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Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.


Introduction and main results
Let (M n , g) be an n-dimensional noncompact complete Riemannian manifold. The Laplace-Beltrami operator ∆ = ∆ g on (M n , g) is essentially self-adjoint on C ∞ 0 (M n ). We also denote by ∆ its unique self-adjoint extension to L 2 (M n , dv g ).
For compact M n , there is a wealth of results on the relations between the geometry of the manifold and spectral properties of the Laplacian (which in this case only has discrete eigenvalues). These relations are not so well studied for the nocompact case, which is the subject of this paper. We will mention some results later in this article but refer the readers to [6] for a more complete review. The past work has been mostly focused on proofs of the purity of absolutely continuous spectrum and absence of embedded eigenvalues. Here we study the opposite question.
If M has constant radial curvature −K 0 , then, for negative K 0 , −∆ has a complete set of eigenvalues {λ n ≥ 0} with λ n → ∞ [3]. For K 0 ≥ 0, σ(−∆) = σ ess (−∆) = K0 4 (n − 1) 2 , ∞ and there are no eigenvalues [3]. For perturbations of the latter case it is natural to expect that whether there are eigenvalues will depend on the size of the perturbation. Perturbations on a compact set can only create eigenvalues below the essential spectrum. Thus the question whether one can embed eigenvalues in the essential spectrum will depend on the rate of approach of −K 0 by the radial curvature K(r) of the perturbation, at infinity. Manifolds with K(r) → −K 0 as r → ∞ are called asymptotically flat if K 0 = 0 and asymptotically hyperbolic if K 0 = 1. The case K 0 > 0 can be rescaled to K 0 = 1 but we find it more useful to keep K 0 and will call all manifolds with K(r) → −K 0 , K 0 > 0, asymptotically hyperbolic.
In this paper we answer the following question. Given any finite or countable (possibly dense) set A ⊂ σ(−∆) = K0 4 (n − 1) 2 , ∞ , can we construct an asymptotically flat or hyperbolic n-dimensional manifold with an embedded (in the absolutely continuous spectrum) eigenvalue at each λ ∈ A? How is it influenced by the asymptotical behavior of the radial curvature?
In particular, this of course implies 1 Corollary 1.2. There exist asymptotically flat and asymptotically hyperbolic n-dimensional Riemannian manifolds with dense point spectrum embedded in the absolutely continuous spectrum.
An interesting question is to study the curvature conditions for possibility to embed an arbitrary countable set in the absolutely continuous spectrum. It turns out, for finite sets it can be done with r|K(r) + K 0 | = O(1), while for countable sets it is enough to require that r|K(r) + K 0 | → ∞, no matter how slowly. We have (1) For any finite A ⊂ K0 4 (n − 1) 2 , ∞ , there exists an n-dimensional manifold with r|K(r) + K 0 | < C, σ ac (−∆) = K0 4 (n − 1) 2 , ∞ , and an embedded eigenvalue at each λ ∈ A.
(1) Theorem 1.3 is sharp in the asymptotically hyperbolic case in the following sense. If r|K(r) + K 0 | → 0 (or even is bounded by a sufficiently small constant), there can be no embedded eigenvalues [15]. We conjecture it is also sharp in a similar sense in the asymptotically flat case. (2) We conjecture that, at least in the hyperbolic case, (2) of Theorem 1.3 is sharp in an even stronger sense. Namely, that given a monotone C(r) > 0, for any countable A ⊂ [0, ∞) there exists an n-dimensional manifold with |K(r) + K 0 | < C(r), σ ac (−∆) = K0 4 (n − 1) 2 , ∞ , and an embedded eigenvalue at each λ ∈ A if and only if lim r→∞ rC(r) = ∞. Given Theorem 1.3 this statement would only require proving that if C(r) is bounded, only eigenvalues below a certain threshold can be embedded, as is the case for n = 1 [15].
Let us present more detail on the history. For the asymptotically hyperbolic case, the sharp transition on a possibility to embed one eigenvalue was given by Kumura [15] based on the arguments of Kato [11]. He excluded eigenvalues greater than K0(n−1) 2 4 under the assumption that K rad +K 0 = o(r −1 ), and also constructed a manifold with the radial curvature K rad +K 0 = O(r −1 ) and with an eigenvalue K0(n−1) 2 4 +1 embedded into the essential spectrum Previous results on absence of embedded eigenvalues under the radial curvature conditions are reviewed in [6]. They go back to Pinsky [25], with a later milestone by Donnelly [4]. Some recent results on the absence of eigenvalues can be found in [19,21].
For the asymptotically flat case, the absence of embedded eigenvalues results go back to [10]. Kumura, Donnelly, and Garofalo [5,7,16] showed the absence of positive eigenvalues of the Laplacian if the curvature K rad = o(r −2 ). As mentioned, we conjecture here that Laplacian has no positive eigenvalues if K rad = o(r −1 ).
Under certain stronger curvature decay assumption on the perturbation, limiting absorption principle, originally from Agmon's theory [1], holds for the Laplacian. See [17,26] and references therein. In this case, Laplacian has purely absolutely continuous spectrum. This paper is the first one in the series where we construct manifolds with unusual spectral properties of the Laplacian and certain sharp curvature bounds. For example, in the upcoming [9] we obtain Riemannian manifolds with singular continuous spectrum embedded in the spectrum of the Laplacian.
