On the bottom of spectra under coverings

Abstract

For a Riemannian covering M 1 → M 0 of connected Riemannian manifolds with respective fundamental groups 1 ⊆ 0 , we show that the bottoms of the spectra of M 0 and M 1 coincide if the right action of 0 on 1 \ 0 is amenable.

On the Spectrum of Differential Operators Under Riemannian Coverings

“…In particular, there exists ϕ ∈ C ∞ (M ∂M) with Sϕ = λ 0 (S)ϕ, which is positive in the interior of M. It is worth to point out that the smooth eigenfunctions of the preceding proposition do not have to be square-integrable. The following corollary is a consequence of Proposition 4.18 (an alternative proof can be found in [2]). The following results describe the behavior of the spectrum of Schrödinger operators under finite sheeted coverings.…”

“…Since ψ u(y) is a composition of ψ u with an element of Γ 1 , we obtain the same uniform estimates up to order d for ψ u(y) , for all u(y). Recall the definition of ψ y in (2). Consider a ball B(z ′ , δ) of the lifted system on M 2 , which intersects supp ψ y , and the corresponding coordinate system.…”

“…Since χ is compactly supported, it follows that Q is finite. The proof of the following lemma is essentially presented in [2], but since we are in a different situation here, we repeat it. Lemma 4.9.…”

“…Bérard and Castillon [4] extended this result by showing that if the covering is amenable and the underlying manifold is complete with finitely generated fundamental group and without boundary, then the bottom of the spectrum of any Schrödinger operator is preserved. Recently, it was proved in [2] that the bottom of the spectrum of a Schrödinger operator is preserved under amenable coverings, without any topological or geometric assumptions.…”

“…A Schrödinger operator on such a manifold does not have a unique self-adjoint extension, when restricted to the above domain, and we are interested in the spectrum of its Friedrichs extension. According to [2], if the covering is amenable, then the bottoms of the spectra of S 1 and S 2 coincide. The amenability is used only to establish λ 0 (S 2 ) ≤ λ 0 (S 1 ), since the inverse inequality holds for any covering, where λ 0 stands for the bottom of the spectrum.…”