Commentarii Mathematici Helvetici
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Spectral convergence of manifold pairs
Karsten Fissmer (1) and Ursula Hamenstädt (2)
(1) Perthesanlage 7a, 53119, Bonn, GERMANY(2) Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, BONN, GERMANY
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 $c\geq 0$ 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.
Keywords: Laplace operator, spectrum, Lipschitz convergence, spectral convergence, multiplicities