Let $Omega$ be a Lipschitz domain in $mathbb{R}^n$, and let $N(lambda)$ be the number of Dirichlet Laplacian eigenvalues less than or equal to $lambda$. The famous Weyl’s law says that as $lambda$ goes to infinity, the asymptotic growth of $N(lambda)$ is like $C(n)|Omega|lambda^{frac{n}{2}}$, where $C(n)=(2pi)^{-n}omega_n$, $omega_n$ is the volume of the unit ball in $mathbb{R}^n$.

Note that in the definition of $N(lambda)$, the multiplicity of Dirichlet eigenvalues have been considered. How about Weyl’s law for the number of distinct eigenvalues? More precisely, let $N_d(lambda)$ be the number of distict Dirichlet Laplacian eigenvalues less than or equal to $lambda$. What is the growth rate of $N_d(lambda)$? Are there any references?