arXiv subjects
Publications and source records attributed to Panagiotis Polymerakis.
We prove that if a Riemannian covering preserves the bottom of the spectrum of a Schrödinger operator, which belongs to the discrete spectrum of the operator on the base manifold, then the covering is amenable.
For a Riemannian covering $π\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of $M_0$.
For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $D_{1}$ is contained in the essential spectrum of any self-adjoint extension of $D_{2}$. We show that if the deck transformations group of the covering is infinite and $D_{2}$ is essentially self-adjoint (or symmetric and bounded from below), then $D_{2}$ (or the Friedrichs extension of $D_{2}$) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if $M_{1}$ is closed, then $p$ is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.
For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $Γ_1\subseteqΓ_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $Γ_0$ on $Γ_1\backslashΓ_0$ is amenable.