Eigenvalue Spacings and Dynamical Upper Bounds for Discrete One-Dimensional Schrödinger Operators

Transcription

1 Eigenvaue Spacings and Dynamica Upper Bounds for Discrete One-Dimensiona Schrödinger Operators Jonathan Breuer, Yoram Last, and Yosef Strauss Institute of Mathematics, The Hebrew University of Jerusaem, Givat Ram, Jerusaem, Israe. Emai: October 6, 009 Abstract We prove dynamica upper bounds for discrete one-dimensiona Schrödinger operators in terms of various spacing properties of the eigenvaues of finite voume approximations. We demonstrate the appicabiity of our approach by a study of the Fibonacci Hamitonian. 1 Introduction Let H = +V be a discrete, one-dimensiona, bounded Schrödinger operator on Z) or N): Hψ) n = ψ n+1 + ψ n 1 + V n ψ n, 1.1) where V n is a bounded rea vaued function on Z or N and in the case of N, Hψ) 1 = ψ + V 1 ψ 1. We are interested here in the unitary time evoution, e ith, generated by H. A wave packet, ψt) = e ith ψ, which is initiay ocaized in space, tends to spread out in time. Connections between the rate of this spreading and spectra properties of H have been the subject of extensive research in the ast three decades. However, whie various ower bounds have been obtained in many interesting cases using an assortment 1

2 of different approaches, the subject of upper bounds is significanty ess we understood. The purpose of this work is to formuate genera dynamica upper bounds in terms of purey spectra information. However, rather than consider spectra measures of H, the infinite voume object, we sha consider finite voume approximations to H. Among the properties we consider, a centra roe wi be payed by the spacing of eigenvaues of these finite voume approximations. In particuar, we sha show that the rate of spreading of a wave packet can be bounded from above by the strength of eigenvaue custering on finite scaes. The spreading rate of a wave packet may be measured in various different ways for a survey of severa of these see [6]). As is often done in this ine of research, we sha focus in this work on the average portion of the tai of the wave packet that is outside a box of size q after time T, P ψ q, T) = n >q T 0 δ n, ψt) e t/t dt, where ψt) = e ith ψ and, is the inner product. Moreover, as ψ = δ 1 is an idea candidate for a wave packet that is ocaized at the origin, we sha further restrict our attention to that case and denote Pq, T) P δ1 q, T) = n >q T 0 δ n, e ith δ 1 e t/t dt. Another important measure of the spreading rate of ψt) is the rate of growth of moments of the position operator: X m T = n T 0 n m δ n, ψt) e t/t dt, where m > 0. These quantities are reated. In particuar, since for any bounded V the baistic upper bound, X m T C m T m, hods, it foows from known resuts see [15]) that if PT α, T) = OT k ) as T for a k, then X m T C m T αm for a m. As mentioned above, various ower bounds for P ψ q, T) and its asymptotics, using oca properties of µ ψ, the spectra measure of ψ, have been obtained. A basic resut in this area which is often caed the Guarneri- Combes-Last bound [4, 16, 17, 6] says that if µ ψ is not singuar with respect to the α-dimensiona Hausdorff measure, then P ψ q, T), for q T α, is

3 bounded away from zero for a T. Various bounds on other reated properties of µ ψ have aso been shown to impy ower bounds on transport see, e.g., [, 18, 0]). Roughy speaking, a centra idea in a these works says that a higher degree of continuity of the spectra measure impies faster transport. Another idea, not unreated to the one above, is to use growth properties of generaized eigenfunctions to get dynamica bounds. There have been many works in this direction e.g., [7, 8,, 3]), some of which aso combine generaized eigenfunction properties with properties of the spectra measure mentioned above. Especiay reevant to this paper is [7] which shows that it is sufficient to have nice behavior of generaized eigenfunctions at a singe energy in order to get ower bounds on the dynamics. The reevance of this resut here is to Theorem 1.9 beow, which assumes contro of a eigenvaues of H q π/). The resut of [7] quoted above may provide a cue as to whether this restriction is necessary in a certain sense or simpy a side-effect of our proof. Whie there are fairy many resuts concerning ower bounds on P ψ q, T), there seem to be much fewer works concerning genera upper bounds on it. If µ ψ is a pure point measure, it foows from a variant of Wiener s theorem [8, Theorem XI.114] that im q im T P ψ q, T) = 0, showing that the buk of the wave packet cannot spread to infinity. For the Anderson mode where V n is a sequence of i.i.d. random variabes), the tais of the wave packet have been shown to remain exponentiay sma for a times see, e.g., [4]), impying boundedness of X. However, as shown in [11], near-baistic growth T of X is aso possibe for pure point spectrum. If µ T ψ is continuous, there are exampes [3, 3] showing that one may have fast transport e.g., near-baistic spreading of the buk of the wave packet) even for cases where µ ψ is very singuar. It is thus understood that, for continuous measures, there can be no meaningfu upper bounds in terms of continuity properties of the spectra measure aone. Moreover, as bounding the growth rate of quantities ike X from above invoves contro of the fu wave packet whie it is T sufficient to contro ony a portion of the wave packet to bound such growth rates from beow), obtaining upper bounds on Pq, T) that woud be good enough to yied meaningfu bounds on the growth rates of moments of the position operator appears to be a significanty more difficut probem then obtaining corresponding ower bounds. The few cases where upper bounds have been obtained for singuar contin- 3

4 uous measures incude a work of Guarneri and Schuz-Bades [19], who consider certain Jacobi matrices discrete Schrödinger operators with non constant hopping terms) with sef-simiar spectra and formuate upper bounds in terms of parameters of some reated dynamica systems, a work by Kiip, Kiseev and Last [], who obtain a fairy genera upper bound on the spreading rate of some portion of the wave packet, and a recent work of Damanik and Tcheremchantsev [9] to which we sha return ater on), who obtain dynamica upper bounds from properties of transfer matrices. Out of these, the ast mentioned work is the ony one obtaining tight contro over the entire wave packet, thus providing upper bounds on Pq, T) that are good enough to yied meaningfu bounds on the growth rates of moments of the position operator. As mentioned above, our aim in this paper is to derive genera dynamica upper bounds in terms of purey spectra information. Moreover, our bounds achieve tight contro of the entire wave packet, thus yieding meaningfu bounds on the growth rates of moments of the position operator ike those of [9]). It is cear, from the exampes cited above, that oca properties of the spectra measure aone are not sufficient for this purpose. Thus, we sha consider the spectra properties of finite voume approximations to H. Before we describe our resuts, et us introduce some usefu notions. It has become customary to use certain exponents to measure the rates of growth of various parts of the wave packet. Foowing [15] aso see [9]), we define { } og PT α, T) = sup α > 0 im sup = 0 α + α = sup T { α > 0 im inf T og T og PT α, T) og T } 1.) = 0 and { α u + og PT α, T) = sup α > 0 im sup T og T α u = sup {α > 0 im inf T og PT α, T) og T } > } >. 1.3) α ± are interpreted as the rates of propagation of the sow moving part of the wave packet, whie α u ± are the rates for the fast moving part. The foowing notion is usefu for appications. 4

