On the spectrum of differential operators under Riemannian coverings

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.


Introduction
A basic problem in Geometric Analysis is the investigation of relations between the geometry of a manifold and the spectrum of a differential operator on it. In this direction, it is natural to study the behavior of the spectrum under maps which respect the geometry of the manifolds. In this paper, we deal with this problem for Riemannian coverings.
Let p : M 2 → M 1 be a Riemannian covering of connected manifolds with (possibly empty) smooth boundary. A Schrödinger operator S 1 on M 1 is an operator of the form S 1 = ∆ + V , where ∆ is the (non-negative definite) Laplacian and V : M 1 → R is smooth and bounded from below. For such an operator S 1 on M 1 , its lift on M 2 is the operator S 2 = ∆ + V • p. The first results involving possibly infinite sheeted coverings and establishing connections between properties of the covering and the (Dirichlet) spectra of S 1 and S 2 , are related to the change of the bottom (that is, the minimum) of the spectrum and were proved by Brooks [6,7]. He showed that if the underlying manifold is complete, of finite topological type, without boundary and the covering is normal and amenable, then the bottom of the spectrum of the Laplacian is preserved. Bérard and Castillon [4] extended this result by showing that if the covering is amenable and the underlying manifold is complete with finitely generated fundamental group and without boundary, then the bottom of the spectrum of any Schrödinger operator is preserved. Recently, it was proved in [2] that the bottom of the spectrum of a Schrödinger operator is preserved under amenable coverings, without any topological or geometric assumptions.
In this paper, we prove a global result about this problem in a more general context. Instead of comparing the bottoms of the spectra, we prove inclusion of spectra under some reasonable assumptions. Moreover, our context allows us to impose various boundary conditions on the operators (for instance, Dirichlet, Neumann, mixed and Robin), while the former results involve only Dirichlet conditions. Furthermore, our theorems are applicable to a broad class of differential operators, including Schrödinger operators with magnetic potential (that is, first order term), Dirac operators and Schrödinger (or Laplace-type) operators on vector bundles. It is worth to point out that the Hodge-Laplacian is a special case of the latter ones.
In order to simplify the statements of our results, we need to set up some notation. Consider a Riemannian or Hermitian vector bundle E 1 → M 1 endowed with a (not necessarily metric) connection ∇. Let D 1 be a (not necessarily elliptic) differential operator of arbitrary order on E 1 . We consider the pullback bundle E 2 := p * E 1 → M 2 endowed with the corresponding metric and connection, and the lift D 2 of D 1 .
As the domain of D 1 we consider the space of compactly supported smooth sections η, which (when M 1 has non-empty boundary) satisfy a boundary condition of the form a∇ n η + bη = 0, where n is the inward pointing normal to the boundary and a, b are functions on the boundary. The domain of D 2 is the space of compactly supported smooth sections, which (when the boundary of M 1 is non-empty) satisfy analogous boundary conditions to the sections in the domain of D 1 . We consider the operators D i restricted to the above domains as densely defined operators in L 2 (E i ), i = 1, 2.
For sake of simplicity, we present here special versions of our main results involving self-adjoint operators. The results are stated for infinite sheeted coverings, since this is the interesting case of amenable coverings. However, we also prove the analogous results for finite sheeted coverings. Our first result provides inclusion of the spectrum σ(D 1 ) of the closure of D 1 , as long as it is self-adjoint, in the essential spectrum σ ess (D ′ 2 ) of any self-adjoint extension D ′ 2 of D 2 .
Theorem 1.1. Assume that D 1 is essentially self-adjoint and let D ′ 2 be a self-adjoint extension of D 2 . If the covering is infinite sheeted and amenable, then σ(D 1 ) ⊂ σ ess (D ′ 2 ).
Recall that a Schrödinger operator on a complete manifold is essentially self-adjoint on the space of compactly supported smooth functions vanishing on the boundary (if it is non-empty). Therefore, in the context of Schrödinger operators, it follows that if the underlying manifold is complete and the covering is infinite sheeted and amenable, then the spectrum of S 1 is contained in the essential spectrum of S 2 .
An important case where the above theorem cannot be applied is that of Schrödinger operators on non-complete Riemannian manifolds. A Schrödinger operator on such a manifold does not have a unique self-adjoint extension, when restricted to the above domain, and we are interested in the spectrum of its Friedrichs extension. According to [2], if the covering is amenable, then the bottoms of the spectra of S 1 and S 2 coincide. The amenability is used only to establish λ 0 (S 2 ) ≤ λ 0 (S 1 ), since the inverse inequality holds for any covering, where λ 0 stands for the bottom of the spectrum. This motivates us to establish the following theorem, which compares the bottom λ 0 (D In particular, for Schrödinger operators, it follows that if the covering is infinite sheeted and amenable, then the bottom of the spectrum of S 1 is equal to the bottom of the essential spectrum of S 2 , without any topological or geometric assumptions. The above results involve amenable coverings. However, the deck transformations group of a (possibly non-amenable) covering provides information about the group of isometries of the covering space. This motivates us to work in a more general context than Riemannian coverings and prove that under some symmetry assumptions, an essentially self-adjoint differential operator does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we show the analogous result for the Friedrichs extension of a symmetric and bounded from below differential operator. In the context of Riemannian coverings, we obtain the following immediate consequences. Corollary 1.3. Assume that D 2 is essentially self-adjoint. If the deck transformations group of the covering is infinite, then D 2 does not have eigenvalues of finite multiplicity and in particular, σ(D 2 ) = σ ess (D 2 ). Corollary 1.4. Assume that D 2 is symmetric and bounded from below, and denote by D For Schrödinger operators, it follows that if the deck transformations group of the covering is infinite, then the spectrum of S 2 is essential, without any assumptions on the manifolds.
All the above results provide information about the spectra from properties of the covering (amenability or infinite deck transformations group). In the converse direction, Brooks [7] proved that if a normal Riemannian covering of a closed manifold (that is, compact without boundary) preserves the bottom of the spectrum of the Laplacian, then the covering is amenable. In this paper, we extend this result to Schrödinger operators and to not necessarily normal coverings. In the following theorem, we denote by h ess (M) the supremum of the Cheeger's constants over complements of compact and smoothly bounded domains of M. Theorem 1.5. Let p : M 2 → M 1 be a Riemannian covering with M 1 closed. Then the following are equivalent: (i) p is infinite sheeted and amenable, (ii) σ(S 1 ) ⊂ σ ess (S 2 ) for some/any Schrödinger operator S 1 on M 1 and its lift S 2 , (iii) λ 0 (S 1 ) = λ ess 0 (S 2 ) for some/any Schrödinger operator S 1 on M 1 and its lift S 2 , (iv) h ess (M 2 ) = 0.
