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Journal Article

On the spectrum of differential operators under Riemannian coverings


Polymerakis,  Panagiotis
Max Planck Institute for Mathematics, Max Planck Society;

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Polymerakis, P. (2020). On the spectrum of differential operators under Riemannian coverings. Journal of Geometric Analysis, 30(3), 3331-3370. doi:10.1007/s12220-019-00196-1.

Cite as: http://hdl.handle.net/21.11116/0000-0006-8E5F-1
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 Schr\"{o}dinger operator, extending a result due to Brooks.