Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain

Abstract

Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary.

10 pages; dedicated to Ari Laptev on the occasion of his 60th birthday

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