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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 $C(n)=(2\pi)^{-n}\omega_n$, $\omega_n$ 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?

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  • $\begingroup$ the growth rate of $N_d(\lambda)$ remains the same, whether or not you count the multiplicity; identical eigenvalues have measure zero, unless they are due to some discrete symmetry of $\Omega$, but that only changes $N$ by some $\lambda$ independent factor. $\endgroup$
    – Carlo Beenakker
    Commented Jul 17, 2021 at 6:53
  • $\begingroup$ @CarloBeenakker, I don't think so. The multiplicity can be very large. For example, for rectangular domain, $\lambda_k$ has multiplicity as $\sqrt{k}$. Actually Nadirashvili proved that for general planar domain in $\mathbb{R}^n$, the multiplicity of $\lambda_k$ has upper bound $2k-1$. $\endgroup$
    – student
    Commented Jul 17, 2021 at 11:50

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