Existence of Spectral Gaps, Covering Manifolds and Residually Finite Groups

Abstract

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, gε) → (M, gε) such that the spectrum of the Laplacian ∆ (Xε,gε) has at least a prescribed finite number of spectral gaps provided ε is small enough.If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec ∆ (X,gε) has, in addition, band-structure and there is an asymptotic estimate for the number N (λ) of components of spec ∆ (X,gε) 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.

An overview of periodic elliptic operators

“…In more general situations, many issues of spectral theory of periodic Laplacians (and more general operators) on cocompact coverings have been studied, such as relations to the amenability of Γ [55], existence of a band-gap structure [60,373], density of states [2,119], Shnol -Bloch theorems [230], presence of pure point spectrum [230], gaps creation [181,182,232,233,270,281], absence of a singular continuous spectrum [184], representation of solutions [238,241,242], etc. See also [55,59,183,185,371] for related considerations.…”

Eigenvalue bracketing for discrete and metric graphs

“…As in the case of manifolds and Schrödinger operators (see e.g. [11,15,16]) we expect that the number of gaps should be large if the fundamental domain has "small" boundary ∂ V compared to the number of vertices V (H) and edges E(H) inside. In other words, a "high contrast" between the different copies of a suitable fundamental domain is necessary in order that our method works.…”

“…For the next statement, the metric graph need not to be equilateral. Abelian groups, finite extensions of Abelian groups (so-called type-I-groups) and free groups are examples of the large class of residually finite groups (see [16] for more details). For Abelian groups, the Floquet-Bloch decomposition can be used in order to calculate the spectrum of the operator on the covering, leading to a detailed analysis in certain models, see e.g.…”

“…For more details on residually finite groups we refer to [16] and the references therein. The next proposition is provided by Adachi [1] (see also [16,Section 5]).…”

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

“…Therefore it is often possible to compare different form domains while the domains of the representing operators remain unrelated. This fact allows, e.g., to develop spectral bracketing techniques in very different mathematical and physical situations using the language of quadratic forms [33,34].…”