It is worth to point out that Brooks proved his theorem in a quite complicated way, relying heavily on geometric measure theory. Our proof of the above theorem is significantly simpler and avoids the use of geometric measure theory. Furthermore, Brooks [6] proved that under some more general (but still quite restrictive) assumptions, if the bottom of the spectrum of the Laplacian is preserved, then the covering is amenable. In particular, these assumptions imply that the bottom of the spectrum of the Laplacian on M 1 is not in the essential spectrum. Moreover, he provided examples demonstrating that without these conditions, the bottom of the spectrum of the Laplacian may be preserved even if the covering is non-amenable. This suggests that under some assumptions on the geometry and the spectrum of the Laplacian on M 1 , the bottom of the spectrum is preserved under a weaker assumption than amenability of the covering. In this direction we prove the following result. Corollary 1.6. Let p : M 2 → M 1 be a Riemannian covering with M 1 complete. Let S 1 be a Schrödinger operator on M 1 with λ 0 (S 1 ) ∈ σ ess (S 1 ), and S 2 its lift on M 2 . If there exists a compact K ⊂ M 1 , such that the image of the fundamental group of any connected component of M 1 K in π 1 (M 1 ) is amenable, then λ 0 (S 1 ) = λ 0 (S 2 ).
The paper is organized as follows: In Section 2, we give some preliminaries. In Sections 3 and 4, we present the construction which is used in order to prove Theorem 1.2 and a more general result (Theorem 4.1) than Theorem 1.1. The proofs are given in Section 4, where we also present the analogous results for finite sheeted coverings. In Section 5, we study manifolds with high symmetry and establish Corollaries 1.3 and 1.4. In Section 6, we present an alternative proof of Brooks' theorem [7], extending it to not necessarily normal Riemannian coverings. In Section 7, we introduce the notion of renormalized Schrödinger operators, which is used to prove Theorem 1.5. Moreover, in this section we establish Corollary 1.6 and we present a simple example demonstrating that the behavior of the bottom of the spectrum of the connection Laplacian under a covering depends on the corresponding metric connection. Therefore, a main point in our results is the independence from the vector bundles, the connections and the differential operators.
Acknowledgements. I would like to thank Werner Ballmann and Dorothee Schüth for some very enlightening discussions and helpful remarks. I am also grateful to the Max Planck Institute for Mathematics in Bonn for its support and hospitality.

Preliminaries
We first recall some basic facts from functional analysis. For more details, see [13]. Let A : D(A) ⊂ H → H be a closed (linear) operator on a separable Hilbert space H over a field F, where F = R or F = C. The spectrum of A is given by The essential spectrum of A is defined as Recall that an operator is called Fredholm if its kernel is finite dimensional and its range is closed and of finite codimension. The discrete spectrum of A is the complement of the essential spectrum in the spectrum of A, that is, The approximate point spectrum of A, denoted by σ ap (A), is defined as the set of all where "⇀" denotes the weak convergence in H. The Weyl spectrum of A, denoted by σ W (A), is the set of all λ ∈ F, such that there exists a Weyl sequence for A and λ.
The following proposition is the characterization of the spectrum of a self-adjoint operator as the set of approximate eigenvalues and the well-known Weyl's criterion for the essential spectrum.
consists of isolated eigenvalues of A of finite multiplicity.
Since we are interested in closures of operators, we need the following elementary lemma, characterizing the approximate point spectrum and the Weyl spectrum of the closure in terms of the initial operator.
, is finite. In this case, this infimum is called the lower bound of B.
The spectrum of a self-adjoint operator A is contained in R and the bottom (that is, the minimum) of the spectrum and the bottom of the essential spectrum of A are denoted by λ 0 (A) and λ ess 0 (A), respectively. The following characterization of the bottom of the spectrum is due to Rayleigh.
If, in addition, A is the closure of an operator B : D(B) ⊂ H → H, then the bottom of the spectrum of A is given by Throughout the paper, manifolds are connected, with possibly empty, smooth and not necessarily connected boundary, unless otherwise stated. Let p : M 2 → M 1 be a Riemannian covering of m-dimensional manifolds, E 1 → M 1 a Riemannian or Hermitian vector bundle of rank ℓ and D 1 : Γ(E 1 ) → Γ(E 1 ) a differential operator of order d. Consider the pullback bundle E 2 := p * E 1 on M 2 , y ∈ M 2 and set x := p(y). Let U 2 be an open neighborhood of y, such that the restriction p| U 2 is an isometry onto its image U 1 . The lift D 2 : Γ(E 2 ) → Γ(E 2 ) of D 1 is the differential operator defined by U 2 ) * η)(p(z))), for any η ∈ Γ(E 2 ) and z ∈ U 2 . Without loss of generality, we may assume that U 1 is contained in a coordinate neighborhood and there exists a trivialization E 1 | U 1 → U 1 × F ℓ . With respect to this coordinate system and trivialization, D 1 is expressed as where A α are smooth maps defined on U 1 , with values ℓ × ℓ matrices with entries in F. Then, with respect to the lifted coordinate system and the corresponding trivialization E 2 | U 2 → U 2 × F ℓ , D 2 has the local expression is densely defined and its adjoint satisfies D ⊂ (D ad ) * . Since the adjoint is closed, it follows that D is closable.
A Schrödinger operator on a possibly non-connected Riemannian manifold M is an operator of the form S := ∆ + V , where ∆ is the Laplacian and V : M → R is smooth and bounded from below. If M is complete and without boundary, then S is essentially self-adjoint on C ∞ c (M), that is, the closure of S : If M is complete with non-empty boundary, then S is essentially self-adjoint on {f ∈ C ∞ c (M) : f = 0 on ∂M}. If M is non-complete, then S restricted to the above domain, does not have a unique self-adjoint extension, and we are interested in the Friedrichs extension of S. By abuse of notation, the spectrum and the essential spectrum of the above described self-adjoint operator are denoted by σ(S) and σ ess (S), respectively, and their bottoms by λ 0 (S) and λ ess 0 (S), respectively. These sets and quantities for the Laplacian on M are denoted by σ(M), σ ess (M) and λ 0 (M), λ ess 0 (M), respectively. Let p : M 2 → M 1 be a Riemannian covering of complete manifolds without boundary. For x ∈ M 1 and y ∈ p −1 (x), the fundamental domain of p centered at y is defined by Some basic properties of these fundamental domains are presented in [2]. It is clear that D y is closed and M 2 is the union of D y , with y ∈ p −1 (x). Moreover, ∂D y and the cut locus Cut(x) of x are of measure zero and p : D y ∂D y → M 1 C 0 is an isometry, where C 0 is a subset of Cut(x). The following two lemmas are proved in [2]. The lemma after these is proved similarly to Lemma 2.6. In these lemmas and in the sequel, we denote open and closed balls by B and C, respectively. Lemma 2.5. If K ⊂ B(x, r), then p −1 (K)∩D y ⊂ B(y, r). In particular, if K is compact, then p −1 (K) ∩ D y is compact. Lemma 2.6. For any r > 0, there exists N(r) ∈ N, such that any z ∈ M 2 is contained in at most N(r) of the balls C(y, r), with y ∈ p −1 (x). Lemma 2.7. Consider the universal coverings p i :M → M i , i = 1, 2. For any r, r 0 > 0, there existsÑ(r, r 0 ) ∈ N, such that Finally, we recall the notions of amenable right action and amenable covering. For more details on amenable left actions, which are completely analogous to right actions, see [4,Section 2]. A right action of a countable group Γ on a countable set X is called amenable if there exists a Γ-invariant mean on L ∞ (X). The following characterization is due to Følner.
