WEYL’S LAW
MATT STEVENSON
Abstract. These are notes for a talk given in the Student Analysis Seminar at the University of Michigan. The n Laplacian on a bounded domain in R has a discrete set of Dirichlet eigenvalues, accumulating only at ∞. Let N(λ) be the number of eigenvalues less than λ, then Weyl’s law asserts that the first term of the asymptotic expansion of N(λ) depends only on the dimension and the volume of the domain. Here we sketch the 1912 proof of Weyl, following the expositions of [1], [5], [6].
1. Introduction Let Ω ⊂ Rn be a bounded domain, then we are interested in solving the eigenvalue problem ∆ϕ = λϕ on Ω (with either Dirichlet or Neumann boundary conditions). The spectral theorem asserts that there exists an 2 orthonormal basis {un} of L (Ω) consisting of (Dirichlet or Neumann) eigenfunctions, where the corresponding eigenvalues λ1 ≤ ... ≤ λn ≤ ... → +∞ accumulate only at +∞ and each have finite multiplicity. Let N(λ) := #{λj ≤ λ} be the eigenvalue counting function, then it is natural to consider the asymptotic behaviour of N(λ) as λ → +∞. Consider the following simple examples. Example. Let Ω = [0, a], then the Dirichlet eigenvalue problem becomes the following:
( d2 dx2 ϕ(x) = λϕ(x), ϕ(0) = ϕ(a) = 0.