The Spectrum of the Weakly Coupled Fibonacci Hamiltonian

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.


Introduction
The Fibonacci Hamiltonian is a central model in the study of electronic properties of one-dimensional quasicrystals. It is given by the following bounded selfadjoint operator in ℓ 2 (Z), [H V,ω ψ](n) = ψ(n + 1) + ψ(n − 1) + V χ [1−α,1) (nα + ω mod 1)ψ(n), where V > 0, α = √ 5−1 2 , and ω ∈ T = R/Z. This operator family has been studied in many papers since the early 1980's and numerous fundamental results are known. Let us recall some of them and refer the reader to the survey articles [D00, D07, S95] for additional information.
The spectrum is easily seen to be independent of ω and may therefore be denoted by Σ V . That is, σ(H V,ω ) = Σ V for every ω ∈ T. Indeed, this follows quickly from the minimality of the irrational rotation by α and strong operator convergence. It was shown by Sütő that Σ V has zero Lebesgue measure for every V > 0; see [S89]. Moreover, it is compact (since it is the spectrum of a bounded operator) and perfect (because the irrational rotation by α is ergodic). Thus, Σ V is a zeromeasure Cantor set. This result was recently strengthened by Cantat [Can] who showed that the Hausdorff dimension of Σ V lies strictly between zero and one.
Naturally, one is interested in fractal properties of Σ V , such as its dimension, thickness, and denseness. While such a study is well-motivated from a purely mathematical perspective, we want to point out that there is significant additional interest in these quantities. In particular, it has recently been realized that the fractal dimension of the spectrum is intimately related with the long-time asymptotics of the solution to the associated time-dependent Schrödinger equation, that is, i∂ t φ = H V,ω φ; see [DEGT].
Date: January 28, 2009. D. D. was supported in part by NSF grant DMS-0800100.
Fractal properties of Σ V are by now well understood for large values of V . Work of Casdagli [Cas] and Sütő [S87] shows that for V ≥ 16, Σ V is a dynamically defined Cantor set. It follows from this result that the Hausdorff dimension and the upper and lower box counting dimension of Σ V all coincide; let us denote this common value by dim Σ V . Using this result, Damanik, Embree, Gorodetski, and Tcheremchantsev have shown upper and lower bounds for the dimension; see [DEGT]. A particular consequence of these bounds is the identification of the asymptotic behavior of the dimension as V tends to infinity: The paper [DEGT] also discusses some of the implications for the dynamics of the Schrödinger equation; let us mention [DT07,DT08] for further recent advances in this direction for the strongly coupled Fibonacci Hamiltonian. By contrast, hardly anything about Σ V (beyond it having Hausdorff dimension strictly between zero and one) is known for small values of V . The largeness of V enters the proofs of the existing results in critical ways. Consequently, these proofs indeed break down once the largeness assumption is dropped. The purpose of this note is to announce results about Σ V for V sufficiently small that are shown by completely different methods. We will indicate some of the main ideas that are used to prove these results, but we defer full details to a future publication.
We would like to emphasize that quantitative properties of regular Cantor sets such as thickness and denseness are widely used in dynamical systems (see [N70, N79], [PT], [M]), found an application in number theory (see [H]), but to the best of our knowledge, these kinds of techniques have never been used before in the context of mathematical physics.

Statement of the Results
In this section we describe our results for small coupling V . Clearly, as V approaches zero, H V,ω approaches the free Schrödinger operator which is a well-studied object whose spectral properties are completely understood. In particular, the spectrum of H 0 is given by the interval [−2, 2]. It is natural to ask which spectral features of H V,ω approach those of H 0 . It follows from Sütő's result that the Lebesgue measure of the spectrum does not extend continuously to the case V = 0. Given this situation, one would at least hope that the dimension of the spectrum is continuous at V = 0.
It was shown by us in [DG] (and independently by Cantat [Can]) that Σ V is a dynamically defined Cantor set for V > 0 sufficiently small (i.e., the small coupling counterpart to Casdagli's result at large coupling). A consequence of this is the equality of Hausdorff dimension and upper and lower box counting dimensions of Σ V in this coupling constant regime. Our first result shows that the dimension of the spectrum indeed extends continuously to V = 0.
More precisely, there are constants C 1 , C 2 > 0 such that Theorem 1 is a consequence of a connection between the Hausdorff dimension of a Cantor set and its denseness and thickness, along with estimates for the latter quantities. Since these notions and connections may be less familiar to at least a part of our intended audience, let us recall the definitions and some of the main results; an excellent general reference in this context is [PT].
Let C ⊂ R be a Cantor set and denote by I its convex hull. Any connected component of I\C is called a gap of C. A presentation of C is given by an ordering U = {U n } n≥1 of the gaps of C. If u ∈ C is a boundary point of a gap U of C, we denote by K the connected component of I\(U 1 ∪ U 2 ∪ . . . ∪ U n ) (with n chosen so that U n = U ) that contains u and write τ (C, U, u) = |K| |U | .
With this notation, the thickness τ (C) and the denseness θ(C) of C are given by and they are related to the Hausdorff dimension of C by the following inequalities (cf. [PT,Section 4.2]), .
Due to these inequalities, Theorem 1 is a consequence of the following result: More precisely, there are constants C 3 , C 4 > 0 such that for V > 0 sufficiently small.
Bovier and Ghez described in their 1995 paper [BG] the then-state of the art concerning mathematically rigorous results for Schrödinger operators in ℓ 2 (Z) with potentials generated by primitive substitutions. The Fibonacci Hamiltonian belongs to this class; more precisely, it is in many ways the most important example within this class of models. One of the more spectacular discoveries is that, in this class of models, the spectrum jumps from being an interval for coupling V = 0 to being a zero-measure Cantor set for coupling V > 0. That is, as the potential is turned on, a dense set of gaps opens immediately (and the complement of these gaps has zero Lebesgue measure). It is natural to ask about the size of these gaps, which can in fact be parametrized by a canonical countable set of gap labels; see [BBG92]. For some examples, these gap openings were studied in [B] and [BBG91]. However, for the important Fibonacci case, the problem remained open. In fact, Bovier and Ghez write on p. 2321 of [BG]: It is a quite perplexing feature that even in the simplest case of all, the golden Fibonacci sequence, the opening of the gaps at small coupling is not known! Our next result resolves this issue completely and shows that, in the Fibonacci case, all gaps open linearly: Theorem 3. For V > 0 sufficiently small, the boundary points of a gap in the spectrum Σ V depend smoothly on the coupling constant V . Moreover, given any such one-parameter family {U V } V >0 , where U V is a gap of Σ V and the boundary points of U V depend smoothly on V , we have that exists and belongs to (0, ∞).

