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LOG-DIMENSIONAL SPECTRAL PROPERTIES OF ONE-DIMENSIONAL QUASICRYSTALS DAVID DAMANIK1 AND MICHAEL LANDRIGAN2 2 0 0 2 1DepartmentofMathematics253–37,CaliforniaInstituteofTechnology,Pasadena, CA 91125,USA n a 2 Department of Mathematics, Idaho State University, Pocatello, ID 83209 J 3 E-mail: [email protected], [email protected] 1 2000 AMS Subject Classification: 81Q10, 47B80 v Keywords: Schrödingeroperators,Hausdorffdimensionalspectralproperties,Stur- 9 0 mian potentials 0 1 Abstract. We consider discrete one-dimensional Schro¨dinger operators on 0 the whole line and establish a criterion for continuity of spectral measures 2 withrespecttolog-Hausdorffmeasures. Weapplythisresulttooperatorswith 0 Sturmianpotentials andthereby provelogarithmicquantum dynamical lower / h bounds forallcoupling constants andalmostallrotationnumbers,uniformly p inthephase. - h t a 1. Introduction m We are interested in discrete one-dimensional Schrödinger operators in ℓ2(Z) : v given by i X r a (1) (Hφ)(n)=φ(n+1)+φ(n 1)+V(n)φ(n) − with potential V:Z R. To each such whole-line operator we associate two half- → line operators, H =P∗HP and H =P∗HP , where P denote the inclusions + + + − − − ± P :ℓ2( 1,2,... )֒ ℓ2(Z) and P :ℓ2( 0, 1, 2,... )֒ ℓ2(Z). + − Fore{achz }C→Rwedefineψ±(n;z{)to−be−theu}niq→uesolutionstothedifference ∈ \ equation (2) ψ(n+1)+ψ(n 1)+V(n)ψ(n)=Eψ(n) − with ∞ ψ±(0;z)=1 and ψ±( n;z)2 < . | ± | ∞ nX=0 D.D.wassupportedinpartbytheNationalScienceFoundationthroughGrantDMS–0010101. 1 2 D. DAMANIK, M. LANDRIGAN With this notation we can define the Weyl functions by m+(z)= δ (H z)−1δ = ψ+(1;z)/ψ+(0;z) 1 + 1 h | − i − m−(z)= δ (H z)−1δ = ψ−(0;z)/ψ−(1;z) 0 − 0 h | − i − for eachz C R. Here and elsewhere, δ denotes the vector in ℓ2 supported at n n ∈ \ with δ (n) = 1. For the whole-line problem, the m-function role is played by the n 2 2 matrix M(z): × [a]†M(z)[a]= (aδ +bδ ) (H z)−1(aδ +bδ ) . b b 0 1 − 0 1 (cid:10) (cid:12) (cid:11) Or, more explicitly, (cid:12) 1 ψ+(0)ψ−(0) ψ+(1)ψ−(0) M = ψ+(1)ψ−(0) ψ+(0)ψ−(1)(cid:20)ψ+(1)ψ−(0) ψ+(1)ψ−(1)(cid:21) − 1 m− m+m− = − 1 m+m− (cid:20) m+m− m+ (cid:21) − − with z dependence suppressed. We define m(z) = tr M(z) , that is, the trace of M. These definitions relate the m-functions to reso(cid:0)lvents(cid:1)and hence to spectral measures. By pursuing these relations, one finds that: 1 m±(z)= dρ±(t), Z t z − 1 (3) m(z)= dΛ(t), Z t z − whereρ+,ρ− arethespectralmeasuresforthepairs(H ,δ ),(H ,δ ),respectively, + 1 − 0 and Λ is the sum of the spectral measures for the pairs (H,δ ) and (H,δ ). It is 0 1 known that the pair of vectors δ ,δ is cyclic for H. 0 1 { } Our goal is to find a criterion for Λ to be absolutely continuous with respect to logarithmicHausdorffmeasures. LetusfirstrecallthenotionofHausdorffmeasure, and logarithmic Hausdorff measure in particular. Given a function h : [0, ) ∞ → [0, )whichis continuouswith h(0)=0,a so-calleddimensionfunction, define for S a∞subset of R, ∞ µ (S)= lim inf h(b a ), h i i δ→0δ−coversXi=1 − whereaδ-coverisacoverofS byintervals(a ,b ),i Noflengthatmostδ. When i i ∈ restricted to Borel sets, this gives rise to a measure µ , called Hausdorff measure h corresponding to the dimension function h. For example, h(x) = xα, 0 < α < 1 or h(x) = logb(x) = (log 1)−b, b > 0. For Hausdorff measures µ , a criterion x xα for absolute continuity of Λ was found in [5]. This criterion is based on power law