Almost Everything About the Fibonacci

Operator

David Damanik

Abstract We consider the Fibonacci operator and discuss results that have been obtained for the spectrum, the spectral measures, and the rate of wavepacket spreading. Our presentation is centered around a distortion result that describes the preimage of balls under the trace of the transfer matrix associated with sites given by Fibonacci numbers.

1 Introduction

This paper is concerned with the Fibonacci operator, which is a discrete one-dimen- sional Schrödinger operator

in 2(Z) with potential V : Z → R given by

√

Here, λ > 0 is the coupling constant, φ = 5−1 is the inverse of the golden mean,

2

and θ [0, 1) is the phase. By strong approximation it may be shown that the spectrum of H does not depend on the phase θ . The spectrum of H does, however, depend on the value of the coupling constant λ and we will henceforth denote it by Σλ.

The Fibonacci operator plays a prominent role in the context of one-dimensional quasicrystals and it has been studied since the early 1980’s in numerous physics and mathematics papers. In this paper, we will focus on the mathematical results that

David Damanik

Department of Mathematics, Rice University, Houston, TX 77005, USA, e-mail: damanik@rice.edu

have been obtained for this operator. References to the relevant physics literature can be found, for example, in the review articles [4, 30].

Natural questions concern the location and size of Σλ, the type of the spectral measures, and the long-time asymptotics of the solutions to the time-dependent Schrödinger equation, i∂t ψ = H ψ . Many of these questions have been answered completely, while others have been partially answered. Central to all of these results is the so-called trace map, which is a dynamical system that describes the evolution of the traces of the transfer matrices associated with H along the sequence of Fibonacci numbers.

We describe the trace map and the crucial distortion result in Sect. 2. The location of the spectrum can be described with the help of the trace map. In Sect. 3, we discuss the size of the spectrum as a subset of the real line; namely, it always has zero Lebesgue measure and, for coupling sufficiently large, we give upper and lower bounds on its fractal dimension. These bounds agree in the λ → ∞ limit. The spectral measures are always purely singular continuous, as discussed in Sect. 4. We sketch the Gordon two-block method that allows one to exclude eigenvalues for all parameter values. Absence of absolutely continuous spectrum is of course a consequence of zero-measure spectrum. Finally, in Sect. 5, we consider the rate at which an initially localized wavepacket spreads out in space if the time-evolution is given by the unitary group generated by the Fibonacci operator. If the coupling constant is sufficiently large, there are upper and lower bounds for the spreading rate that show behavior which is different from the one that occurs for periodic or random potentials. In other words, the intermediate spectral type is indeed reflected by intermediate wavepacket spreading behavior.

2 The Trace Map

where the transfer matrices Φ(n, z) are given by Φ(n, z) = T (n, z) × · · · × T (1, z) and

T (m, z) = z − V (m) −1 .

1 0

For z C and k ≥ 1, set Mk (z) = Φ(Fk , z), where the potential V is given by (2) with θ = 0 and Fk is the k-th Fibonacci number, that is, F0 = F1 = 1 and Fk+1 = Fk + Fk−1 for k ≥ 1. These matrices obey

Letting x−1(z) = 1 and x0(z) = 2z , the recursion (6) holds for all k ≥ 0. See, for example, [28] for these results.

The trace map is defined as follows,

T : (x, y, z) → (2xy − z, x, y),

and it may be defined on either R3 or C3, depending on the context. The forward

for some N ≥ 0; see [10]. This N is unique. Moreover, in this case we have |xN +k (z)| ≥ (1 + δ)Fk for k ≥ 0. Thus, it is natural to consider the sets σkδ =

In particular, {xk (z)} is bounded for every z Σλ. The invariant (7) yields a λ- dependent upper bound that works uniformly for all such z’s.