Proposition 2.8. The right action of a countable group Γ on a non-empty, countable set X is amenable if and only if for any finite G ⊂ Γ and ε > 0, there exists a non-empty, finite F ⊂ X, such that for all g ∈ G. Such a set F is called a Følner set for G and ε.
A countable group Γ is called amenable if the right action of Γ on itself is amenable. In this case, the right action of Γ on any countable set X is amenable. Moreover, it is clear that any right action on a non-empty, finite set is amenable.
A Riemannian covering p : M 2 → M 1 is called amenable if the right action of π 1 (M 1 ) on π 1 (M 2 )\π 1 (M 1 ) (that is, the set of right cosets of π 1 (M 2 ) in π 1 (M 1 ), when considered as deck transformations groups of the universal coverings) is amenable. Clearly, a normal covering is amenable if and only if its deck transformations group is amenable. Furthermore, finite sheeted coverings are amenable.
The following criteria for amenability of groups are immediate consequences of the definition and Proposition 2.8. Corollary 2.9. Any finitely generated group of subexponential growth is amenable. Corollary 2.10. A countable group Γ is amenable if and only if any finitely generated subgroup of Γ is amenable Corollary 2.11. Any countable solvable group is amenable.
Proof: From Corollaries 2.9 and 2.10, it follows that any countable abelian group is amenable. From the definition, it is clear that an extension of an amenable group by an amenable group is also amenable.

Coverings of manifolds with boundary
The aim of this section is to show the following proposition, according to which, any Riemannian covering of manifolds with boundary can be "extended" to a Riemannian covering of manifolds without boundary. In order to prove this proposition, we need to establish some auxiliary lemmas.  Proof: Since there exists a strong deformation retraction of N onto M, every loop in N can be homotoped to a loop in M. This implies that for any x ∈ M and y 1 , y 2 ∈ q −1 (x), there exists a path in q −1 (M) from y 1 to y 2 . Since M is connected, it follows that so is q −1 (M) and the restriction q : q −1 (M) → M is a covering of (connected) manifolds.

Spectrum of operators under amenable coverings
Throughout this section, we work in the following context, which is briefly described in the Introduction.
Let p : M 2 → M 1 be a Riemannian covering, E 1 → M 1 a Riemannian or Hermitian vector bundle endowed with a connection ∇ and D 1 : Γ(E 1 ) → Γ(E 1 ) a differential operator on E 1 . Let E 2 → M 2 be the pullback bundle, endowed with the corresponding metric and connection ∇, and D 2 : Γ(E 2 ) → Γ(E 2 ) the lift of D 1 . If M 1 has empty boundary, we consider the space of compactly supported smooth sections of E i as the domain of D i , that is, where n i is the inward pointing normal to ∂M i , i = 1, 2, a 1 , b 1 are real or complex valued functions (depending on whether the bundles are Riemannian or Hermitian) on ∂M 1 , and It is worth to point out that we do not impose any assumptions on a 1 and b 1 . When we refer to closability, symmetry or essential self-adjointness of D i , we consider the operator D i : From Lemma 2.4, the operator D i is closable and we denote by D i its closure, i = 1, 2.
Our aim in this section is prove Theorem 1.2 and the following more general version of Theorem 1.1.
If the covering is infinite sheeted and amenable, then

Partition of unity
In this subsection, we construct a special partition of unity, which is used in the sequel to obtain cut-off functions on M 2 .
Consider the universal coverings p i :M → M i and denote by Γ i the deck transformations group of p i , i = 1, 2. If M 1 has empty boundary, consider a Riemannian metric h, conformal to the original metric g, such that (M 1 , h) is complete. If M 1 has non-empty boundary, consider a Riemannian manifold N 1 containing M 1 , as in Proposition 3.1, and a Riemannian metric h, conformal to the original metric g, such that (N 1 , h) is complete. From now on, geodesics are considered with respect to h and its lifts. We denote by grad f and grad h f the gradient of a function f with respect to g and h (or their lifts), respectively. If M 1 has empty boundary, distances are considered with respect to h or its lifts. In this case, we denote the open (respectively, closed) ball of radius r around a point z by B(z, r) (respectively, C(z, r)). If M 1 has non-empty boundary, the distance between two points is considered in (N 1 , h) or its corresponding covering space. In this case, B(z, r) and C(z, r) stand for the corresponding balls in There exists a non-negative ψ u ∈ C ∞ c (M), such that supp ψ u ⊂ C(u, r + 1) and ψ u = 1 in C(u, r + 1/2). Moreover, if M 1 has non-empty boundary, ψ u can be chosen such that grad ψ u is tangential to ∂M .
, it follows that supp ψ u ⊂ C(u, r +1). Since ε < 1/8, the points where ν is not smooth are not in C(u, r + 1), which yields that In particular, grad h ψ u is tangential to ∂M , and so is grad ψ u , since g and h are conformal.
Let ψ u be a function as in the above lemma and for any y ∈ p −1 (x), fix u(y) ∈ p −1 2 (y) and g(y) ∈ Γ 1 , such that u(y) = g(y)u. Consider the functions ψ u(y) := ψ u • g(y) −1 inM and ψ y in M 2 defined by ψ y (z) := It is clear that ψ y ∈ C ∞ c (M 2 ), supp ψ y ⊂ C(y, r + 1) and ψ y ≥ 1 in C(y, r + 1/2), for any y ∈ p −1 (x). Moreover, if M 1 has non-empty boundary, then grad ψ y is tangential to ∂M 2 , for all y ∈ p −1 (x). From Lemma 2.6, there exists N(r + 2) ∈ N, such that for any z ∈ M 2 , the ball B(z, 1) intersects at most N(r +2) of the supports of ψ y , with y ∈ p −1 (x). Therefore, y∈p −1 (x) ψ y is locally a finite sum and hence, well-defined and smooth. If Consider the smooth partition of unity consisting of the functions with y ∈ p −1 (x).
and ϕ y > 0 in C(y, r + 1/2), for any y ∈ p −1 (x). If M 1 has non-empty boundary, then for any y, y ′ ∈ p −1 (x), we have that grad ψ y is tangential to ∂M 2 and ψ 1 = 0 in B(y ′ , r). This yields that grad ϕ y is tangential to and r > 0, such that supp η ⊂ B(x, r). If M 1 has non-empty boundary, we choose r large enough, so that B(u, r) ∩ ∂M = ∅. Consider a partition of unity associated to x, u and r as in (3) and for a finite P ⊂ p −1 (x), set χ := y∈P ϕ y .
