On the Spectrum of Differential Operators Under Riemannian Coverings

Abstract

For a Riemannian covering p : M 2 → M 1 , we compare the spectrum of an essentially self-adjoint differential operator D 1 on a bundle E 1 → M 1 with the spectrum of its lift D 2 on p * E 1 → 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.

Bottom of Spectra and Amenability of Coverings

“…Under the assumption that the Ricci curvature of M 0 is bounded from below, this is the main result in [3]. Finally, the second author of this article established Theorem C in full generality [23]. We review this development in more detail in Section 5.1.…”

“…Based on Theorem 5.8, the second author was able to establish Theorem C in full generality [23,24]. An important step of the proof is the establishment of an analogue of Brooks's Theorem 5.1 for manifolds with boundary, where we are interested in the Neumann spectrum of (the Laplacian of) the manifold.…”

“…For r > 0 we denote by G r the finite subset of g ∈ π 1 (M 0 , x) such that g contains a loop of length at most r. Strictly speaking, allowing that M 1 need not be connected is not contained in [22]. It is, however, only a trivial extension of [22, Theorem 6.1] and will be used in our argument below.…”

“…Polymerakis, Theorem 6.1 in[22]). Let p : M 1 → M 0 be a Riemannian covering with M 0 closed and M 1 possibly non-connected.…”

Spectral estimates and discreteness of spectra under Riemannian submersions

“…where S (F ) is the Friedrichs extension of S considered as in (2), and D(·) denotes the domain of the operator. More details on the renormalization of Schrödinger operators may be found in [14,Section 7]. Given f ∈ C ∞ c (M), it is straightforward to compute…”