5 Definition 1.1. We ca a sequence of nonnegative numbers, {a n } n=1, exponentiay growing if sup n+1 a a n a n < and inf n+1 n a n > 1. We are finay ready to describe our resuts. We first discuss the resuts for H on N, since the whoe ine case wi be reduced to this case. For q > 1, q N, k [0, π], et H q k) be the restriction of H to {1,...,q} with boundary conditions ψq + 1) = e ik ψ1), namey, V e ik 1 V H q k) = ) e ik V q As is we known see, e.g., [9, Section 7]), for any k 0, π), H q k) has q simpe eigenvaues, E q,1 k) < E q, k) <... < E q,q k). E q,j k) are continuous monotone functions of k and, as k varies over [0, π], they trace out bands. An idea that goes back to Edwards and Thouess [13] aso see [30]) is to use the width of the bands, b q,j E q,j π) E q,j 0) which can be identified as a measure of what is often caed Thouess energy in physics iterature), as a measure of the system s sensitivity to a variation of boundary conditions. Faster spreading of the wave packet is intuitivey associated with a greater degree of extendedness of the eigenstates of H and thus with a greater sensitivity to a change in k. Our first theorem is motivated by this intuitive picture: Theorem 1.. For 1 j q, et b q,j E q,j π) E q,j 0). Then Pq, T) 4e ) 1 + V ) T 6 sup b q,j ) j q Coroary 1.3. Assume that there exist β > 3 and a sequence, {q } =1, such that sup 1 j q b q,j < q β. Then α 3/β. If, moreover, the sequence {q } =1 above is exponentiay growing, then α+ 3/β. The bands traced by E q,j k) make up the spectrum of the whoe ine operator with q-periodic potentia, V per, defined by V per nq+j = V j 1 j q). An eementary argument shows that [1, Theorem 1.4] impies that sup b q,j π 1 j q q. 5

6 Of course, we need stronger decay to get a meaningfu bound. The 1 + V ) T 6 factor in Theorem 1. is party due to the effective approximation of V by V per, which underies our anaysis. It can be repaced by T if V happens to be periodic of period q and it can be improved, for some q s, in cases where V has good repetition properties that make it cose to being periodic for some scaes. As seen in Coroary 1.3, this OT 6 ) factor impies that for Theorem 1. to be meaningfu, sup j b q,j ) the squared maxima Thouess width ) must decay quite fast. By considering additiona information, one can do better. Let Ẽq,j = E q,j π ). We wi show that contro of the custering properties of the Ẽq,j eads to exponentia bounds on Pq, T). Definition 1.4. We say that the set E q {Ẽq,j} q j=1 is ε, ξ)-custered if there exists a finite coection {I j } k j=1 k q) of disjoint cosed intervas, each of size at most ε, such that E k j=1 I j and such that every I j contains at east q ξ points of the set {Ẽq,j} q j=1. When we want to be expicit about the cover, we say that E q is ε, ξ)-custered by {I j } k j=1. Theorem 1.5. Let b q,j E q,j π) E q,j 0) for 1 j q. For any /3 < ξ 1 and 0 < α < 1, there exist constants δ > 0 and q 0 > 0 such that, if q q 0 and the set E q {Ẽq,1, Ẽq,,...,Ẽq,q} is q 1/α, ξ)-custered, then for T q 1/α, Pq, T) 4e 1 + V ) T 4 sup b q,j ) e Cqδ, 1.5) 1 j q where C is some universa constant. In particuar, Pq, q 1/α ) 4e 1 + V ) q 4/α sup b q,j ) e Cqδ. 1.6) 1 j q Coroary 1.6. Assume that there exist /3 < ξ 1, 0 < α < 1 and a sequence {q } =1, such that the set E q is q 1/α, ξ)-custered for a N. Then αu α. If, moreover, the sequence {q } =1 above is exponentiay growing, then α u + α. Thus, we see that the spreading rate of a wave packet can be bounded by the strength of eigenvaue custering at appropriate ength scaes. The fina ingredient in our anaysis comes from considering different ength scaes at once. The oca bound of Theorem 1.5 can be improved if the custers at different scaes form a structure with a certain degree of sef-simiarity. We first need some definitions. As above, for any q N, we et E q = {Ẽq,j} q j=1. 6

7 Definition 1.7. We say that the sequence {E q } =1 is uniformy custered if E q is {q 1/α, ξ }-custered by U = {Ij} k j=1 and the foowing hod: i) If 1 < then q 1/α 1 1 > q 1/α. ii) There exist µ 1 and a constant C 1 > 0 so that inf Ij C 1 q µ/α. 1 j k iii) There exists a δ > 0 so that δ < ξ < 1 δ) and δ < α < 1 for a. iv) Define ξ = og sup 1 j k # E I j)) / og q, then there exists a constant C so that ξ ξ C og q. When we want to be expicit about the cover and the reevant exponents, we say that {E q } =1 is uniformy custered by the sequence {U } =1 U = {Ij} k j=1 ) with exponents {α, ξ, µ}. { } Definition 1.8. Let U = {Ij }k j=1 be a sequence of sets of intervas =1 such that ε Ij Cε µ for a monotonicay decreasing sequence {ε } =1 and constants C > 0 and µ 1. Let 0 < ω < 1. We say that {U } =1 scaes nicey with exponents µ and ω if for any 1 > ε > 0 there exists a set of intervas, U ε = {Ij ε}kε j=1, of ength at most ε and no ess than Cεµ, such that U ε = U with the foowing properties: i) If ε 1 > ε then for any 1 j k ε there exists an 1 m k ε1 such that I ε j I ε 1 m. ii) There exists a constant C 3 > 0 such that if ε 1 > ε then for any 1 m k ε1, #{j I ε j I ε 1 m } C 3 ε 1 /ε ) ω. In simpe words, {U } =1 scaes nicey if the sequence may be extended to a continuous famiy parameterized by the interva engths) in such a way that intervas of one ength scae are contained in and, to a certain extent, nicey distributed among the intervas of a arger ength scae. Remark. We emphasize that, whie the famiies U in Definition 1.7 are assumed to consist of disjoint intervas, we do not assume this disjointness about the famiies U ε in Definition 1.8. Remark. An exampe of a nicey scaing sequence is given by the sequence {U } =1 where U 1 = {[0, 1/3], [/3, 1]}, U = 7

8 {[0, 1/9], [/9, 1/3], [/3, 7/9], [8/9, 1]} and so on U is the set of intervas obtained by removing the midde thirds of the intervas comprising U 1 ). It is not hard to see that this sequence scaes nicey with exponents µ = 1 and ω = og og 3 setting as the vaue of C 3 in Definition 1.8). Theorem 1.9. Assume { that {q } =1} is a sequence such that {E q } =1 is uniformy custered by U = {Ij }k j=1 with exponents {α, ξ, µ}. Suppose, moreover, that {U } =1 scaes nicey with exponents µ and ω for some =1 0 < ω < 1. Assume aso that, for some ζ > 0, ) µ 1 ω + ζ < ξ α. 1.7) µ ω Then for any m > 0 there exists C m > 0 such that for any N and T q 1/α. Pq, T) C m q m 1.8) Coroary Under the assumptions of Theorem 1.9, αu im inf α. If, moreover, the sequence {q } =1 in the theorem is exponentiay growing, then α u + im sup α. Note that 1.7) says that one may trade strong custering for greater degree of uniformity over different ength scaes: When µ = 1 so custer sizes are very uniform), 1.7) says that the custering strength parameter ξ ony has to be positive. On the other hand, when µ = 1.7) says ξ α > ω. Now note that, since there are at east q ξ eigenvaues in each interva, there are at most q 1 ξ = ε α 1 ξ ) intervas, which shows that the assumption ω α 1 ξ ) is a natura one. But this, combined with ξ α > ω, immediatey impies ξ > /3, which is the condition of Theorem 1.5. Now et H = + V be a fu ine Schrödinger operator. We sha treat H by reducing the anaysis to that of two corresponding haf-ine cases. Let V ± V ± H ±. = V±n