Moreover, assuming λ > λ0(δ) = [12(1+δ)2 +8(1+δ)3 +4]1/2, the invariant (7) implies that σkδ ∩ σkδ+1 ∩ σkδ+2 = and that the set σkδ has exactly Fk connected components. Each of them is a topological disk that is symmetric about the real axis.

All roots of xk are real. Consider such a root z and define ω(z) = #{0 ≤ l ≤ k − 1 : |xl (z)| ≤ 1}. Let ck,m = #{roots of xk with ω(z) = m}. An explicit formula for ck,m was found in [14, Lemma 5] (noting that our ck,m equals ak,m + bk,m in the notation of that paper). In particular, it follows from [14] that, for every k ≥ 2, ck,m is non-zero if and only if k2 ≤ m ≤

Let {zk(j )}1≤j ≤Fk be the roots of xk and write ωk(j ) = ω(zk(j )) for 1 ≤ j ≤ Fk . Denote by B(z, r) the open ball in C that is centered at z and has radius r. Define

Then, we have the following distortion result [10, 11].

Theorem 1. Fix k ≥ 3, δ > 0, and λ > max{λ0(2δ), 8}. Then, there are constants cδ , dδ > 0 such that

where rk(j ) = cδ Su(λ)−ωk(j ) , and Rk(j ) = dδ Sl (λ)−ωk(j ) . The first inclusion in (11) only needs the assumption λ > λ0(2δ).

The proof relies on the Koebe Distortion Theorem and upper and lower bounds for the derivative of xk at the roots. Such bounds were shown in [8, 22, 27]. We point out that Raymond’s paper [27] developed a number of tools that have proved to be useful in the quantitative study of the trace map, some of which have been refined and extended in later papers such as [14, 8, 22].

3 The Cantor Structure and the Dimension of the Spectrum

In this section we discuss properties of the spectrum of the Fibonacci operator as a subset of the real line.

The location of the spectrum can be conveniently described with the help of the trace map; compare (9). While (11) is a useful tool in the study of the size of the spectrum at large values of the coupling constant, there is actually a result that holds for all couplings.

Theorem 2. For every λ > 0, the spectrum of the Fibonacci operator has zero Lebesgue measure, Leb(Σλ) = 0.

Since the spectrum is closed and does not contain any isolated points by general principles, it follows from Theorem 2 that the spectrum of the Fibonacci operator is always a Cantor set.

This result was shown by Süto˝ in 1989 [29]; see also [1] for a contemporaneous proof by Bellissard, Iochum, Scoppola and Testard of zero measure spectrum for potentials (2) with φ replaced by an arbitrary irrational number and a more recent paper by Lenz [25] who develops a completely different approach to zero measure spectrum that is even more general in scope.

Let us describe the main steps in the proof of zero-measure spectrum based on the trace map. The Lyapunov exponent is defined by

Here we choose θ = 0 and leave the dependence of γ (z) on λ implicit. It was shown by Hof that the limit exists and is the same as the one obtained when one averages over the phase [16]. Consider the set Zλ = {z : γ (z) = 0}. It follows by general principles that Zλ Σλ. In the Fibonacci case, we even have equality. This was shown by Süto˝ [29] who used the boundedness of {xk (z)} for z Σλ to show a subexponential upper bound for Φ(n, z) . This result was later improved by Iochum and Testard [18] who proved a power-law upper bound for Φ(n, z) . Both works proved these bounds for the case θ = 0; see [7] for an extension to general θ . On the other hand, Kotani showed in a much more general context (aperiodic ergodic potentials taking finitely many values) that Zλ has zero Lebesgue measure. Combining these two results, it follows that Σλ has zero Lebesgue measure for every

λ > 0.

As a next step, it is natural to study the dimension of the spectrum. Recall that for S R bounded and infinite, the following two dimensions are of interest. For α [0, 1], let

and then define the Hausdorff dimension of S by

dimH (S) = inf{α : hα (S) < ∞} = sup{α : hα (S) = ∞}.

