Weyl's_law
In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain . In particular, he proved that the number, , of Dirichlet eigenvalues (counting their multiplicities) less than or equal to satisfies
where is a volume of the unit ball in .[1] In 1912 he provided a new proof based on variational methods.[2][3] Weyl's law can be extended to closed Riemannian manifolds, where another proof can be given using the Minakshisundaram–Pleijel zeta function.