9 be the restrictions of H to the positive and negative haf-ines, with the restriction to the negative haf-ine rotated to act on N) for notationa convenience. Note that V 0 is not present in either H + or H ). Furthermore, et and P δ0 q, T) = n >q P ± δ 1 q, T) = n>q T T 0 0 δ n, e ith δ 0 e t/t dt δ n, e ith± δ 1 e t/t dt, so that the sum in the second formua is restricted to N. Then Proposition For any q > 1 and any T > 0, P δ0 q, T) T P + δ 1 q, T) + P δ 1 q, T) ) 1.9) Thus, as remarked above, the fu-ine probem may be reduced to two haf-ine probems abeit with an extra factor of T ). Accordingy, Theorems 1., 1.5 and 1.9 above impy corresponding theorems and coroaries simiar to Coroaries 1.3, 1.6 and 1.10 for a fu ine operator. Formuating these resuts is straightforward, so we eave that to the interested reader. We ony note that, in the case of the appication of Theorems 1.5 and 1.9 and the corresponding coroaries, the T factor is of minor significance because of the existence of exponentia bounds. In the appication of Theorem 1., however, this factor is ceary significant. To demonstrate the appicabiity of our resuts we use them to obtain a dynamica upper bound for the Fibonacci Hamitonian, which is the most studied one-dimensiona mode of a quasicrysta. This is the operator with V given by: 5 1 VFib;n λ = λχ [1 θ,1)nθ mod1), θ = 1.10) with a couping constant λ > 0 and where χ I is the characteristic function of I. For a review of some of the properties of the Fibonacci Hamitonian, see [6]. A subbaistic upper bound for the fast-spreading part of the wave packet under the dynamics generated by the Fibonacci Hamitonian has been recenty obtained by Damanik and Tcheremchantsev in [9, 10]. They used ower bounds on the growth of transfer matrices off the rea ine to obtain an 9

10 upper bound on α u +. In particuar, they confirm the asymptotic dependence α u + 1 as λ ) of the transport exponent on the couping constant og λ as predicted by numerica cacuations see [1, 1]). In Section 5 beow, using purey spectra data for the Fibonacci Hamitonian, we appy our above resuts to get the same asymptotics for α u + λ) abeit with worse constants than those of [9]). We note that our primary purpose in this paper is to estabish genera bounds which are based on custering and sef-simiar mutiscae custering of appropriate eigenvaues. In particuar, we aim to highight the connection between sef-simiar Cantor-type spectra and what is often caed anomaous transport. Roughy speaking, our resuts show that as ong as the sef-simiar Cantor-type structure manifests itsef in a corresponding tight behavior of the eigenvaues of appropriate finite-voume approximations of the operator, meaningfu upper bounds can indeed be obtained and combined with existing ower bounds to estabish the occurrence of anomaous transport). We further note that appying our genera resuts to the Fibonacci Hamitonian is done mainy to demonstrate their appicabiity in their present form. In cases where one is interested in obtaining bounds for concrete modes, it is ikey that by using some of our core technica ideas beow aong with the most detaied reevant data avaiabe for the concrete mode in question one wi be abe to estabish stronger bounds than those obtained by direct appication of our above resuts. The rest of this paper is structured as foows. Section has some preiminary estimates that wi be used throughout the paper. The proof of Theorem 1. is aso given there. Section 3 has the proof of Theorems 1.5 and 1.9. Section 4 has the proofs of coroaries 1.3, 1.6 and 1.10 and of Proposition Section 5 describes the appication of our resuts to the Fibonacci Hamitonian. Acknowedgment. This research was supported by The Israe Science Foundation Grant No. 1169/06). Preiminary Estimates and Proof of Theorem 1. Let H = +V be a discrete Schrödinger operator on N with V, the potentia, a bounded rea-vaued function. For q N, k [0, π], et H q k) be defined as 10

11 in 1.4). Finay, et Hper q = +V per q where V per q j +nq) = V j) for 1 j q and n 0. We start by deriving an inequaity that wi be centra to a subsequent deveopments. Let Gk, n; z) = δ n, H z) 1 δ k and G q per k, n; z) = δ n, H qper z ) 1 δk. We define D q z) = G q per1, q; z) + 1 G q per1, q; z)..1) It is not hard to see, by recognizing D q z) as the trace of a transfer matrix, that it is a monic poynomia of degree q. This poynomia is caed the discriminant for Hper q. It has been studied extensivey in connection with the spectra anaysis of periodic Schrödinger operators see e.g. [9, Section 7] the discriminant there is haf ours). Among its properties that wi be usefu for us are: 1. The zeros of D q are precisey the q distinct eigenvaues of H q π ), i.e., the set {Ẽq,1, Ẽq,,...,Ẽq,q} and so are rea and simpe).. The essentia spectrum of Hper q of the set [, ]. is given by the inverse image under Dq 3. The restriction of D q to R takes the vaues and precisey at the eigenvaues of H q 0) and of H q π), and for any 1 j q, D q E q,j π)) D q E q,j 0)) = D q ) E) = 0 impies E R and D q E). Our starting point is Lemma.1. For q > 1 and for any positive T, ) Pq, T) P δ1 q, T) 4T V ) inf E R Dq E + i/t).) Proof. We start by recaing the formua []: 0 δ n, e ith δ k e t/t dt = 1 π δ n, H E i/t) 1 δ k de.3) 11

12 which is key in many recent works on this topic this work incuded). By.3) we see that Pq, T) = 1 1 δ n, H E i/t) 1 δ 1 de π T n>q = 1 de ε G1, n; E + iε), π R n>q.4) where we use ε = 1. T Now, et Hq = H δ q+1, δ q δ q, δ q+1 and et Gq k, n; z) = ) 1 δ n, Hq z δk. Then, by the resovent formua G1, n; z) = G q 1, n; z) G1, q; z) G q q + 1, n; z) G1, q + 1; z) G q q, n; z) = G1, q; z) G q q + 1, n; z),.5) if n > q since G is a direct sum. Moreover, note that ε n>q G q q+1, n; E + iε) = Im G q q + 1, q + 1; E + iε), this can be seen by noting that fn) G q q + 1, n; E + iε), satisfies fn + 1) + fn 1) + V n)fn) = E + iε)fn) for a n > q; now mutipy this by fn), sum up and take imaginary parts). Thus, we get from.4) that Pq, T) = 1 G1, q; E + iε) Im π G q q + 1, q + 1; E + iε)de,.6) which, by 1 π R R Im G q q + 1, q + 1; E + iε)de = 1, impies immediatey that Pq, T) sup G1, q, E + iε)..7) E R Next, we approximate G by G q per. Let H q per = Hq per δ q+1, δ q δ q, δ q+1 and et G q per k, n; z) = δ n, Hq per z ) 1 δk. Note that.5) hods with G repaced by G q per and G q repaced by G q per. Again, by the resovent formua 1