Remark 4.4. Since P is finite, it follows that χ ∈ C ∞ c (M 2 ) and χθ ∈ Γ c (E 2 ). Since supp η ⊂ B(x, r), we have that supp θ is contained in the union of the balls B(y, r), with y ∈ p −1 (x). Therefore, if M 1 has non-empty boundary, from Remark 4.3, χθ satisfies analogous boundary conditions to η, that is, χθ ∈ D(D 2 ). Proposition 4.5. There exists a constant C, independent from P , such that for any z ∈ M 2 , we have D 2 (χθ)(z) ≤ C.

Proof:
Consider δ > 0, such that for any x ′ ∈ C(x, r + 1), the ball B(x ′ , 2δ) is evenly covered and contained in a coordinate neighborhood, and 2δ), D 1 has a local expression of the form (1), with A α smooth. This yields that in B(x i , δ), D 1 is expressed in the form (1), with A α smooth and bounded. For any such ball, we fix a coordinate system (which can be extended to the corresponding ball of radius 2δ) and a trivialization. Since C(x, r + 1) is covered by finitely many such balls, it follows that there exists C 1 > 0, such that in any of these balls, we have A α ≤ C 1 , for all multi-indices α of absolute value less or equal to the order d of D 1 .
Since η is smooth and compactly supported in B(x, r), there exists C 2 > 0, such that in any of these balls, denoting by (η (1) , . . . , η (ℓ) ) the local expression of η, we have that for all multi-indices α of absolute value less or equal to d, that is, we have uniform estimates up to order d for η (with respect to this system of trivializations). We lift these balls and the corresponding coordinate systems and trivializations to M 2 andM . Similarly, if ψ 1 = 0, we obtain uniform estimates up to order d for f 1 , which yield uniform estimates up to order d for ψ 1 (with respect to the lifted system on M 2 ).
Since ψ u is smooth and compactly supported in C(u, r + 1), which intersects finitely many balls of the lifted system onM , there exist uniform estimates up to order d for ψ u . Since ψ u(y) is a composition of ψ u with an element of Γ 1 , we obtain the same uniform estimates up to order d for ψ u(y) , for all u(y). Recall the definition of ψ y in (2). Consider a ball B(z ′ , δ) of the lifted system on M 2 , which intersects supp ψ y , and the corresponding coordinate system. It is clear that for any w ∈ p −1 2 (z ′ ), the lifted system onM contains the ball B(w, δ) and the corresponding coordinate system. From Lemma 2.7, there exists N(r + 1, δ) ∈ N, independent from y and z ′ , such that at mostÑ(r + 1, δ) such balls intersect the support of ψ u(y) . Since we have uniform estimates up to order d for ψ u(y) , which are independent from y ∈ p −1 (x), we obtain the same uniform estimates up to order d for ψ y , for all y ∈ p −1 (x). From Lemma 2.6, it follows that at most N(r + 1 + δ) of the supports of ψ y , with y ∈ p −1 (x), intersect the open ball B(z, δ), for any z ∈ M 2 . This yields that there exist uniform estimates up to order d for ψ 1 + y∈p −1 (x) ψ y .
Recall the definition of ϕ y in (3). Since the denominator is greater or equal to 1 and we have uniform estimates (independent from y) up to order d for the numerator and the denominator, we obtain the same uniform estimates up to order d for ϕ y , for all y ∈ p −1 (x). From Lemma 2.6, at most N(r + 1 + δ) of the supports of ϕ y , with y ∈ p −1 (x), intersect the ball B(z, δ), for any z ∈ M 2 . Therefore, we obtain uniform estimates up to order d for χ, which are independent from P Clearly, for z ∈ supp(χθ), we have that z ∈ B(y, r), for some y ∈ p −1 (x), and in particular, z is contained in a ball of the system. With respect to the corresponding coordinate system and trivialization, denoting by (θ (1) , . . . , θ (ℓ) ) the local expression of θ, we have where C 3 is the uniform bound up to order d for χ (which is independent from P ) and C(d, ℓ) is a constant depending only on d and ℓ.
Proof: Follows immediately from Proposition 4.5.

Amenable coverings
In this subsection we continue to work in the setting of the previous subsection, that is, we extend the covering p : M 2 → M 1 to a Riemannian covering p : N 2 → N 1 according to Proposition 3.1 (if needed) and consider conformal Riemannian metrics, such that the manifolds become complete. If M 1 has empty boundary, for x ∈ M 1 and y ∈ p −1 (x), we denote by D y the fundamental domain of p : M 2 → M 1 centered at y, with respect to these conformal Riemannian metrics. If M 1 has non-empty boundary, we denote by D y the part of the fundamental domain of p : N 2 → N 1 that lies in M 2 . Furthermore, volumes, integrals and L 2 -norms are with respect to the original Riemannian metrics. As in the previous subsection, consider the universal coverings p i :M → M i , denote by Γ i the deck transformations group of p i , i = 1, 2, and fix x ∈ M 1 and u ∈ p −1 1 (x). It is quite convenient to identify Γ 2 \Γ 1 with p −1 (x), that is, Γ 2 γ is identified with p 2 (γu), and study induced right action of Γ 1 on p −1 (x). Clearly, if y = p 2 (γu), for some γ ∈ Γ 1 , then y · g = p 2 (γgu), for any g ∈ Γ 1 . It is worth to point out that p is amenable if and only if this right action of Γ 1 on p −1 (x) is amenable.
For r > 0, consider the finite set G r := {g ∈ Γ 1 : d(u, gu) < r} and the subgroup G r of Γ 1 generated by G r . We are interested in the right action of G r on p −1 (x). The next remark is a simple description of the orbits of this action.
Proof: Assume to the contrary that the statement does not hold. Then there exists r 0 > 0, such that the action of G r 0 on p −1 (x) has only finitely many finite orbits O 1 , . . . , O k , for some k ∈ N. Since p is infinite sheeted, there exists also an infinite orbit O. Since the action of Γ 1 on p −1 (x) is transitive, for y i ∈ O i , there exists g i ∈ Γ 1 , such that y i · g i ∈ O, for i = 1, . . . , k. Then there exists R > 0, such that G r 0 ∪ {g 1 , . . . , g k } ⊂ G R and the action of G R on p −1 (x) has only infinite orbits. It is clear that so does the action of G r on p −1 (x), for any r ≥ R, which is a contradiction.
Let r > 0 large enough, so that B(u, r) ∩ ∂M = ∅, if M 1 has non-empty boundary. If p is infinite sheeted, we choose r ≥ R, where R is the constant from Lemma 4.8. Consider a partition of unity consisting of the functions ϕ 1 and ϕ y , with y ∈ p −1 (x), associated to x, u and r as in (3). For a finite P ⊂ p −1 (x), let χ := y∈P ϕ y and consider the sets Clearly, χ = 0 in B(y, r), for any y ∈ p −1 (x) Q. Since χ is compactly supported, it follows that Q is finite. The proof of the following lemma is essentially presented in [2], but since we are in a different situation here, we repeat it. Lemma 4.9. If p is amenable, then for any ε > 0, there exists a non-empty, finite P ⊂ p −1 (x), such that #(Q − ) #(Q + ) < ε.
