Spectra of manifolds less a small domain (Q1103243)

Let M be a compact connected \(C^{\infty}\) Riemannian manifold with Laplacian \(\Delta\). Then \(\Delta\) has pure point spectrum consisting of eigenvalues \(\lambda_ i\), \(i\geq 0\). Suppose M *\(\subset M\) is a compact submanifold, of codimension at least two, with tubular neighborhood \(B_{\epsilon}\) of radius \(\epsilon >0\). The associated Laplacian acts on L \(2(M-B_{\epsilon})\), where Dirichlet boundary conditions are imposed. Let \(\lambda_{i,\epsilon}\), \(i\geq 1\), denote the eigenvalues of \(\Delta_{\epsilon}\). It is well known that \(\lambda_{j,\epsilon}\downarrow \lambda_{j-1}\), as \(\epsilon\downarrow 0\). The authors compute the first correction term \(\beta (\epsilon)=\lambda_{j,\epsilon}-\lambda_{j-1}\), in terms of the eigenfunction \(\phi_{j-1}\) associated to \(\lambda_{j-1}\).