The lower box counting dimension of S is given by

where NS (ε) = #{j Z : [j ε, (j + 1)ε) ∩ S = }. The upper box counting dimension, dim+B (S), is defined with a lim sup in place of the lim inf. When the lower and upper box counting dimensions coincide, we say that the box counting dimension exists and denote it by dimB (S).

The following result about these dimensions was obtained for the set Σλ in [14].

Theorem 3. Suppose that λ ≥ 16. Then, the box counting dimension of Σλ exists and obeys dimB (Σλ) = dimH (Σλ).

The assumption λ ≥ 16 comes from a paper of Casdagli [2], who works under

this assumption and proves the hyperbolicity of the trace map, restricted to the in-

variant surface Ωλ = {(x, y, z) R3 : x2 + y2 + z2 − 2xyz − 1 = λ42 }; compare (7). It is expected (see [2]) that such a result extends to all λ > 0. The statement of

Theorem 3 then follows from known results for dynamically defined Cantor sets; see [14] for a discussion and references.

In order to describe the large coupling asymptotics of the dimension of the spectrum, let us introduce the function

Theorem 5. For every λ > 0 and every θ [0, 1), the Fibonacci operator has purely singular continuous spectrum.

One half of the result, the absence of absolutely continuous spectrum, is an immediate consequence of Theorem 2. The set Σλ is too small to support an absolutely continuous spectral measure. Historically, however, it was first shown by Kotani that the absolutely continuous spectrum is empty for every λ > 0 and Lebesgue almost every θ (this is equivalent to Leb(Zλ) = 0 for every λ > 0—the result quoted from [23] in the previous section) and the insight contained in Kotani’s short paper [23] was the key to all zero measure spectrum results mentioned above [1, 25, 29]. The other half of the result, the absence of eigenvalues, was established in [6] (see [1, 3, 17, 21] for earlier results in this direction).

Let us briefly sketch how eigenvalues, and in fact the existence of decaying solutions to (3) for z’s in the spectrum, can be excluded. A crucial realization is that the restriction to the positive half-line of every Fibonacci potential “begins with infinitely many squares”. Formally, the result is the following. Choose any θ [0, 1) (and any λ > 0; we will see that the latter choice is irrelevant for what follows) and consider the transfer matrices Φ(n, z) associated with the potential (2). Then, for infinitely many values of k, we have that

In particular, once such a property holds for one value of λ > 0, it holds for all values. The subsequence of {Fk } for which this statement holds is θ -dependent. The paper [6] develops a partition based approach to Fibonacci potentials from which the result just quoted follows quickly. These partitions are derived from an iteration of the rule (5) and a globalization procedure. In fact, the squares obtained in (13) have the additional property

By (7) and (9), |xk (z)| with z Σλ and k Z+ is uniformly bounded from above by a constant depending only on λ. Moreover, since transfer matrices always have determinant one, the Cayley-Hamilton Theorem says that

Thus, by (4) and (13)–(15) we see that every non-zero solution u of (3) with z Σλ must have lim supn→∞ |u(n)| > 0. Consequently, there are no decaying (and particularly no 2) solutions.

The proof sketched here contains three main ideas. The use of the Cayley-Ha- milton Theorem to exclude decaying solutions goes back to Gordon [15]. Using the uniform upper bounds for transfer matrix traces to be able to work on a half-line only is an idea of Süto˝ [28]. The partition approach that allows one to treat all phases was developed by Damanik and Lenz in [6].

