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Mathematics, Differential Geometry, Spectral Theory
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.
