Program for computing Weyl's law

Question

I have calculated the solution to the Laplacian eigenvalue problem on the unit square
$qquad - Delta u(x,y) = lambda u(x,y) text{ on } {[0,1]}^2$
with the Dirichlet's boundary condition ($u = 0$).
My question is as follows:
Is it possible to write a program which calculate $N(t)$, the number of eigenvalues less than or equal to $t$?
Please help me.

Daniel Huber · Answer

Note that you should not name symbols with capital letters, because the is danger to overwrite a system symbol. In your case N is used by the system to coerce a machine  number. So let's call the function selEV. As there are an infinity of Eigenvalues, we must restrict the number of Eigenvalues to calculate. You must guess an upper bound of expected eigen values: maxn. For an example I choose maxn=10 :
{ℒ, ℬ} = {-Laplacian[u[x, y], {x, y}],
    DirichletCondition[u[x, y] == 0, True]};

maxn = 10;
selEV[t_] :=  Select[DEigenvalues[{ℒ, ℬ},  u[x, y], {x, 0, π}, {y, 0, π}, maxn], # <= t &] // Length

And e.g. for t==7we get:
selEV[7]

(* 3 *)

Program for computing Weyl's law

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