How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law

Question

I am trying to prove the Weyl's asymptotic law for eigenvalues. In the document Weyl's law of p. $4$, I have managed to go up to the step $$\tilde{\nu_k} \leq \nu_k \leq \mu_k \leq \tilde{\mu}_k \implies N_{\tilde{\nu_k}}(\lambda) \leq N_{\nu_k}(\lambda) \leq N_{\mu_k}(\lambda) \leq N_{\tilde{\mu}_k}(\lambda).$$ Right after, it is indicated that $N_{\mu, i} (\lambda)=c_n \text{vol}(Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$. Does someone could explain to me how it is possible to find an explicit function for $N_{\mu, i} (\lambda)$?

Answer 1

You're not going to find an explicit function for $N$. (For example, $N$ is not even continuous.)

To prove the asymptotic expansion $$N(\lambda) = c_n|\Omega|\lambda^{n/2} + o(\lambda^{n/2})$$ for the Dirichlet counting function on a domain $\Omega$, the simplest way in bounded Euclidean domains --- and one that I assume you're to use, since you appear to have the inequality between Neumann and Dirichlet eigenvalues --- is to appeal to Dirichlet-Neumann bracketing.

This idea stems from Weyl's original papers (1911,1912) and is described very well in volume 1 of Courant-Hilbert, Methods of Modern Mathematical Physics. The basic idea is to cover $\Omega$ with small $\epsilon$-cubes $\{\chi_j\}$. Denote by $N_*^D$ the Dirichlet eigenvalue counting function and $N_*^N$ the Neumann eigenvalue counting function for the domain $*$. Let $\mathcal{S} = \{\chi\ |\ \chi\subset\Omega\}$ and let $\mathcal{T} = \{\chi\ |\ \chi\cap\Omega\neq\emptyset\}$.

Then argue that as $$ S = \bigcup_{\mathcal{S}} \chi \subset \Omega \subset \bigcup_{\mathcal{T}}\chi = T $$ by Dirichlet monotonicity and by the inequality you give in the question above, we have the following inequalities among counting functions: $$ N_S^D \leq N_\Omega^D \leq N_T^D \leq N_T^N $$

Next, use the fact that all the $\chi$ are isometric, Dirichlet monotonicity, and Neumann monotonicity to argue that the following holds: $$ (\#\mathcal{S})N_\chi^D \leq N_\Omega^D \leq (\#\mathcal{T})N_\chi^N $$ (Here $\#$ denotes the cardinality operator.)

Finally, use the explicit upper bounds you already know for the Dirichlet and Neumann counting functions of cubes to produce explicit upper and lower bounds for $N_\Omega^D$. The last step is to choose how $\epsilon\to 0$ as $x\to\infty$.

(There is a bit of a subtlety here, for as $\epsilon\to 0$, the bounds at a fixed $\lambda$ becomes very, very bad. This is an expression of the uncertainty principle: When the wavelength is larger than the size of the square, you can't capture the eigenfunction anymore. However, by letting $\lambda\to\infty$ as the squares shrink, you are able to keep the eigenfunctions "in resolution," as it were, and this lets you prove the theorem.)