2005/03/03 by Fernando Lledó, Lledó, Fernando, Olaf Post +1
Computer Science · Mathematics · #20E26 #35P15 #57M10 #58J50 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math-ph/0503005
openalex publication_date 2005/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper we consider Riemannian coverings (X,g) → (M,g) with residually finite covering group Γ and compact base space (M,g). In particular, we give two general procedures resulting in a family of deformed coverings (X,g_\eps) → (M,g_\eps) such that the spectrum of the Laplacian Δ(X_\eps,g_\eps) has at least a prescribed finite number of spectral gaps provided \eps is small enough. If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that \spec Δ(X,g_\eps) has, in addition, band-structure and there is an asymptotic estimate for the number N(λ) of components of \spec \laplacian (X,g_\eps) that intersect the interval [0,λ]. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.