# Generating spectral gaps by geometry

June 15, 2004

Motivated by the analysis of Schr\"odinger operators with periodic potentials
we consider the following abstract situation: Let $\Delta_X$ be the Laplacian
on a non-compact Riemannian covering manifold $X$ with a discrete isometric
group $\Gamma$ acting on it such that the quotient $X/\Gamma$ is a compact
manifold. We prove the existence of a finite number of spectral gaps for the
operator $\Delta_X$ associated with a suitable class of manifolds $X$ with
non-abelian covering transformation groups $\Gamma$. This result is based on
the non-abelian Floquet theory as well as the Min-Max-principle. Groups of type
I specify a class of examples satisfying the assumptions of the main theorem.

Keywords:

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