upper and lower bounds for solutions to (2) and the proof uses the Jitomirskaya- Last [8] extension of Gilbert-Pearson theory [6]. We will prove a similar criterion for absolute continuity of Λ with respect to µ which is based only on power logb law lower bounds for solutions of (2). This is motivated by our application of this criterion to operators with Sturmian potentials where the lower bounds can be shown for almost all rotation numbers, whereas upper bounds are known only for a zero-measure set of rotation numbers. Explicitly, we will show the following: LOG-DIMENSIONAL SPECTRAL PROPERTIES OF ONE-DIMENSIONAL QUASICRYSTALS 3 Theorem 1. SupposeV isboundedandthereisγ >0suchthatforΛ-almostevery energy E, every solution of (2) with (4) ψ(0)2+ ψ(1)2 =1 | | | | obeys the estimate (5) ψ C Lγ L E k k ≥ for L > 0 sufficiently large and some E-dependent constant C . Then Λ is abso- E lutely continuous with respect to µ for b=2γ. logb Remarks. 1. The norm in (5) is defined as in [5, 8], that is, L k·k 1/2 ⌊L⌋ 2 2 ψ L = ψ(n) + (L L ) ψ( L +1)  . k k −⌊ ⌋ ⌊ ⌋ nX=0(cid:12) (cid:12) (cid:12) (cid:12)  (cid:12) (cid:12) (cid:12) (cid:12)  2. Thistheoremanditsproofhavequantumdynamicalconsequenceswhichwillbe discussed at the end of the paper. 3. Our proof shows that one can draw even stronger continuity conclusions if the constant C in (5) can be chosen uniformly in the energy. E 4. We do not need that V is bounded. Exponential upper bounds on solutions of (2) are sufficient. This of course holds for bounded V and since we have an application of Theorem 1 to Sturmian potentials (which are bounded) in mind, we give the theorem in this simplified form. The organization is as follows. We will prove Theorem 1 in Section 2 and then apply it to Sturmian potentials in Section 3. Quantum dynamical applications are discussed in Section 4. 2. Log-continuity of whole-line spectral measures TheproofofTheorem1followsastrategysimilartotheoneusedin[5]. Namely, we will first deduce m-function properties on a half-line from the assumptions on solutions to (2), uniformly in the boundary condition. In a second step we use the maximum modulus principle to show a similar property for the whole-line m- function which then implies the assertion of the theorem. Proposition 2.1. Fix E R. Suppose every solution of (2) with (4) obeys the ∈ estimate (6) C Lγ ψ CL 1 ≤k kL ≤ 2 for constants γ,C ,C and for L > 0 sufficiently large. Then there exists C such 1 2 3 that for ǫ>0 small enough, sin(ϕ)+cos(ϕ)m+(E+iǫ) logb(ε) (7) sup C , (cid:12)cos(ϕ) sin(ϕ)m+(E+iǫ)(cid:12)≤ 3 ǫ ϕ (cid:12) − (cid:12) (cid:12) (cid:12) where b=2γ. (cid:12) (cid:12) 4 D. DAMANIK, M. LANDRIGAN Proof. Denote sin(ϕ)+cos(ϕ)m+(E+iǫ) m+(E+iǫ)= ϕ cos(ϕ) sin(ϕ)m+(E+iǫ) − and let ψ1/2 be the solutions to (2) with ϕ ψ1(0)=sin(ϕ), ψ1(1)=cos(ϕ) ϕ ϕ and ψ2(0)= cos(ϕ), ψ2(1)=sin(ϕ). ϕ − ϕ Given ǫ>0, let L (ǫ)>0 be defined by ϕ 1 ψ1 ψ2 = . k ϕkLϕ(ǫ)k ϕkLϕ(ǫ) 2ǫ Then the Jitomirskaya-Lastinequality [8] reads 5 √24 ψ1 5+√24 (8) − < k ϕkLϕ(ǫ) < . m+(E+iǫ) ψ2 m+(E+iǫ) | ϕ | k ϕkLϕ(ǫ) | ϕ | From (6) and (8) we get for ǫ>0 small enough, ǫ ǫ ψ2 sup m+(E+iǫ) sup(5+√24)k ϕkLϕ(ǫ) logb(ǫ) | ϕ | ≤ logb(ǫ) ψ1 ϕ ϕ k ϕkLϕ(ǫ) ψ2 = sup(5+√24)k ϕkLϕ(ǫ) ψ1 × ϕ k ϕkLϕ(ǫ) 1 1 ×2 ψ1 ψ2 logb( 1 ) k ϕkLϕ(ǫ)k ϕkLϕ(ǫ) 2kψϕ1kLϕ(ǫ)kψϕ2kLϕ(ǫ) (5+√24) log(2 ψ1 ψ2 )b = sup k ϕkLϕ(ǫ)k ϕkLϕ(ǫ) 2 ψ1 2 ϕ k ϕkLϕ(ǫ) (5+√24) log(2C2Lϕ(ǫ))b sup 2 ≤ 2 C2L (ǫ)2γ ϕ 1 ϕ C 3 ≤ if b=2γ. 2 Proposition 2.2. Given a Borel set