13 with z = E + iε), G q per 1, q; z) G1, q; z) = G q per 1, n; z) Vper q n) V n)) Gn, q; z) n>q = G q per 1, q; z) G q perq + 1, n; z) Vpern) q V n) ) Gn, q; z) n>q G q per 1, q; z) V Im G q perq+1,q+1;z) Im Gq, q; z) ε ε V ε G q per 1, q; z),.8) where the first inequaity foows by appying Cauchy-Schwarz and the second inequaity foows from Gk, ; z) 1. This immediatey impies Im z G1, q; E + iε) 1 + V ) G q ε per 1, q; E + iε) 1 + V G q ε per 1, q; E + iε).9). Now, since G q per 1, n; z) is the exponentiay decaying soution to ψn+1)+ ψn 1)+Vpern)ψn) q = zψn) for n > ), it foows that G q per 1, q; z) < 1 and so, by.1), that G q per 1 D q z). This impies G q per 1, q; z) D q z)..10) Combining.10),.9), and.7) remembering that ε = T 1 ) finishes the proof. Thus, our probem is reduced to the anaysis of D q z). As D q is monic we know q ) D q z) = z Ẽq,j..11) j=1 13

14 Moreover, as the zeros of D q ) are simpe, any point E R ies between two extrema points of D q or between an extrema point and ±. Thus, each point E R ies in an interva of the form [xe), ye)), or [xe), ), or, ye)) where xe), ye) are two extrema points of D q and the interva contains no other extrema points. Any such interva contains a unique zero of D q. In this way we associate with each point E R a unique zero Ẽq,jE) of D q. Using this notation, for any E R D q E + iε) = q E + iε Ẽq,j j=1 = E + iε Ẽq,jE) j je) E Ẽq,j j je) j je) ε E Ẽq,j j je) E + iε Ẽq,j j je) j je) E Ẽq,j = ε D q E) j je) E + iε Ẽq,j E Ẽq,jE) j je) E Ẽq,j. E + iε Ẽq,j E Ẽq,j.1) In a sense, a our theorems foow from ower bounds on the right hand side of.1). In particuar, Theorem 1. foows from noting j je) E+iε E q,j j je) E E q,j 1 and studying the other terms, whie Theorems 1.5 j je) and 1.9 foow from a more detaied anaysis of E+iε E q,j. j je) E E q,j We proceed now with the proof of Theorem 1.. We first need a emma. Lemma.. Reca b q,j E q,j π) E q,j 0). For any E [Ẽq,1, Ẽq,q] D q E) D e E Ẽq,jE) q ) Ẽq,jE)).13) b q,j where, for E = Ẽq,j for some j, the eft hand side is interpreted as the derivative.) Remark. The proof of this emma is essentiay contained in the proof of Lemma 1 of [5], athough it is stated somewhat differenty there aso see 14

15 [7, Theorem 5.4]). We repeat it here with some detais omitted) for the reader s convenience. Proof. We start with the upper bound. For this it suffices to show that for any E [Ẽq,j, Ẽq,j+1] 1 j q 1)) D q E) D e E Ẽq,j q ) Ẽq,jE)).14) Enumerate the zeros of D q ) by Eq,1, 0..., Eq,q 1. 0 Ceary, Ẽ q,j < Eq,j 0 < Ẽ q,j+1 for any 1 j q 1), and any E [Ẽq,j, Ẽq,j+1] is contained either in [Ẽq,j, Eq,j 0 ] or in [E0 q,j, Ẽq,j+1]. As the zeros of D q ) are simpe, Eq,j 0 is either a oca maximum or a oca minimum of D q. We sha prove.14) for E [Ẽq,j, Eq,j] 0 with Eq,j 0 a oca maximum. A the other cases are simiar. Let fe) = d og de Dq E). It is straightforward to see that f E) < 1 E E q,j ) and so E 0 q,j fe) = f x)dx > E 1 E Ẽq,j 1 Eq,j 0 Ẽq,j.15) note fe 0 q,j) = 0). Fix some E Ẽq,j, E). It foows that og Dq E) E D q E ) = og Dq E) og D q E ) = fx)dx > og E Ẽq,j E E Ẽq,j 1. Thus which impies D q E) D q E ) > 1 E Ẽq,j, e E Ẽq,j D q E) e E Ẽq,j > D q E ) D q Ẽq,j) E Ẽq,j. The estimate.14) foows from this by taking the imit E Ẽq,j. We now turn to prove the ower bound. First, to fix notation, define the intervas B q,j [Ẽ q,j, Ẽr q,j ] = { [Eq,j 0), E q,j π)] if E q,j 0) < E q,j π) [E q,j π), E q,j 0)] if E q,j π) < E q,j 0) 15.16)

16 so that b q,j = B q,j. We sha refer to the B q,j as the bands. Let further Bq,j = [Ẽ q,j, Ẽq,j] and Bq,j r = [Ẽq,j, Ẽr q,j ] be the two parts of the band B q,j and et b i q,j = Bq,j i i =, r). Note that, as D q ) is a q 1) degree poynomia with simpe zeros, D q ) has a singe maximum in each interva [Eq,j 0, E0 q,j+1 ] 1 j q ). Since, for 1 j q ), B q,j+1 [Eq,j, 0 Eq,j+1], 0 it foows that D q ) has a singe maximum in each interva B q,j 1 j q). If this maximum is in Bq,j then, by monotonicity D q ) Ẽq,j) D q ) E) for a E Bq,j r from which it foows that b r q,j Otherwise, the maximum is in B r q,j = Dq Ẽq,j) D q Ẽr q,j ) Ẽq,j Ẽr q,j D q ) Ẽq,j). b q,j and we get D q ) Ẽq,j). Assume that j q 1), and that D q is increasing on B q,j and consider the poynomia ge) = D q E) + which has a zero at Ẽ q,j. An anaysis simiar to the one eading to.15) shows that for E Ẽ q,j, E0 q,j ) g E) ge) = d de og ge) > 1 E Ẽ q,j which impies putting E = Ẽq,j) 1 Eq,j 0 Ẽ q,j D q ) Ẽq,j) = g Ẽq,j) gẽq,j) > 1 b q,j 1 b q,j + br q,j = b r q,j b q,j b q,j + br q,j ). Letting t q,j =, it foows that D q ) E q,j ) b q,j > t q,jb r q,j b q,j +..17) br q,j By considering the function D q E) and performing a simiar anaysis, we can get the same inequaity with and r interchanged: b r q,j > t q,jb q,j b q,j +..18) br q,j 16

17 We showed above that either b r q,j t q,j or b q,j t q,j. Assume b r q,j t q,j. Then.17) impies that b q,j ) + b q,j t q,j t q,j ) 0, from which it foows that b q,j 5 1t q,j. Simiary, b q,j t q,j impies that b r q,j 5 1t q,j so that in any case we have Thus b q,j = b q,j + br q,j t q,j. D q ) Ẽq,j) b q,j,.19) for any j q 1). For the case of j = 1, q ony one of the inequaities.17) or.18) can be obtained, but by monotonicity it foows that b r q,j t q,j corresponds to the case where.17) hods and vice versa and so we get.19) for a 1 j q, which concudes the proof. Proof of Theorem 1.. Note first that by the first equaity in.1), inf E R D q E + iε) inf E [ Eq,1, E q,q] Dq E + iε). By Lemma. we see that for any E [Ẽq,1, Ẽq,q] D q E) E Ẽq,jE) e ) b q,je)..0) Combining.),.1) and.0) with ε = 1/T), and noticing that j je) E+iε E q,j j je) E E q,j 1, we get Pq, T) 4T V ) sup D q E) E [ E q,1, E q,q] E Ẽq,jE) = 4e 5 + 1) T6 1 + V ) sup 4e 5 + 1) T6 1 + V ) b q,je) E [ E q,1, E q,q] ) sup b q,j. j 17