Since Q is the disjoint union of Q + and Q − , for δ#(G 2r+2 )N(2r + 1) < 1, we have This completes the proof, since δ > 0 is arbitrarily small. Proof: First assume that the second statement of Lemma 4.8 holds. Then the action of G 2r+2 on p −1 (x) has infinitely many finite orbits O n , with n ∈ N. Clearly, for P := O n , we have that Q − is empty. Indeed, if there exists y 0 ∈ Q − , then there exist z ∈ B(y 0 , r), y 1 ∈ P and y 2 ∈ p −1 (x) P , such that ϕ y i (z) > 0, i = 1, 2. It follows that d(z, y i ) < r + 1, i = 1, 2, which yields that d(y 1 , y 2 ) < 2r + 2. From Remark 4.7, there exists g ∈ G 2r+2 , such that y 2 = y 1 · g, which is a contradiction, since P is an orbit of the action of G 2r+2 on p −1 (x). For a compact K ⊂ M 2 , the set P K := p −1 (x) ∩ B(K, r + 2) is finite and in particular, intersects only finitely many orbits O n . Let P be an orbit that does not intersect P K . Since supp ϕ y ⊂ C(y, r + 1), for any y ∈ p −1 (x), it is clear that for such P , the support of χ does not intersect K.
If the first statement of Lemma 4.8 holds, then the action of G r on p −1 (x) has only infinite orbits. For a compact K ⊂ M 2 , consider the finite set P K := p −1 (x) ∩ B(K, r + 2). From Lemma 4.9, for any ε > 0, there exists a non-empty, finite P ⊂ p −1 (x), such that where N(2r + 1) is the constant from Lemma 2.6. Since the action of G r on p −1 (x) has only infinite orbits, it follows that Q − is nonempty. Indeed, since P is non-empty and this action has only infinite orbits, there exists an infinite orbit O and z 1 ∈ P ∩ O. Since P is finite, there exists z 2 ∈ O P , and from Remark 4.7, there exist k ∈ N and y 1 , . . . , y k ∈ p −1 (x), with y 1 = z 1 , y k = z 2 and d(y i , y i+1 ) < r, for i = 1, . . . , k − 1. Since y 1 ∈ P and y k / ∈ P , there exists 1 ≤ j < k, such that y j ∈ P and y j+1 / ∈ P . Since d(y j , y j+1 ) < r, it follows that 0 < χ(y j+1 ) < 1 and in particular, y j ∈ Q − . Evidently, Q + is contained in P . Since Q − is non-empty, it is clear that which yields that #(P ) > #(P K ), from the choice of δ. In particular, the finite set P ′ := P P K is non-empty. Consider the function χ ′ and the sets Q ′ + , Q ′ − and Q ′ corresponding to P ′ as in (4). Clearly, the support of χ ′ does not intersect K, since supp ϕ y ⊂ C(y, r + 1), for any y ∈ p −1 (x). From Lemma 2.6, it follows that for any y 0 ∈ p −1 (x), the support of ϕ y 0 intersects at most N(2r + 1) open balls B(y, r), with y ∈ p −1 (x). Hence, we have that from the choice of δ.
Remark 4.11. After endowing M 1 or N 1 with h (depending on whether M 1 has empty boundary or not) and the covering space with its lift, we have that p : D y → M 1 is an isometry up to sets of measure zero, for any y ∈ p −1 (x). Thus, for f ∈ C c (M 1 ), we have where Vol h i (respectively, Vol g i ) is the measure on M i induced by h (respectively, g) or its lift, i = 1, 2. Since g and h are conformal, there exists a positive ϕ v ∈ C ∞ (M 1 ), such that For simplicity of notation, we omit d Vol g i in the integrals and the index of Vol g i . From (5), we have Dy f • p = M 1 f , for any f ∈ C c (M 1 ) and y ∈ p −1 (x). Similarly, for a compact K ⊂ M 1 , we have Vol(K) = Vol(p −1 (K) ∩ D y ), for any y ∈ p −1 (x).
Proposition 4.12. Let p : M 2 → M 1 be an infinite sheeted, amenable Riemannian covering. Consider η ∈ D(D 1 ) with η L 2 (E 1 ) = 1 and λ ∈ F. Then for any ε > 0 and K ⊂ M 2 compact, there exists ζ ∈ D(D 2 ) with ζ L 2 (E 2 ) = 1, such that supp ζ ⊂ p −1 (supp η), Proof: Let p 1 :M → M 1 be the universal covering of M 1 and fix x ∈ M 1 , u ∈ p −1 1 (x) and r ≥ R (from Lemma 4.8), such that supp η ⊂ B(x, r) and B(u, r) ∩ ∂M = ∅, if M 1 has non-empty boundary. Consider a partition of unity consisting of the functions ϕ 1 and ϕ y , with y ∈ p −1 (x), associated to x, u and r as in (3), and let θ be the lift of η. From Remark 4.4, for any finite set P ′ ⊂ p −1 (x) and χ ′ := y∈P ′ ϕ y , we have that χ ′ θ ∈ D(D 2 ). From Proposition 4.5, there exists C > 0, independent from P ′ , such that D 2 (χ ′ θ)(z) ≤ C, for any z ∈ M 2 . Hence, we obtain that max z∈M 2 From Proposition 4.10, there exists a non-empty, finite P ⊂ p −1 (x), such that the support of χ := y∈P ϕ y does not intersect K and where Q + , Q − and Q are the sets corresponding to P as in (4).
Since χθ is in the domain of D 2 , so is the normalized section ζ := (1/ χθ L 2 (E 2 ) )χθ. Evidently, ζ L 2 (E 2 ) = 1 and supp ζ ⊂ p −1 (supp η). From Lemma 2.5, we have that supp ζ ∩ D y ⊂ B(y, r), for any y ∈ p −1 (x), which yields that supp ζ is contained in the union of the fundamental domains D y , with y ∈ Q. Clearly, we have from the definition of Q + and Remark 4.11. Therefore, we obtain that For y ∈ Q + , we have χ = 1 in B(y, r), which is a neighborhood of supp θ ∩ D y . This implies that Since (D 2 − λ)(χθ)(z) ≤ C 0 , for any z ∈ M 2 , it follows that Proof of Theorem 4.1: Let λ ∈ σ ap (D 1 ). From Lemma 2.2, there exists (η k ) k∈N ⊂ D(D 1 ), such that η k L 2 (E 1 ) = 1 and (D 1 − λ)η k → 0 in L 2 (E 1 ). Consider an exhausting sequence (K k ) k∈N of M 2 . From Proposition 4.12, for any k ∈ N, there exists ζ k ∈ D(D 2 ), such that Therefore, (D 2 − λ)ζ k → 0 in L 2 (E 2 ) and for any compact K ⊂ M 2 , there exists k 0 ∈ N, such that supp ζ k ∩ K = ∅, for all k ≥ k 0 . It follows that (ζ k ) k∈N is a Weyl sequence for D ′ 2 and λ, and in particular, λ ∈ σ W (D ′ 2 ). Proof of Theorem 1.1: Follows immediately from Theorem 4.1 and Proposition 2.1.
