On the spectrum of differential operators under Riemannian coverings

2018/03/08 by Polymerakis, Panagiotis · 1 citation
#35P15 #53C99 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1803.03223

Abstract

For a Riemannian covering p \colon M₂ → M₁, we compare the spectrum of an essentially self-adjoint differential operator D₁ on a bundle E₁ → M₁ with the spectrum of its lift D₂ on p^*E₁ → M₂. We prove that if the covering is infinite sheeted and amenable, then the spectrum of D₁ is contained in the essential spectrum of any self-adjoint extension of D₂. We show that if the deck transformations group of the covering is infinite and D₂ is essentially self-adjoint (or symmetric and bounded from below), then D₂ (or the Friedrichs extension of D₂) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M₁ 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.

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