18 3 Proof of Theorems 1.5 and 1.9 We present in this section more refined ower bounds for the poynomias D evauated at a distance 1/T from the rea ine. In particuar, we examine the consequences of custering of the zeros of these poynomias. The bounds deveoped here are for fairy genera poynomias, and we beieve they may be interesting in other contexts as we. For this reason we depart from D for most of the anaysis and present our resuts for genera poynomias under the assumptions described beow). We return to D for the proofs of Theorems 1.5 and 1.9. To fix notation, et Q be a monic poynomia of degree q with rea and simpe zeros. Denote the set of zeros of Q by ZQ) = {z 1,...,z q }. As in the previous section see the discussion foowing.11)), any point E R ies between two extrema points of Q or between an extrema point and ±. Any such interva contains a unique zero of Q. In this way we associate with each point E R a unique zero ze) of Q. Our first emma iustrates why custering of zeros impies ower bounds away from R. Definition 3.1. Let Q be a poynomia of degree q and et ε > 0. We sha say that Q is ε-covered if Q is monic with rea and simpe zeros, and ZQ) is covered by a finite coection U ε Q) = {I j } k j=1 k q) of disjoint cosed intervas of size not exceeding ε such that every I j contains, in addition to at east one point of ZQ), a point x for which Qx) =. When we want to be expicit about the famiy of covering intervas, we sha say that Q is ε-covered by U ε Q) = {I j } k j=1. Remark. In the anaysis beow, Qx) = is not essentia. With obvious modifications, it can be carried out just as we with any other constant. Since the reevant constant for the appications is, we use it here. Let Q be an ε-covered poynomia and et {x j } be the points where Qx) =. Let b Q,j = x j zx j ) and b Q = sup j { b Q,j }. For E R, et I E be an eement of U ε Q) containing ze). Finay, et de) = E ze). We have the foowing emma: Lemma 3.. Let ε > 0 and assume Q is an ε-covered poynomia. Let A ε = {E R de) 8ε} and et B ε = R \ A ε. 18

19 Then for E A ε we have QE + iε) εq ze)) e ) #ZQ) IE ), 3.1) and for E B ε QE + iε) ) ε 9 #ZQ) IE) bq ) q 1. 3.) 4 Proof. We begin with 3.1). Let E A ε and write, as in.1), QE + iε) ε QE) z j ze) E z j) + ε E ze) z j ze) E z. j) The argument for the upper bound in.13) transates into this more genera setting and we get QE + iε) εq ze)) e εq ze)) e εq ze)) e z j ze) E z j ) + ε E z j ) E z j ) + ε E z j ) z j ze), z j I E ) ε #ZQ) IE 1 + ε + 8ε). Equation 3.1) foows. To prove 3.), fix E B ε and assume E > ze) with obvious changes, everything works simiary for E < ze)). Now, since ZQ) R, QE + iε) QE). Next, it foows from the definition of ze) that for any ze) < y < E, Qy) QE). Let Ê I E be a point for which QÊ) =. Ceary ze) < Ê + 4ε < E. Therefore, QE + iε) QE) QÊ + 4ε). 19

20 Now QÊ + 4ε) QÊ + 4ε) QÊ) = Ê + 4ε zê) Ê zê) Ê + 4ε z j Ê + 4ε z j Ê z j z j / I E Ê z j Ê + 4ε z j Ê z j, z j I E, z j zê) 3ε bq,j 3 #ZQ) I E) 1 z j / I E since for any y I E y Ê ε. As for the rest of the eements of ZQ), it is cear that those of them that are ocated to the eft of Ê contribute a factor that is greater than 1 to the right hand side. So we sha assume they are a ocated to the right of Ê. Since those that are in I E were aready taken into account, it foows that we are considering ony roots that are to the right of E. Thus we may assume that z j Ê > 8ε. Pugging this into the right hand side of the ast inequaity and remembering that #ZQ) = q, 3.) foows. The estimate 3.1) impies that crowding together of many zeros of Q in I E s eads to an exponentia ower bound on QE + iε) for appropriate E s. If not a the zeros are covered by a singe interva of size ε, the estimate 3.) might not be usefu. The foowing modification is taiored to dea with this probem. Lemma 3.3. Let 1/5 > ε > 0. Assume Q is an ε-covered poynomia and et 0 < ϕ < 1 be such that ε ϕ 1 > 5. Let and et A ϕ = {E R de) ε ϕ } B ϕ = R \ A ϕ. Then for E A ϕ we have QE + iε) εq ) ze)) #ZQ) IE e 1 + ε ϕ ), 3.3) 4 and for E B ϕ QE + iε) ) ε 9 #ZQ) IE) 1 4ε 1 ϕ ) q. 3.4) bq 0

21 Proof. To prove 3.3) we simpy repeat the proof of 3.1) to obtain QE + iε) εq ze)) e εq ze)) e εq ze)) e z j ze) E z j ) + ε E z j ) E z j ) + ε E z j ) z j ze), z j I E ) ε #ZQ) IE 1 + ε + ε ϕ ), which impies 3.3) since ϕ < 1. To prove 3.4), fix E B ϕ and assume E > ze) again, obvious changes shoud be made for E < ze)). As before, et Ê I E be a point for which QÊ) =. Since εϕ > 5ε, ze) < Ê + 4ε < E. Thus and as before QE + iε) QE) QÊ + 4ε), QÊ + 4ε) ε 3 #ZQ) I E) bq,j z j / I E Ê + 4ε z j Ê z j. As in the proof of 3.), assuming that z j / I E satisfy z j Ê > E Ê εϕ and remembering that #ZQ) = q, we get 3.4). Definition 3.4. Let 0 < ε < 1 and 0 < ξ 1. We say that the poynomia Q is ε, ξ)-custered if Q is ε-covered by some U ε Q) = {I j } k j=1, so that for any I j U ε Q), # ZQ) I j ) q ξ. When we want to be expicit about the covering set, we sha say that Q is ε, ξ)-custered by U = {I j } k j=1. Lemma 3.5. Let 0 < α < 1 and /3 < ξ 1. Then there exist δ = δα, ξ) > 0, q 0 = q 0 α, ξ, δ) > 0 and a universa constant C > 0 such that any q 1/α, ξ)-custered poynomia Q with q degq) q 0 satisfies inf QE + E R iq 1/α ) 1 { ) }) 1 min min e z ZQ) Q z), bq e Cqδ 3.5) 1

22 Proof. If ξ = 1 then by assumption there exists a singe interva of size q 1/α containing a the zeros of Q. It then foows from Lemma 3. that QE + iq 1/α ) q min 1/α min z ZQ) Q ) ) z) q 8, e 81 q 1/α bq ) ) q 9. 4 It foows that 3.5) hods with any δ < 1 for sufficienty arge q. Assume now that 1 > ξ > /3 and et ε q 1/α. Pick δ > 0 so that δ < min 3 ξ 1, 1 ξ). It foows that 0 < 3 ξ 1 < ξ δ < 1. Let ϕ = 1 αξ + αδ. Then 0 < ϕ < 1 and, for q sufficienty arge, Since ε ϕ 1 = q 1/α ϕ/α = q ξ δ > q 3ξ 1 > 5. By Lemma 3.3 we have, for E A ϕ, q QE + iε) 1/α min z ZQ) Q ) ) z) q ξ e 4q ϕ)/α ) < we may use og1 + x) x to obtain 4q ϕ)/α 1+x q QE + iε) 1/α min z ZQ) Q ) z) ) e 1/8 q ξ+ϕ/α /α) e q 1/α min z ZQ) Q ) z) = e qδ 8. e In case E B ϕ we have from Lemma 3.3 QE +iε) q 1/α bq ) 9 qξ 1 4q ϕ/α 1/α ) q = q 1/α bq ) 9 qξ 1 4q δ ξ/ ) q.