Assume now that the operator D i : D(D i ) ⊂ L 2 (E i ) → L 2 (E i ) is symmetric and bounded from below, and let D (F ) i be its Friedrichs extension, i = 1, 2. For more details on the Friedrichs extension of a symmetric, bounded from below and densely defined linear operator on a Hilbert space, see [18]. It is well-known that the Friedrichs extension of an operator preserves its lower bound. In particular, for i = 1, 2, we have Recall the following proposition for the essential spectrum of a self-adjoint operator. 1 ) + 2/k and supp ζ k ∩ supp ζ k ′ = ∅, for all k, k ′ ∈ N, with k = k ′ . Evidently, for any ε > 0, there exists k 0 ∈ N, such that R D 2 (ζ k ) < λ 0 (D (F ) 1 ) + ε, for all k ≥ k 0 . Consider the subspace G ε of D(D 2 ) spanned by {ζ k : k ≥ k 0 }. Since the sections ζ k , with k ∈ N, have disjoint supports, the space G ε is infinite dimensional. Clearly, any θ ∈ G ε is of the form θ := k 0 +µ i=k 0 m i ζ i , for some µ ∈ N and m k 0 , . . . , m k 0 +µ ∈ F. Therefore, we have For sake of completeness, we also present the analogous results for finite sheeted coverings. It is clear that they cannot be improved in order to obtain as strong statements as in the case of infinite sheeted amenable coverings. Proposition 4.16. Let D ′ 2 be a closed extension of D 2 . If p is a finite sheeted Riemannian covering, then σ ap (D 1 ) ⊂ σ ap (D ′ 2 ) and σ W (D 1 ) ⊂ σ W (D ′ 2 ). Proof: If η is in the domain of D 1 , then its lift is in the domain of D 2 . For λ ∈ σ W (D 1 ), from Lemma 2.2, there exists a Weyl sequence (η k ) k∈N ⊂ D(D 1 ) for D 1 and λ. Then, the sequence consisting of the normalized (in L 2 (E 2 )) lifts of η k , k ∈ N, is a Weyl sequence for D ′ 2 and λ. Hence,  In particular, there exists ϕ ∈ C ∞ (M ∂M) with Sϕ = λ 0 (S)ϕ, which is positive in the interior of M. It is worth to point out that the smooth eigenfunctions of the preceding proposition do not have to be square-integrable. The following corollary is a consequence of Proposition 4.18 (an alternative proof can be found in [2]).  The following results describe the behavior of the spectrum of Schrödinger operators under finite sheeted coverings.
The following characterization of the bottom of the essential spectrum of a Schrödinger operator follows from the Decomposition Principle ([3, Proposition 1]) and Propositions 2.3 and 4.14. Recall that this quantity is infinite when the spectrum is discrete.  where n is the inward pointing normal to ∂M and a, b are real or complex valued functions (depending on whether E is Riemannian or Hermitian) defined on ∂M. It is worth to point out that we do not impose any assumptions on a and b. From Lemma 2.4, the operator D is closable and denote by D its closure.
Theorem 5.1. Let Γ be a group of automorphisms of E preserving the metric of E, such that the induced action on M is isometric and D(g * η) = g * Dη, for any g ∈ Γ and η ∈ Γ(E). If M has non-empty boundary, assume that ∇, a and b are Γ-invariant along the boundary. If for any compact K ⊂ M, there exists g ∈ Γ, such that gK ∩ K = ∅, then σ ap (D) = σ W (D) and D does not have eigenvalues of finite multiplicity.
Proof: Let λ ∈ σ ap (D). From Lemma 2.2, there exists (η k ) k∈N ⊂ D(D), such that η k L 2 (E) = 1 and (D − λ)η k → 0 in L 2 (E). Since η k is compactly supported, there exists an exhausting sequence (K k ) k∈N of M, such that supp η k ⊂ K k , for all k ∈ N. For any k ∈ N, consider g k ∈ Γ, such that g k K k ∩ K k = ∅, and set ζ k := (g k ) ⋆ η k . Then ζ k ∈ Γ c (E) and if M has non-empty boundary, then ζ k satisfies the same boundary conditions with η k , since via isometries the boundary is mapped to itself and so does the inward pointing normal. It follows that ζ k ∈ D(D), ζ k L 2 (E) = 1 and (D − λ)ζ k → 0 in L 2 (E). Clearly, supp ζ k = g k (supp η k ), which yields that for any compact K ⊂ M, there exists k 0 ∈ N, such that supp ζ k ∩ K = ∅, for all k ≥ k 0 . This implies that ζ k ⇀ 0 in L 2 (E), that is, (ζ k ) k∈N is a Weyl sequence for D and λ. Hence, λ ∈ σ W (D). Assume that there exists an eigenvalue λ of D of finite multiplicity, and consider θ ∈ D(D) with θ L 2 (E) = 1 and Dθ = λθ. Then there exists (η k ) k∈N ⊂ D(D), such that η k → θ and Dη k → Dθ. Clearly, for g ∈ Γ, we have g * η k ∈ D(D),g * η k → g * θ and D(g * η k ) → g * (Dθ), which yields that g * θ ∈ D(D) and D(g * θ) = λ(g * θ).
Let (K k ) k∈N be an exhausting sequence of M and consider (g k ) k∈N ⊂ Γ, such that g k K k ∩ K k = ∅, for any k ∈ N. It is clear that the sections θ k := (g k ) * θ satisfy Dθ k = λθ k and θ k L 2 (E) = 1, for all k ∈ N. Since the eigenspace corresponding to λ is finite dimensional, after passing to a subsequence, we may assume that θ k → θ 0 in L 2 (E), for some θ 0 , with θ 0 L 2 (E) = 1. Consider a non-zero ζ ∈ Γ c (E) and set ζ k := (g −1 k ) * ζ. Then Let ε > 0 and consider a compact K ⊂ M, such that M K θ 2 < ε 2 / ζ 2 L 2 (E) . Since supp ζ and K are eventually subsets of K k , there exists k 0 ∈ N, such that supp ζ k ∩K = ∅, for all k ≥ k 0 . Therefore, for k ≥ k 0 , we have supp ζ k ⊂ M K, and in particular, | θ k , ζ L 2 (E) | < ε. This yields that θ k ⇀ 0 in L 2 (E), which is a contradiction, since θ k → θ 0 in L 2 (E) and θ 0 L 2 (E) = 1.
