Weyl law

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

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

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

This is a very amazing achievement of mathematicians, The realist meaning we can actually charge with the spectrum assessment.

In fact, we know, the only thing we know is that the eigenfunction with different spectra is orthogonal and we have a condition named maximum-minimum principle for the k eigenvalue. but how could we charge with the symptom of them? It seems not to be chargeable, though we have a $latex L^2$ isometry, the spectrum expansion, but it still not seems to be chargeable The main difficult comes from the compactness this just means a divide of the whole space, and we consider the X-ray transition from every point to the whole space, it needs to be passion kernel, or we change it to be a pare matrix.

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

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