23 Again, by og1 + x) QE + iε) x 1+x and for q sufficienty arge q 1/α bq q 1/α bq q 1/α bq ) 9 qξ e 8q1+δ ξ/ 1 4q δ ξ/ ) ) e qξ e 16q1+δ ξ/ ) e qξ 1 16q 1+δ 3ξ/ ) q 1/α bq ) e qξ, since 1 + δ 3ξ/ < 0. Thus, since δ < ξ, we see that for q sufficienty arge QE + iq 1/α ) 1 { ) }) 1 min min Q z), bq e qδ e 8 z ZQ) and we are done. Proof of Theorem 1.5. By Lemma.1, Pq, T) 4T V ) inf E R Dq E + i/t). Since T 1 q 1/α, D q E + i/t) D q E + iq 1/α ). By assumption, there exists a coection UD q ) = {I 1,...,I k } of disjoint intervas, each of size not exceeding q 1/α such that each one of these intervas contains at east q ξ zeros of D q. Since q ξ for q arge enough, we see from property 4 of D q quoted at the beginning of Section ) that D q is q 1/α, ξ)-custered. Therefore, by Lemma 3.5, we see that inf E R Dq E+iq 1/α ) 1 e { min min D q ) Ẽq,j), j bd q ) 1 }) e Cqδ, 3.6) where C is universa and δ depends on α and ξ. Now, ceary b D q,j b q,j so obviousy b Q sup 1 j q b q,j. By Lemma., this impies that This is 1.5). Pq, T) 4e T V ) sup b q,j ) e Cqδ. 1 j q 3

24 A serious imitation of Theorem 1.5 is the assumed ower bound on the custering strength, that is, the requirement that ξ > /3. The reason for this requirement is the fact that one needs to overcome an exponentiay decreasing factor in the estimate of QE + iε) for E B ϕ see 3.) and 3.4)). The issue here is the absence of a ower bound on the distance of the zeros of Q that are to the right of E. Such a ower bound, however, coud be obtained by assuming an upper bound on the custering strength and a ower bound on the custer sizes. The foowing emma shows that the existence of such bounds simutaneousy on different scaes aows us to consider any ξ > 0. We first reca Definition 1.7 in the context of genera poynomias. Definition 3.6. We say that a sequence of poynomias, {Q } =1, with degq ) q, is uniformy custered if Q is {q 1/α, ξ }-custered by U = {Ij }k j=1 and the foowing hod: i) If 1 < then q 1/α 1 1 > q 1/α. ii) There exists µ 1 and a constant C 1 > 0 so that inf Ij C 1q µ/α. 1 j k iii) There exists δ > 0 so that δ < ξ < 1 δ), and δ < α for a. iv) If we define ξ by sup 1 j k # ZQ ) Ij) q ξ, then there exists a constant C so that ξ ξ C. og q As above, when we want to be expicit about the cover and the reevant exponents we sha say that {Q } =1 is uniformy custered by U = {Ij }k j=1 with exponents {α, ξ, µ}. Lemma 3.7. Let {Q } =1 be a sequence of poynomias with degq ) q, that is uniformy custered by U = {Ij }k j=1 with exponents {α, ξ, µ}. Suppose, moreover, that {U } =1 scaes nicey with exponents µ and ω for some 0 < ω < 1. Assume aso that, for some ζ > 0 ) µ 1 ω + ζ < ξ α. 3.7) µ ω 4

25 Finay, suppose that im inf infz ZQ ) Q z) ) > 0 and that b Q is bounded from above. Then for any m > 0 ) inf Q E + iq 1/α ) q m/α =. 3.8) E R im Proof. Let ε = q 1/α ξ. Choose δ > 0 so that im inf δ > 0 and ζ α > δ for a, and et ϕ = 1 α ξ + α δ. It foows that 0 < ϕ < 1. As in Lemma 3.3, et and et Q E + iε ) A ϕ = {E R de) ε ϕ } B ϕ = R \ A ϕ. Then for arge enough and for E A ϕ, as in the proof of Lemma 3.3, 1/α q min z ZQ ) Q z) e ) e qδ 8. Thus, since α and min z ZQ ) Q z) are bounded away from zero, we ony need to obtain a ower bound for E B ϕ for sufficienty arge. So et E B ϕ. As before, since ϕ is bounded away from 1, if is arge enough ε ϕ > 5ε. Let {zj} q j=1 denote the zeros of Q and et z E) be the unique zero associated with E according to the discussion at the beginning of this section. Assume z E) < E and et Ê I E be a point with QÊ) =. As before Q E + iε ) QÊ + 4ε ) q 1/α 3 bq #ZQ ) IE) Ê + 4ε z j Ê z zj / I j E q 1/α e bq qξ Ê + 4ε z j Ê z. zj / I j E 3.9) We sha use the fact that {Q } is uniformy custered with nicey scaing covering sets to obtain ower bounds on Ê+4ε z zj j / I E. As in the proof 5 Ê z j

26 of Lemma 3.3, we sha assume that the zj that are not in I E are ocated to the right of E. ϕ Let M be an integer so arge that < ξ α M + 1 α 1 η for some η > 0 for a. Such an M exists since α < 1 and ξ is bounded away from 0. Let ˆε = ε ϕ /Mµ). Then, since {U } =1 scae nicey, there is a coection of intervas Uˆε of ength at most ˆε and at east C 1 ε ϕ /M C 1 some positive constant) covering the eements of U as in Definition 1.8. In particuar, it is possibe to cover J k j=1 I j [E, E + C 1 ε ϕ /M ] by using no more than two eements of Uˆε. We proceed to anayze the possibe distribution of zeros of Q in J. Let 1 r < M. Note that, by the nice scaing property, there exists a constant C 3 > 0 such that any eement of Uˆε r contains no more than C 3 ˆε r /ε ) ω = C 3 ε ωϕ r)/mµ) 1) eements of U ε. By the uniform custering, each set in U ε hods no more than q ξ zeros of Q. Therefore, each interva of Uˆε r hods no ω more than C 3 q ξ εωϕ r)/mµ) 1) ϕ α r)/mµ) 1) = C 3 q ξ zeros of Q. ) ω Now, there are no more than C 3 ˆεˆε = C3 ε ϕ ω)/mµ) eements of Uˆε in each interva of Uˆε. Therefore, it takes no more than C 3 ε ϕ ω)/mµ) eements of Uˆε to cover J. Each of these has ength at east C 1ˆε µ = C 1 ε ϕ /M and hods no more than C 3 q ξ ω ϕ α )/Mµ) 1) zeros of Q. Take the two intervas cosest to E so that at east one of them ies competey to the right of E). Each of these intervas contains no more than C 3 ε ϕ ω)/mµ) eements of Uˆε 3, each of ength at east C 1 ε 3ϕ ω /M 3ϕ α )/Mµ) 1) and containing at most C 3 q ξ zeros of Q. Of these eements of Uˆε 3 contained in the two intervas above), take the two cosest to E again, so that at east one ies competey to the right of E) and decompose them as above with the eements of Uˆε 4. Continue in this manner up to Uˆε M. From the picture we have just described we get the foowing estimate: Ê + 4ε z j Ê zj Θ 1)Θ ), 3.10) z j / I E 6

