Generating spectral gaps by geometry

2004/06/15 by Fernando Lledó, Lledó, Fernando, Olaf Post +1
Mathematics · Physics and Astronomy · #58J50 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:58J50

paper · pdf · doi:10.48550/arxiv.math-ph/0406032

Some mistakes corrected (still 12 pages, 1 figure)

arxiv created 2005/06/23 · arxiv updated 2009/12/01

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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