Σ, suppose that the estimate (6) holds for every E σ(H) with C ,C independent of E. Then, given any function m− : 1 2 C+ C+∈, and any E Σ, → ∈ m+(E+iǫ)+m−(E+iǫ) logb(ε) (9) m(E+iǫ) = C | | (cid:12)1 m+(E+iǫ)m−(E+iǫ)(cid:12)≤ 3 ǫ (cid:12) − (cid:12) (cid:12) (cid:12) for all ǫ > 0, where b = 2γ(cid:12). Consequently, Λ(E) is uni(cid:12)formly logb-Lipschitz continuous at all points E Σ. In particular, Λ is absolutely continuous with respect ∈ to µ on Σ. logb If (6) holds only with E-dependent constants C ,C , but with a uniform γ, we 1 2 can still deduce absolute continuity of Λ with respect to µ . logb LOG-DIMENSIONAL SPECTRAL PROPERTIES OF ONE-DIMENSIONAL QUASICRYSTALS 5 Proof. Fix E Σ and ǫ > 0. Let z = e2iϕ and µ = (m+ i)/(m++i). We may ∈ − then rewrite (7) as 1+µz logb(ε) sup C . (cid:12)1 µz(cid:12)≤ 3 ǫ |z|=1(cid:12) − (cid:12) (cid:12) (cid:12) By Im(m+) > 0 we have µ <(cid:12) 1 and(cid:12)so (1 +µz)/(1 µz) defines an analytic | | − functionon z : z 1 . The pointz =(i m−)/(i+m−)lies inside the unit disk { | |≤ } − since Im(m−)>0. We have 1+µ i−m− i+m− m=i (cid:16) (cid:17) · 1 µ i−m− − (cid:16)i+m−(cid:17) as can be checked by direct calculation. The estimate (9) thus follows from the maximum modulus principle. This estimate and the representation (3) provide Λ [E ǫ,E+ǫ] 2ǫIm m(E+iǫ) 2C logb(ε) for all E Σ, ǫ>0, 3 − ≤ ≤ ∈ (cid:0) (cid:1) (cid:0) (cid:1) from which Λ(E) is uniformly logb-Lipschitz continuous on Σ. IfwepermitC ,C todependonE,theonlyconsequenceisthatnowC depends 1 2 3 on E and so Λ need not be uniformly Lipschitz continuous. However, absolute continuity is still guaranteed. 2 Proof of Theorem 1. The assertion follows from Propositions 2.1 and 2.2. 2 3. Application to Sturmian Potentials In this section we discuss the case where V is given by (10) V(n)=λχ (nθ+β mod 1). [1−θ,1) The non-trivial situation (i.e., non-periodic) is when we assume that the coupling constant λ is nonzero and the rotation number θ (0,1) is irrational. Operators ∈ H with such potentials are standardmodels for one-dimensionalquasicrystals;see, for example, [1, 4]. It is quite easy to see that the spectrum of H does not depend onβ andcanhence be denotedby Σ . Itis knownthat forallparameterchoices, λ,θ subject to the above conditions, the operator H has purely singular continuous spectrum [1, 5]. More detailed studies of the singular continuous spectral type can be found in [3, 5, 9] where absolute continuity with respect to xα Hausdorff measures is established for certain parameter values. Explicitly, it is known that foreveryλandeveryboundeddensitynumberθ,thereis α>0suchthatforevery θ,thespectralmeasuresofH areabsolutelycontinuouswithrespecttoµ . Recall xα that θ is called a bounded density number if the coefficients a in the continued n fraction expansion of θ, 1 θ = 1 a + 1 1 a + 2 a + 3 ··· satisfy limsup 1 n a < . The set of bounded density numbers is small n→∞ n i=1 i ∞ in Lebesgue sense: itPhas measure zero. Let us define the associated rational ap- proximants pn of θ by qn 6 D. DAMANIK, M. LANDRIGAN p =0, p =1, p =a p +p , 0 1 n n n−1 n−2 q =1, q =a , q =a q +q . 0 1 1 n n n−1 n−2 Our goal here is to establish absolute continuity of spectral measures with respect to logarithmic Hausdorff measures for almost every θ. Theorem 2. For every λ and almost every θ, there is b>0 such that for every β, Λ is absolutely continuous with respect to µ . logb This theorem follows from Theorem 1 and the following proposition from [5]: Proposition 3.1. Let θ be such that for some B < , q Bn for every n N. n ∞ ≤ ∈ Then for every λ, there exist 0<γ,C < such that for every E Σ and every λ,θ ∞ ∈ β, every normalized solution u of (2) obeys (11) u CLγ L k k ≥ for L sufficiently large. Note that the assumption of this proposition is obeyed by almost every θ [10]. 