27 where etting C 1 = min1, C 1 ) ) and Θ 1 ) = C 3 ε ϕ ω)/mµ) +1 j=1 C 3 ε ϕ ω)/mµ) +1 j=1 C 3 ε ϕ ω)/mµ) +1 j=1 C 3 ε ϕ ω)/mµ) +1 j=1 1 4ε j C 1 ε ϕ 4ε 1 j C 1 ε M 1 M 4ε 1 j C 1 ε M M ξ ) α ω C3 q ϕ µ 1) ϕ ϕ 1 4ε j C 1 ε M ϕ ) Θ ) = 1 4ε q. C 1 ε ϕ M ξ ) ω M 1)ϕ α 1 C3 q Mµ ξ ) ω M )ϕ α 1 C3 q Mµ ξ ) ω )ϕ α C3 q Mµ 1, The term Θ ) comes from the zeros of Q outside J. The term Θ 1 ) comes from the contribution of the zeros inside J. The first term in the product defining Θ 1 ), i.e. C 3ε ϕω)/mµ) ξ +1 ) α ω j=1 1 4ε C3 q ϕ µ 1) j C 1 ε ϕ, comes from the contribution of the zeros inside the two intervas of Uˆε M 1) that are cosest to E from the right one of them may contain E and extend to its eft as we). The zeros inside each of these intervas have to be distributed among [ at most C 3 ε ϕ ω)/mµ) ]+1 eements of Uˆε M with at most C 3 q ξ ω α ϕ µ 1) each eement. In the same way, the second term in the product comes from estimating the contribution of the zeros inside the two intervas of Uˆε M ) that are cosest to E from the right. Continuing in this manner we obtain Θ 1 ). The factor in the exponent C 3 q ξ ω α ϕ µ 1) in ) comes from the fact that intervas in the various U ε may intersect each other. Once again, we reca that we assume that a zeros outside IE are to the right of E. 7

28 By 3.9) we ony have to show im inf og Θ 1 ) q ξ 0, og Θ im inf ) 0. A straightforward computation shows that the q ξ choice of M insures that this indeed is the case for Θ ) so we are eft with og Θ estimating im inf 1 ) 0. Now, for arge enough, q ξ og Θ 1 ) q ξ 16C 3 q ξ C M s=0 M s=0 C 3 ε ϕ ω)/mµ) +1 1 α 1 M s q C og q ) q ξ ξ C og q ) q ξ ξ Cq ξ ξ j=1 M ϕ ) ε M s 1 M ϕ ) j C 1 ω M s)ϕ ξ 1 α Mµ q ξ ϕ 1 + ω ωϕ M α q α α µα s=0 ϕ 1 + ω ωϕ α q α α µα 1 α ϕ 1 ω µ)+ω 1) q ogq ) 1 q ω M s)ϕ 1 α Mµ q ξ C 3 q ϕ ω)/α Mµ) +1 j=1 q ϕ Mα + ωϕ α µm s 1 ϕ α M ω 1) µ 1 j 3.11) where C, C, C are some positive constants. By iv) in Definition 3.6 we see that ξ q ξ Ĉ for some positive constant Ĉ. Moreover, a straightforward computation shows that 3.7), together with ζ α > δ, impies that ) ) 1 im sup α ϕ 1 ω + ω 1 < 0. This, together with 3.11), impies µ og Θ that im inf 1 ) 0 and we are done. q ξ Proof of Theorem 1.9. By Lemma.1, ) Pq, T) 4T V ) 1. inf E R D q E + i/t) If T 1 q 1/α, D q D E + i/t) q E 1/α + iq ). Thus, we ony need to check that D q satisfies the assumptions of Lemma 3.7. We know see the discussion after Coroary 1.3) sup 1 j q b q,j is bounded from above, which 8

29 by Lemma., impies that min 1 j q D q ) Ẽ q,j) is bounded away from zero. The rest of the properties are cear from the assumptions of Theorem Proof of Coroaries 1.3, 1.6, 1.10 and Proposition 1.11 The coroaries foow from Proposition 4.1. Let α > 0. Then 1. If there exist a monotone increasing sequence {q } =1, a constant C > 0 and an ε > 0 such that Pq, q 1/α ) Cq ε, then α α. Moreover, if {q } =1 is exponentiay growing and Pq, T) Cq ε for any T q 1/α, then α + α.. If there exists a monotone increasing sequence {q } =1 such that Pq, q 1/α ) = Oq m ) for a m, then αu α. Moreover, if {q } =1 is exponentiay growing and Pq, T) = Oq m ) for a m uniformy in T q 1/α, then α u + α. Proof. The proof of the bounds for α ± part 1 in the Proposition) and α u ± part in the proposition) are amost identica so we give the detais ony for part Let T = q 1/α. Then im inf T og PT α, T) og T im inf og PT α, T ) og T εα < 0 so α α by definition. Now assume {q } is exponentiay growing. We sha show that α > α + for any α > α. Our proof here foows the strategy impemented in [9, Proof of Theorem 3] for the construction of NT). Let α > α. Let q q = sup +1 q q and q = inf +1 q and r = og q. Finay, for any sufficienty og q arge T et T) be the unique index such that q T) T α < q T)+1 9

30 and et qt) = q T)+ T). Note that qt) q T) q r T) q T) and q T) Cq T) CT α for some constant C > 0, so qt) Cq T) T αr/ T) CT α T αr/ T). Since T), it foows that for any δ > 0 there exists a constant C δ such that qt) C δ T α+δ. Pick δ so that α + δ < α. Thus, for sufficienty arge T, PT α, T) PC δ T α+δ, T) PqT), T) Pq T)+1, T) and we get reca q 1/α T) T < q 1/α T)+1 ) im sup T og PT α ) og T im sup εα og q T)+1 og q T) = εα < 0. Therefore α > α + and we are done.. Repeat the proof of part 1 with the changes: α to α u, α+ to α + u, and repace εα by. Proof of Coroary 1.3. Let α > 3/β. Ceary, by Theorem 1., for any T q 1/α, Pq, T) CV )q 6 α β, where CV ) is a constant that depends on the 6 potentia V. By the assumption, β < 0 so we may appy part 1 of α Proposition 4.1 to concude that α α and so that α 3/β. If {q } =1 is exponentiay growing we get α + α which impies that α + 3/β. Proof of Coroary 1.6. The coroary foows immediatey from Theorem 1.5 and Proposition 4.1 above. Proof of Coroary The coroary foows immediatey from Theorem 1.9 and Proposition 4.1 above. We concude this section with the proof of Proposition 1.11 reducing the fu ine case to the haf ine cases treated above. 30