Theorem 5.2. Assume that D is symmetric and bounded from below, and denote by D (F ) its Friedrichs extension. Under the assumptions of Theorem 5.1, the spectrum of D (F ) is essential and D (F ) does not have eigenvalues of finite multiplicity.
Proof: Let η ∈ D(D (F ) ) and g ∈ Γ. From the invariance of D(D) and D under the action of Γ, it follows that g * η ∈ D(D (F ) ) and D (F ) (g * η) = g * (D (F ) η). As in the proof of Theorem 5.1, it follows that D (F ) does not have eigenvalues of finite multiplicity. From Proposition 2.1, we obtain that σ(D (F ) ) = σ ess (D (F ) ).
The above theorems can be applied to Riemannian coverings with infinite deck transformations group. In the context of the previous section, we obtain the following consequences. Besides Riemannian coverings, the above theorems can be applied to manifolds with high symmetry. For instance, it follows that the spectrum of the Laplacian on a noncompact homogeneous space is essential. Moreover, we obtain the analogous statement, if there exists a non-compact Lie group acting on the manifold properly discontinuously via isometries.

Coverings of closed manifolds
The Cheeger's constant of a Riemannian manifold M is defined by where the infimum is taken over all compact and smoothly bounded domains K of M which do not intersect ∂M. It is related to λ 0 (M) via Cheeger's inequality (cf. [9]): Brooks [7] actually proved that a normal Riemannian covering of a closed manifold is amenable if and only if the Cheeger's constant of the covering space is zero. The following result is an extension of that of Brooks, to not necessarily normal coverings. In order to prove this theorem, we need the following proposition. In the sequel, for a subset W of M, we denote by B(W, r) the tubular neighborhood  It follows that This completes the proof, since Vol g (U) ≤ Vol g (U B g (∂U, r)) + Vol g (B g (∂U, r)).
Consider the finite sets Recall that r > 2diam(M 1 ) ≥ diam(D y ), for all y ∈ p −1 (x), and M 2 is covered by the fundamental domains D y , with y ∈ p −1 (x). Evidently, U B(∂U, 2r) is contained in the union of D y , with y ∈ F . Furthermore, B(∂U, 2r) contains the union of D y , with y ∈ F ′ . From (7), since the intersection of different fundamental domains is of measure zero, and Vol(D y ) = Vol(M 1 ), for any y ∈ p −1 (x), it follows that Let g ∈ G r and y ∈ F F g. Then y ∈ U, d(y, ∂U) ≥ r and y · g −1 / ∈ F . From Remark 4.7, it follows that d(y, y · g −1 ) < r. Therefore, y · g −1 ∈ U and d(y · g −1 , ∂U) < r, which yields that y · g −1 ∈ F ′ . Hence, F F g ⊂ F ′ g and in particular, we obtain that For any finite G ⊂ π 1 (M 1 ), there exists r > 2diam(M 1 ), such that G ⊂ G r . The above arguments imply that for any finite G ⊂ π 1 (M 1 ) and ε > 0, there exists a Følner set for G and ε. From Proposition 2.8, it follows that p is amenable.

Applications and examples
Throughout most of this section we restrict ourselves to Schrödinger operators and present some consequences of our main results in this context. We begin with some auxiliary considerations.
We first introduce the notion of renormalized Schrödinger operators. This notion was introduced by Brooks in [6] for the Laplacian on complete manifolds without boundary. Let S be a Schrödinger operator on a possibly non-connected Riemannian manifold M without boundary, and let ϕ ∈ C ∞ (M) be a positive λ-eigenfunction of S. It is worth to point out that we do not require ϕ to be square-integrable or M to be complete. Consider ) and S (F ) is the Friedrichs extension of S. Clearly, the following diagram is commutative In particular, S ϕ is self-adjoint and σ(S ϕ ) = σ(S) − λ. From Proposition 2.3, it follows that , Hence, the bottom of the spectrum of S ϕ can be approximated with Rayleigh quotients of compactly supported smooth functions in M. With a simple computation of the Rayleigh quotient of such a function (as in [6, Section 2], using the Divergence Theorem, instead of the * -operator), we obtain the following expression for λ 0 (S) − λ.
The modified Cheeger's constant of M is defined by where the infimum is taken over all compact and smoothly bounded domains K of M. From the preceding proposition, it is easy to establish an analogue of Cheeger's inequality. From Corollary 7.2, we have that for any k ∈ N. After taking the limit with respect to k, the statement follows from Proposition 4.22.
Remark 7.4. The above arguments can be easily modified in order to obtain analogous results for manifolds with boundary. In that case, it suffices to consider a λ-eigenfunction of S which is positive and smooth only in the interior of M. Then, in Proposition 7.1, the infimum is taken over smooth functions with compact support in the interior of M.
Proof of Theorem 1.5: From Corollary 4.20, the first statement implies the second. From Corollary 4.19, the third statement follows from the second. Assume that λ 0 (S 1 ) = λ ess 0 (S 2 ), for some Schrödinger operator S 1 on M 1 . From Proposition 4.18, there exists a positive λ 0 (S 1 )-eigenfunction ϕ ∈ C ∞ (M 1 ) of S 1 , and its liftφ ∈ C ∞ (M 2 ) is a positive λ 0 (S 1 )-eigenfunction of S 2 . From Corollary 7.3, it follows that h esŝ ϕ (M 2 ) = 0. Since ϕ is positive and M 1 is closed, this yields that h ess (M 2 ) = 0. Assume that h ess (M 2 ) = 0. Then h(M 2 ) = 0 and Theorem 6.1 yields that p is amenable. Assume that p is finite sheeted. Then M 2 is closed. Consider a smoothly bounded domain U of M 2 , such that M 2 U is connected. Evidently, M 2 U is a compact manifold with boundary. From the definition of the Cheeger's constant, it is clear that h(M 2 U ) = h(M 2 U). From [9], it follows that h ess (M 2 ) ≥ h(M 2 U) > 0, which is a contradiction. Hence, p is infinite sheeted.
Proof: If the covering is finite sheeted, the inclusion of spectra follows from Corollary 4.21. In this case, M 2 is closed, which yields that the spectrum of S 2 is discrete. From Corollary 4.19, the second statement implies the third.
Assume that the fourth statement holds. Since h(M 2 ) = 0, from Theorem 6.1, p is amenable. Since h ess (M 2 ) = 0, from Theorem 1.5, it follows that p is finite sheeted.
The following characterization for points of the essential spectrum of a Schrödinger operator is an immediate consequence of the Decomposition Principle.
Our second application is motivated by Corollary 3.8 of the arXiv version of [1].
Theorem 7.7. Let p : M 2 → M 1 be a Riemannian covering with M 2 simply connected and complete. Let S 1 be a Schrödinger operator on M 1 and S 2 its lift on M 2 . If there exists a compact K ⊂ M 1 , such that the image of the fundamental group of any connected component of M 1 K in π 1 (M 1 ) is amenable, then σ ess (S 1 ) ⊂ σ ess (S 2 ).
