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Journal Article

#### On the spectrum of differential operators under Riemannian coverings

##### External Resource

https://doi.org/10.1007/s12220-019-00196-1

(Publisher version)

##### Fulltext (public)

arXiv:1803.03223.pdf

(Preprint), 350KB

Polymerakis_On the spectrum of differential operators under Riemannian coverings_2020.pdf

(Publisher version), 525KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

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

##### Abstract

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.