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Generating spectral gaps by geometry
, 2004
"... Abstract. Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let ∆X be the Laplacian on a noncompact Riemannian covering manifold X with a discrete isometric group Γ acting on it such that the quotient X/Γ is a compact manifold. ..."
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Abstract. Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let ∆X be the Laplacian on a noncompact Riemannian covering manifold X with a discrete isometric group Γ acting on it such that the quotient X/Γ is a compact manifold. We prove the existence of a finite number of spectral gaps for the operator ∆X associated with a suitable class of manifolds X with nonabelian covering transformation groups Γ. This result is based on the nonabelian Floquet theory as well as the MinMaxprinciple. Groups of type I specify a class of examples satisfying the assumptions of the main theorem. 1.