4. Dynamical Implications Continuitypropertiesofspectralmeasuresimplyquantumdynamicalbounds,as demonstrated by works of Guarneri [7], Combes [2], and Last [12]; among others. In particular, Last derives dynamical bounds from the non-singularity of spectral measures with respect to xα–Hausdorff measures. More recently, Landrigan [11] has observed that these results hold for general Hausdorff measures, including the logarithmicHausdorffmeasureswhichareofprimaryinterestinthepresentarticle. Let us briefly recall the results of [11] and discuss their consequences for Sturmian models. Let H be as in (1) and let φ ℓ2(Z). The Schrödinger time evolution is given ∈ by φ(t) = e−itHφ and the “spreading” of φ(t) is usually measured by considering two quantities, the survival probability 2 φ,φ(t) 2 = e−itxdµ (x) = µˆ (t)2, |h i| (cid:12)Z φ (cid:12) | φ | (cid:12) (cid:12) (cid:12) (cid:12) whereµφ isthe spectralmeasurec(cid:12)orrespondingto(cid:12) the pair(H,φ), andexpectation values X m (t)= e−itHφ, X me−itHφ h| | i h | | i of moments of the position operator X m = nm δ , δ . n n | | | | h ·i nX∈Z In the case of singular continuous spectral measures it is natural to consider time averagedquantities. Then,intuitively,thefasterthespreading,thefasterthedecay of µˆ 2 and the faster the increaseof X m , where the time average is φ T T T h| | i hh| | ii h·i defined by f = 1 T f(t)dt. These relations are made explicit by the following h iT T 0 pair of propositions wRhich are simplified versions of Lemma 12 and Theorem 6 of [11],respectively. Inparticular,theyshowthatthedynamicalboundonecanprove isnaturallyrelatedtothemaximaldimensionfunctiononecanpicktogetadesired continuity property. LOG-DIMENSIONAL SPECTRAL PROPERTIES OF ONE-DIMENSIONAL QUASICRYSTALS 7 Proposition 4.1. If µ is uniformly h–Lipschitz continuous, then there is C >0 φ φ such that 1 µˆ 2 <C h . φ T φ h| | i · (cid:18)T(cid:19) Proposition 4.2. Ifµ isabsolutelycontinuouswithrespecttotheHausdorffmea- φ sure µ , then for each m>0, there is D >0 such that h φ,m −m 1 X m >D h . T φ,m hh| | ii · (cid:18)T(cid:19) Remark. The assumption can be relaxed. It suffices that µ is not singular with φ respect to µ . h Letus nowstatethe dynamicalbounds weobtainforSturmianpotentials. Note that since δ ,δ is cyclic for H, absolute continuity of Λ with respect to µ is 0 1 h inheritedby{µ fo}rall φ ℓ2(Z)anduniformh-Lipschitzcontinuityisinheritedby φ µ for all φ ℓ2(Z) of co∈mpact support. φ ∈ Corollary 4.3. For every λ and almost every θ, there is b>0 such that for every β, the Sturmian operator corresponding to the triple (λ,θ,β) satisfies the following: (a) For every φ ℓ2(Z), there is C >0 such that φ ∈ µˆ 2 <C (logT)−b. φ T φ h| | i · (b) For every φ ℓ2(Z) of compact support and every m > 0, there is D such φ,m ∈ that X m >D (logT)bm. T φ,m hh| | ii · Proof. This follows from Theorem 2 along with Propositions 4.1 and 4.2. 2 References [1] J.Bellissard,B.Iochum,E.Scoppola,andD.Testard,Spectralpropertiesofone-dimensional quasi-crystals,Commun. Math. Phys.125(1989), 527–543 [2] J. M. 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