Proof: Let λ ∈ σ ess (S 1 ). From Proposition 7.6, there exists (f k ) k∈N ⊂ C ∞ c (M), such that f k = 0 on ∂M 1 , f k L 2 (M 1 ) = 1, (S − λ)f k → 0 in L 2 (M 1 ) and for every compact K 0 ⊂ M 1 , there exists k 0 ∈ N, such that supp f k ∩ K 0 = ∅, for all k ≥ k 0 . Without loss of generality, we may assume that the supports of f k are connected, since we may restrict each f k to a connected component of its support and obtain a sequence with the same properties.
Consider a compact K ⊂ M 1 , such that the image of the fundamental group of any connected component of M 1 K in π 1 (M 1 ) is amenable. Clearly, after passing to a subsequence, we may assume that the functions f k are supported in M 1 K. Since for any k ∈ N, the support of f k is connected, it follows that supp f k ⊂ U k , where U k is a connected component of M 1 K. From the Lifting Theorem, it follows that the inclusion U k ֒→ M 1 can be lifted to the covering space M ′ k := M 2 /Γ k , where Γ k is the image of π 1 (U k ) in π 1 (M 1 ). In particular, any f k can be lifted to some f ′ k ∈ C ∞ c (M ′ k ).
Since the covering q k : M 2 → M ′ k is normal with deck transformations group Γ k , it follows that it is amenable. If q k is finite sheeted, letf k be the normalized (in L 2 (M 2 )) lift of f ′ k on M 2 . If q k is infinite sheeted, from Proposition 4.12, there existsf k ∈ C ∞ c (M 2 ), such that f k L 2 (M 2 ) = 1, suppf k ⊂ q −1 k (supp f ′ k ) and where S ′ k is the lift of S 1 on M ′ k . In particular, (S 2 − λ)f k → 0 in L 2 (M 2 ) and suppf k is contained in p −1 (supp f k ). From Proposition 7.6, it follows that λ ∈ σ ess (S 2 ).
Remark 7.8. In the proof of Theorem 7.7, the only properties of Schrödinger operators used are essential self-adjointness and Proposition 7.6, which follows from the Decomposition Principle. Therefore, this proof establishes the analogous result for essentially self-adjoint differential operators, for which the Decomposition Principle holds (cf. [3]). For instance, if M 1 has empty boundary, then the statement of Theorem 7.7 holds for any elliptic differential operator D 1 , such that D 1 and D 2 are essentially self-adjoint on the spaces of compactly supported smooth sections.
Proof of Corollary 1.6: Follows immediately from Theorem 7.7 and Corollary 4.19.
Let p : M 2 → M 1 be a Riemannian covering of complete manifolds. As stated in the Introduction, there are examples where p is non-amenable and λ 0 (M 1 ) = λ 0 (M 2 ). From Theorem 1.1, Propositions 4.16 and 2.1, if p is amenable, then σ(M 1 ) ⊂ σ(M 2 ). It is natural to examine if this inclusion implies amenability of the covering. From Theorem 7.7, it is easy to construct an example of a non-amenable, normal Riemannian covering p : M 2 → M 1 with M 1 complete, with bounded geometry and of finite topological type (that is, M 1 admits a finite triangulation, where the simplices are defined on the standard simplex with possibly some lower dimensional faces removed), such that σ(M 1 ) = σ(M 2 ). Example 7.9. Let M 1 be a 2-dimensional torus with a cusp, endowed with a Riemannian metric, such that M 1 is complete and outside a compact set the metric is the standard metric of the flat cylinder. It is clear that M 1 is of finite topological type and has bounded geometry. From [15,Theorem 1], it follows that σ ess (M 1 ) = [0, +∞). Clearly, there exists a compact subset K of M 1 , such that π 1 (M 1 K) = Z. From Theorem 7.7, it follows that for the simply connected covering space M 2 of M 1 , we have σ ess (M 2 ) = [0, +∞). However, π 1 (M 1 ) is the free group in two generators, which is non-amenable (cf. [4, Section 2]).
The next simple observation, provides a sufficient geometric condition for amenability of coverings.
Proposition 7.10. Let M 1 be a complete Riemannian manifold, without boundary and with non-negative Ricci curvature. Then any covering p : M 2 → M 1 is amenable.
Proof: LetM be the simply connected covering space of M 1 . From the Bishop-Gromov Comparison Theorem, it follows thatM has polynomial growth and hence, every finitely generated subgroup of π 1 (M 1 ) has polynomial growth (cf. [16]). From Corollary 2.9, it follows that every finitely generated subgroup of π 1 (M 1 ) is amenable and Corollary 2.10 yields that so is π 1 (M 1 ). Therefore, any covering p : M 2 → M 1 is amenable.
Next, we present an example of an infinite sheeted amenable covering with trivial deck transformations group. In particular, this implies that the results of Section 5 cannot be applied to arbitrary infinite sheeted amenable coverings.
Example 7.11. Let Γ 1 be the countable group of invertible, upper triangular 2 × 2 matrices with entries in Q and let M 1 be a Riemannian manifold with π 1 (M 1 ) = Γ 1 (cf. [2,Section 5]). Let Γ 2 be the subgroup of Γ 1 consisting of diagonal matrices. Denote bỹ M the simply connected covering space of M 1 and consider M 2 :=M /Γ 2 . It is easy to see that the covering p : M 2 → M 1 is infinite sheeted and does not have non-trivial deck transformations. However, Γ 1 is solvable and in particular, amenable (from Corollary 2.11), which yields that p is an amenable covering.
Recall that in our main results there are no assumptions on the vector bundles, the connections and the differential operators. We end this section with an example which demonstrates that these play a crucial role in the behavior of the spectrum even under finite sheeted coverings. Namely, this example shows that whether or not the bottom of the spectrum of the connection Laplacian is preserved under a Riemannian covering depends on the corresponding metric connection.
If M is a closed Riemannian manifold and E → M is a Riemannian vector bundle endowed with a metric connection ∇, then the (corresponding) connection Laplacian is given by ∆ = ∇ * ∇. It is well-known that ∆ : Γ(E) ⊂ L 2 (E) → L 2 (E) is essentially self-adjoint and its spectrum is discrete (cf. [14]). Example 7.12. Consider S 1 := R/Z and the trivial bundle E 1 := S 1 × R 2 with the standard metric. We can identify smooth sections of E 1 with smooth maps f : R → R 2 with f (x) = f (x + 1), for all x ∈ R. For φ ∈ R, consider the metric connection ∇ φ , defined by for any smooth section f = (f 1 , f 2 ) of E 1 . Since the spectrum of the connection Laplacian ∆ φ is discrete for any φ ∈ R, it is clear that λ 0 (∆ φ , E 1 ) = 0 if and only if there exists a parallel section of E 1 with respect to ∇ φ , or equivalently, φ = 2kπ, for some k ∈ Z.