5 Bounds on Wavepacket Spreading

In this section we discuss the spreading of an initially localized wavepacket that evolves according to the Schrödinger equation i∂t ψ = H ψ . We therefore consider ψ (t ) = e−it H δ0 and study its spreading via the time-averaged moments of the position operator,

where p > 0. To describe the power-law growth of |X|p (T ), consider the lower

δ0

transport exponent

and the upper transport exponent

Both functions β± (p) are nondecreasing in p and hence the following limits exist,

δ0

We consider these dynamical quantities for the Fibonacci operator and hence they depend on the parameters λ and θ . The moments grow no faster than Cε T p+ε for any ε > 0 and therefore the transport exponents take values in the interval [0, 1].1

There are numerous papers that establish bounds for the spreading of a wavepacket for the Fibonacci operator; see, for example, [3, 11, 8–10, 12–14, 20, 22]. The situation is particularly well understood in the regime of large λ and large p.

Here is a result that should be regarded as a companion to Theorem 4.

√

Theorem 6. (a) Suppose λ > 24. Then, we have

1 As background information, it may be useful to remark that (in the one-dimensional situation we consider here) the transport exponents vanish for random potentials, whereas they are equal to one for periodic potentials. Here we are dealing with a potential of intermediate complexity/disorder and we will see that this is reflected by intermediate transport.

Remarks. (i) These bounds are uniform in θ .

(ii)The functions Su(λ) and Sl (λ) are as defined in (10).

(iii)Part (a) is shown in [11], while part (b) is from [10].

(iv)In the limit λ → ∞, we find

The convergence is uniform in θ .

(v) It was shown in [14] that whenever the transfer matrices are bounded by a power law on the spectrum, we have that αu± is bounded from below by the upper/lower box counting dimension of the spectrum. Thus, in the Fibonacci case, we always have

Comparing (12) and (16), we see that for λ sufficiently large, the inequality (17) is strict.

The papers mentioned above also establish lower bounds for β± (p) for every

δ0

λ > 0, θ [0, 1), and p > 0. These bounds show that the transport exponents are always strictly positive. In particular, there is always spreading that is measurable on a power-scale. However, many of these results do not seem to be optimal. This is why (16) is special in this regard as it yields an optimal description of a quantum dynamical quantity in an asymptotic regime. We refer the reader to the original literature for more detailed dynamical lower bounds. As for dynamical upper bounds, Theorem 6(b) is the only known result (see also [5, 22] for earlier weaker results that bound the spreading rate of the slow part of the wavepacket from above). In particular, there are no non-trivial upper bounds for transport exponents for small

values of λ and even for λ ≥ 8, there are no upper bounds for β± (p) that improve

δ0

upon the bound in Theorem 6(b).

Let us describe the mechanisms leading to Theorem 6. The starting point is Kato’s formula,

which follows quickly from the Plancherel Theorem for the Fourier transform; com-

pare, for example, [22, Lemma 3.2]. This allows one to relate |X|p (T ) to decay

δ0

properties of the Green function, which in turn is closely related to the growth of the norm of Φ(·, E + Ti ). To obtain bounds on the Green function or the norm of the transfer matrix, we start again from the distortion result, Theorem 1. If a complex energy in question, z = E + Ti , belongs to a set σkδ , we go “backwards in time” and use (8) to show that up to level k, the traces are bounded by the uniform λ-dependent constant and norms are bounded by a uniform λ-dependent power-law. Conversely, if z = E + Ti does not belong to σkδ σkδ+1, then the escape condition (8) must have held sometime in the past and by now the trace and hence the norm of the transfer matrix must be very large. Since Theorem 1 allows one to link T and k for such

statements to be true, one can derive explicit bounds for the transfer matrix norm in this way.

Acknowledgements The title of the present paper wants to make reference to the pair of articles [19, 24], which at the time provided a similar snapshot of the state of the art for the other very prominent quasi-periodic discrete one-dimensional Schrödinger operator, the almost Mathieu operator. I would like to express my gratitude to Svetlana Jitomirskaya and Yoram Last for their support, especially during my early postdoctoral years. I also want to thank very warmly everyone I collaborated with on the results presented in this paper: Mark Embree, Anton Gorodetski, Rowan Killip, Daniel Lenz, András Süto,˝ and Serguei Tcheremchantsev.

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