31 Proof of Proposition Let R ± = δ ±1, δ 0 + δ 0, δ ±1 and et H ± = H R ±. Appying the resovent formua to H z) 1 with z C \ R, one gets δ±n, H z) 1 δ 0 = δ±n, H z) 1 δ 0 δ ±n, H ± z) 1 δ 0 δ±1, H z) 1 δ 0 δ ±n, H ± z) 1 δ ±1 δ0, H z) 1 δ 0 = δ ±n, H ± z) 1 δ ±1 δ0, H z) 1 δ 0 = δ n, H ± z ) 1 δ0 δ 1, H z) 1 δ 0 for any integer n 1, since H ± are direct sums. Since δ 0, H z) 1 δ 0 1 Im z) it foows that for n > 1 δ ±n, H E i/t) 1 δ 0 T δ n, H ± E i/t ) 1 δ1 which immediatey impies by.3) that 0 δ ±n, e ith δ 0 e t/t dt T δ n, e ith± δ 1 e t/t dt. Pugging the eft hand side into the definition of P δ0 q, T) and appying the above inequaity to the positive and negative n separatey we get 1.9). 0 5 An Anaysis of the Fibonacci Hamitonian In this fina section we appy our method to get an upper bound for the dynamics of the Fibonacci Hamitonian H F. This is the whoe-ine operator with potentia given by 1.10). We sha describe its reevant properties beow. For a more comprehensive review see [6]. We sha concentrate on the appication of Theorem 1.9, but wi aso remark on the possibiity of using Theorems 1. and 1.5 to get weaker resuts with significanty ess effort. The unique spectra properties of H F make it an idea candidate for studying the reationship between spectra properties and dynamics. In particuar, for a λ the spectrum of H F is a Cantor set and the spectra measure is aways purey singuar continuous. Anomaous transport has been indicated 31

32 by various numerica works since the ate 1980 s see e.g. [1, 14, 1]). In particuar, work by Abe and Hiramoto [1, 1] suggested that α ± and α u ± behave 1 ike as λ reca λ is the couping constant in 1.10)). og λ In [] Kiip, Kiseev and Last have proven both a ower bound and an upper bound on the sow moving part of the wave packet whose asymptotic behavior agrees with this prediction. As mentioned in the Introduction, an upper bound for the fast moving part of the wave packet was proven recenty by Damanik and Tcheremchantsev in [9] and in [10] where they proved the same upper bound without time averaging. The Damanik-Tcheremchantsev bound reads: for λ 8, α u + og η, where η = 5+1 and ζλ) = λ 4+ λ 4) 1. og ζλ) We sha aso need rλ) = λ +. Using Theorem 1.9, we sha show Theorem 5.1. Let λ > 8 and et αλ) = 3og rλ) ogζλ)η) ogrλ)η) og η ogζλ)). Then α u + λ) αλ). Remark. We note that for λ 17, αλ) < 1 so this is a meaningfu upper 3og rλ) ogζλ)η) 3og rλ) ogζλ)η) bound. Aso, for λ 8, 3 and as ogrλ)η) ogrλ)η) λ. Fix λ > 8. By Proposition 1.11 and the symmetry of H F VFib; n λ = VFib;n 1 λ for n ), it is enough to consider the one-sided operator H+ F which is the restriction of H F to N. In order to appy Theorem 1.9 we need to choose a sequence {q } =1. As in most works deaing with H F, we sha focus on the Fibonacci sequence: F = F 1 + F and F 0 = F 1 = 1, and et q F. We reca that there exists a constant C η > 0 such that Cη 1 η q C η η so that, in particuar, q is exponentiay growing. We need to show that E q is uniformy custered by a sequence of interva famiies, {U } =1, that scaes nicey. The reevant exponents wi determine αλ). Let H per be the whoe-ine operator with potentia Vq per given by V per q ;nq +j = V λ Fib;j 1 j q, namey, Vq per is the q -periodic potentia whose first q entries coincide with those of VFib λ. As mentioned in the Introduction, the spectrum of Hper, σ, is a set of intervas bands). We wi presenty show that a natura cover for E q is provided by the bands in σ m) and σ m)+1 for some m) to be determined ater. 3

33 Thus, we begin by considering σ. Foowing [], we define a type A band as a band I σ such that I σ 1 so that I σ +1 σ ) = ). We define a type B band as a band I σ such that I σ and so I σ 1 = ). Letting σ 1 = R and σ 0 = [, ], and noting σ 1 = [λ, λ + ], we get that, for λ > 4, σ 0 consists of one type A band and σ 1 consists of one type B band. The structure of the spectrum of the Fibonacci Hamitonian H F can be deduced by using the foowing emma Lemma 5.3 in []): Lemma 5.. Assume λ > 4. Then for any > 0 : 1. Every type A band I σ contains exacty one type B band I + σ + and no other bands from σ +1 or σ +.. Every type B band I σ contains exacty one type A band I +1 σ +1 and two type B bands I +,1 σ + and I +, σ + ocated one on each side of I. Let Ik B be a type B band in σ k. Using Lemma 5. one can construct, for m > k a cass Sk,m B of bands, beonging to σ m, which are contained in Ik B, i.e., if I m σ m and I m Sk,m B then I m Ik B. The same can be done for a type A band Ik A σ k, i.e., one can construct, by a repeated use of Lemma 5., a cass Sk,m A of bands in σ m such that if I m Sk,m A then I m Ik A note that by Lemma 5. for m = k + 1 we have Sk,k+1 A = ). Our anaysis proceeds through the foowing Lemma 5.3. Let I B k σ k be a type B band. Then for m k 1 we have #S B k,m = F m k. Let I A k σ k be a type A band. Then for k 0 and m k+ we have #S A k,m = #SB k+,m = F m k. Proof. We notice that the procedure for the construction of the casses of intervas Sk,m B and SA k,m is such that for fixed Z with k + 1 we have #Sk+,m+ B = #Sk,m B and for k + 0 we have #SA k+,m+ = #Sk,m A. By Lemma 5. we aso have #Sk,m A = #SB k+,m for m k +. Therefore, #Sk,m B = #SB 1,m k+1 and #SA k,m = #SA 0,m k = #SB,m k = #SB 1,m k 1. The proof of Lemma 5.3 proceeds by induction. Note first that #Sm,m+1 B = 1 = F 1 and #Sm,m+ B = = F. Assume that we know that #Sm,m+ B = F for = 0, 1,..., k and consider #Sm,m+k+1 B. Let IB m σ m be a type B band in σ m. By Lemma 5. there is one type A band Im+1 A σ m+1 with Im+1 A IB m 33

34 and two type B bands I B m+, j σ m+, j = 1, with I B m+, j IB m. Therefore, for k 1 we have #S B m,m+k+1 = #SA m+1,m+k+1 + #SB m+,m+k+1 = #SA m,m+k + #SB m,m+k 1 = = #S B m+,m+k + #S B m,m+k 1 = #S B m,m+k + #S B m,m+k 1 = = F k + F k 1 = F k+1. We have obtained the foowing picture. The set σ m is made up of F m disjoint bands Im 1, I m,...,ifm m. These bands are a disjoint to the type B bands of σ m+1 Im+1, B,1 Im+1, B,...,I B,m) m+ whie the type A bands of σ m+1 are a contained in σ m. Thus, by Lemma 5.3 the famiy Ũ m = {Im 1, I m,..., IFm m, IB,1 m+1, Im+1, B,...,I B,m) m+ } is a cover for σ k for a k m. Since E q σ more precisey, each eement of E q is contained in a unique band of σ ), we can take U = Ũm) for a function m) to be defined ater. We need two additiona preiminary resuts. Lemma 5.4 Proposition 5. in []). Assume that λ > 8 and k 3. Then, for every E σ k we have where ζλ) = λ 4+ λ 4) 1. D q k ) E) ζλ) k, 5.1) Lemma 5.5 Equation 57 in [7]). If λ > 4 and k 1. Then for every E σ k we have D q k ) E) Cλ + ) k 5.) where C is some positive constant. Remark. It is a straightforward computation to see that Lemma 5.4 and Coroary 1.3 impy that α + 6og η for og ζλ) H+ F. The bounds we obtain for α+ u are better so we do not eaborate on this point here. We can now prove 34