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Generating spectral gaps by geometry

Fernando Lledó·Olaf Post
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Abstract

Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let $Δ_X$ be the Laplacian on a non-compact 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 $Δ_X$ associated with a suitable class of manifolds $X$ with non-abelian covering transformation groups $Γ$. 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.

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Fernando Lledó, Olaf Post. 2005-06-23. Generating spectral gaps by geometry. https://arxiv.org/abs/math-ph/0406032

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