Spectral convergence of manifold pairs

Abstract

Let $(M_i,A_i{)}_i$ be pairs consisting of a complete Riemannian manifold $M_i$ and a nonempty closed subset $A_i$. Assume that the sequence $(M_i,A_i{)}_i$ converges in the Lipschitz topology to the pair $(M,A)$. We show that there is a number $c\ge 0$ which is determined by spectral properties of the ends of $M_i-A_i$ and such that the intersections with $[0,c)$ of the spectra of $M_i$ converge to the intersection with $[0,c)$ of the spectrum of $M$. This is used to construct manifolds with nontrivial essential spectrum and arbitrarily high multiplicities for an arbitrarily large number of eigenvalues below the essential spectrum.