The Riemannian manifolds (M, g) we construct are rotationally symmetric, and we construct rotationally symmetric eigenfunctions, thus reducing the problem to a one-dimensional Schrödinger operator. Fix some O ∈ M as the origin. Using the radial coordinates (from O) we construct Riemannian manifold with the structure of the form (M, g) = R n , dr 2 + f 2 1 (r)g S n−1 (1) where g S n−1 (1) is the standard Riemannian metric on the unit sphere, and we need to construct f 1 so that the Laplacian has the desired properties. Suppose h(r) is a function on M only depending on the radius r. Then the Laplacian is equivalent to the following one-dimensional Schrödinger operator, where .
In order to make the manifold smooth in the neighborhood of O, f even 1 (0) must vanish at 0 and f ′ 1 (0) = 0. This implies S(r) is singular at 0. Thus we need to deal with one-dimensional Schrödinger operator (1) with singularities at both 0 and ∞.
In the neighborhood of ∞ by the Liouville transformation, Laplacian (1) can be normalized to a Schrödinger operator of the form . Constructing operators (3) with given sets embedded as eigenvalues in the essential spectrum is an old question, going back to the celebrated work of Wigner-von Neuman [28] who constructed an explicit potential with an embedded (given) eigenvalue.
Simon [27] showed that for any rate of decay h(x) that is slower than Coulomb and any countable subset A ⊂ R + , there exists q(x) bounded by h(x), so that operator (3) (wholeline or with the desired boundary conditions) has an embedded eigenvalue at each λ ∈ A. However, the potential in Simon's construction is not continuous and thus cannot be used for our purposes. Previously, Naboko [24] constructed smooth potentials with the same property but only if elements of A are rationally independent.
Naboko's construction starts from the origin. Then he first constructs piecewise-constant potentials with desired properties, which are then smoothed out. Simon uses a different method. He uses the Wigner-von Neumann type to construct the desired potential and the method of L 2 perturbations to guarantee boundary conditions at the origin 0 for the eigenfunctions. His construction starts at ∞, thus it is nontrivial to make it smooth.
In this paper we develop a new construction, based on piecewise Wigner-von Neumann potentials, different from both [24] and [27]. In fact, we view the construction itself as one of the important achievements of this paper. It is robust and fundamental in that it can be applied in a variety of contexts to construct embedded eigenvalues. In the forthcoming work it is adapted by one of the authors and Ong to construct eigenvalues embedded into the spectral band for perturbed periodic operator, in both continuous and discrete settings [20,23], and also to construct eigenvalues embedded into the absolutely continuous spectrum for perturbed Stark type operators [18,22].
First, in order to deal with singularities at both 0 and infinity, it is natural to construct Riemannian metric around 0 and ∞ separately so that the two operators − ∂ 2 ∂r 2 − (n− 1)S(r) ∂ ∂r (one around 0, another around ∞) have the given eigenvalues. We start at O with the standard Euclidean metric for r ≤ 1 2 . Thus the eigenfunctions of the Laplacian (1) are given by the Bessel functions. In the neighborhood of ∞ (r ≥ 3) we use piecewise Wigner-von Neumann type functions to construct the f 1 (r), adding the eigenvalues one (or fewer) at a time. We allow Wigner-von Neumann type potentials to be adapted in the next segment to balance the boundary conditions of the associated eigenfunctions.
When further eigenvalues are taken into consideration, we need to adapt the next segment of Wigner-von Neumann type potential to balance the new boundary conditions. However, Wigner-von Neumann type potential (associated to a fixed eigenvalue) may significantly change other eigenfunctions. As we add new eigenvalues, the change will accumulate. To overcome this difficulty, we use the quantitative analysis to study the relationships for all the Wignervon Neumann type potentials, corresponding eigenfunctions and the other eigenfunctions. In particular, an important building block is a Theorem that allows to construct a Wigner-von Neumann type potential on a sufficiently long interval so that a solution for a given energy with given boundary conditions decays, while solutions for energies from a given finite set, for all boundary conditions, do not grow too much. Then we proceed with double induction, so that at each new step we add an interval with decay for each previously treated energy, to ensure the overall decay.
After the separate construction, we need to connect the Riemannian metric at r < 1 2 and r > 3 smoothly so that the eigenfunctions of the two separate operators on r < 1 2 and r > 3 connect smoothly. This can be done if the boundary conditions of eigenfunctions match at some fixed point r ∈ [ 1 2 , 3]. As we work with the one-dimensional construction, the proof of Theorems 3.1 and 3.2, establishes also the following Theorems.

Preparation
The following result is well known. See [13] or page 93 in [8].
Consider the differential equation −y ′′ + V y = λy with λ > 0. Then the following asymptoticis holds (uniform with respect to φ) as x goes to infinity (1) if κ = ± √ λ, then there exists a fundamental system of solutions are jointly continuous with respect to x, φ. The following theorem is an important building block for our inductive construction. It allows to construct a potential with desired bounds that ensures decay of the solution for a given energy/boundary condition and stabilization (not much growth) of solutions for energies from a given finite set with arbitrary boundary conditions.

Inductive Construction
We reformulate Theorem 1.3 in a more convenient way.
We will first construct f 1 near the origin, and then extend it inductively, adding one segment at a time.
Set for r ≥ 1, Thus we have Remark 3.3. Actually K(r) is the radial curvature and ∆r = (n − 1)S(r).