On the spectrum of differential operators under Riemannian coverings

Abstract

For a Riemannian covering $p:M_2\to M_1$, 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 $p^{\ast }{E}_1\to M_2$. 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.