Our next result concerns the sum set Σ
This set is equal to the spectrum of the so-called square Fibonacci Hamiltonian. Here, one considers the Schrödinger operator [H (2) V ψ](m, n) =ψ(m + 1, n) + ψ(m − 1, n) + ψ(m, n + 1) + ψ(m, n − 1)+ in ℓ 2 (Z 2 ). The theory of tensor products of Hilbert spaces and operators then implies that σ(H V ) = Σ V +Σ V . This operator and its spectrum have been studied numerically and heuristically by Even-Dar Mandel and Lifshitz in a series of papers [EL06,EL07,EL08]. Their study suggested that at small coupling, the spectrum of Σ V + Σ V is not a Cantor set; quite on the contrary, it has no gaps at all.
Our final theorem confirms this observation: Theorem 4. For V > 0 sufficiently small, we have that Σ V + Σ V is an interval.
Notice that Theorem 4 is a direct consequence of Theorem 2 and the famous Gap Lemma, which was used by Newhouse to construct persistent tangencies and generic diffeomorphisms with infinite number of attractors (the so-called "Newhouse phenomenon"): Gap Lemma (Newhouse [N70]). If C 1 , C 2 ⊂ R 1 are Cantor sets such that τ (C 1 ) · τ (C 2 ) > 1, then either one of these sets is contained entirely in a gap of the other set, or C 1 ∩ C 2 = ∅.

Comments on the Proofs
We will exploit the following relation between the spectrum of the Fibonacci Hamiltonian and the dynamical system known as the trace map. By the trace map we mean the following polynomial map: The trace map preserves the following quantity (called Fricke-Vogt invariant): I(x, y, z) = x 2 + y 2 + z 2 − 2xyz − 1.
We denote by S V the surface I(x, y, z) = V 2 4 invariant under T . The relation between the trace map and the spectrum of the Fibonacci Hamiltonian is given by the following statement.
Theorem 5 (Sütő [S87]). An energy E ∈ R belongs to the spectrum Σ V of H V,ω if and only if the positive semiorbit of the point ( E−V 2 , E 2 , 1) under iterates of the trace map T is bounded.
Denote by ℓ V the line It is easy to check that ℓ V ⊂ S V . It is known that the set is a hyperbolic set for all V = 0, see [Cas], [DG], [Can]. Moreover, its stable manifolds are transversal to ℓ V for small V . Therefore the set W s (Λ V ) ∩ ℓ V is affinely equivalent to Σ V . In particular, Σ V is a dynamically defined Cantor set for small values of V . The surface S 0 is the so-called Cayley cubic, it has four conic singularities and can be represented as a union of a two dimensional sphere (with four conic singularities) and four unbounded components. The restriction of the trace map to the sphere is a pseudo-Anosov map (a factor of a hyperbolic map of a two-torus), and its Markov partition can be presented explicitly (see [Cas] or [DG]). For small values of V , the map T : S V → S V "inherits" the hyperbolicity of this pseudo-Anosov map everywhere away from singularities. The dynamics near the singularities needs to be considered separately. Due to the symmetries of the trace map, it is enough to consider the dynamics of T near one of the singularities, say, near the point p = (1, 1, 1). The set P er 2 (T ) of periodic orbits of period two is a smooth curve that contains the point p and intersects S V at two points (denote them by p 1 and p 2 ) for V > 0. Also, this curve is a normally hyperbolic manifold, and we can use the normally hyperbolic theory (see [HPS], [PSW]) to study the behavior of T in a small neighborhood of p; see [DG] for details. In particular, since the distance between p 1 and p 2 is of order V , and the gaps in the spectrum are formed by the points of intersection of W ss (p 1 ) and W ss (p 2 ) with the line ℓ V , the size of a given gap is of order V as V → 0, which implies Theorem 3.
In order to estimate the thickness (and the denseness) of the spectrum Σ V , we notice first that the Markov partition for T : S 0 → S 0 can be continuously extended to a Markov partition for T : Λ V → Λ V , and while the size of the elements of these Markov partitions remains bounded, the size of the distance between them is of order V . The natural approach now is to use the distortion property (see, e.g., [PT]) to show that for the iterated Markov partition, the ratio of the distance between the elements to the size of an element is of the same order. The main technical problem here is again the dynamics of the trace map near the singularities, since the curvature of S V is very large there for small V . Nevertheless, one can still estimate the distortion that is obtained during a transition through a neighborhood of a singularity and prove boundedness of the distortion for arbitrarily large iterates of the trace map. This implies Theorem 2.