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

Let −Δ denote the Dirichlet Laplace operator on a bounded open set in Rd. We study the sum of the negative eigenvalues of the operator −h^2Δ − 1 in the semiclassical limit h → 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.