion kernel, or we change it to be a pare matrix.


Yes, We can just look this phenomenon as there are two different world, one is the real world in the Ecliud space, there other is the a wave function world, in the second world it is composted by the unique of the solution $latex u$ for $latex \Delta u=\lambda u$ for some eigenvalue $latex \lambda$ and all units in this world is the translation and rescaling of u. Then thing become interesting, now how to understand the other guy, i.e. the other eigenvalue and eigenfunctions? They must be the u after some translation combine with rescaling and trslation and rotation!!! so there is a dynamic system action on it! and if we only let it to be affine map,i.e. combine only taslation and rotation, then we just get the all eigenfunctions with the same eigenvalue.

Now let us see what is it, it is just need to be compatible with the boundary condition, so it need to be moduli space that the boundary map is a measure that is arrive able by only affine translation of the function (we look it as a obsevalbel) So it is a restriction from a high dimensional space to the boundary of it.  and very fortunately it could assume a