Hölder Continuity of the Spectra for Aperiodic Hamiltonians
aa r X i v : . [ m a t h . SP ] J a n H ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODICHAMILTONIANS
SIEGFRIED BECKUS, JEAN BELLISSARD, HORIA CORNEAN
Abstract.
We study the spectral location of a strongly pattern equivariant Hamiltonians aris-ing through configurations on a colored lattice. Roughly speaking, two configurations are ”closeto each other” if, up to a translation, they ”almost coincide” on a large fixed ball. The larger thisball is, the more similar they are, and this induces a metric on the space of the correspondingdynamical systems. Our main result states that the map which sends a given configuration intothe spectrum of its associated Hamiltonian, is H¨older (even Lipschitz) continuous in the usualHausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by thedistance of the corresponding dynamical systems. Introduction
This work is a follow-up on a series of papers concerning periodic approximations for Hamiltoniansmodelling aperiodic media [2, 3, 4, 5, 6]. Such Hamiltonians are bounded self-adjoint operatorsdefined as effective models describing the behavior of conduction electrons in a solid. In themost common case the electrons are considered as independent. The first task in considering amaterial of this type is to compute the spectrum as a set and then the density of states (electronicproperties). From a mathematical point of view, the next important task would be to determinethe nature of the spectral measures, and then to investigate the transport properties.In the previous series of papers, systematic methods have been developed to compute the spectrumas a set through a sequence of periodic approximations [2, 3, 4, 5], as well as the density of states[6]. In the present work, a special class of models is investigated for which the speed of convergencecan be evaluated more accurately in terms of the distance of the associated dynamical systems.In order to do so, a metric is introduced on the space of possible atomic configurations and it willbe proved that, at least for the class considered here, the spectrum, as a compact subset of thereal line, is a H¨older (even Lipschitz) continuous function when the set of compact subsets of R isendowed with the Hausdorff metric.1.1. The class of models.
The Euclidean distance on R d is denoted by | x | := qP dj =1 x j andthe max norm on R d is given by | x | max := max ≤ j ≤ d | x j | .The class of systems considered here models a solid with atoms on a lattice L ⊆ R d . Here alattice is a subgroup that is isomorphic to Z d , namely L = M Z d where M is a d × d invertiblematrix with real coefficients. The properties of each atom are encoded by a letter in an alphabet A , where ( A , d A ) is a compact metric space. The alphabet A may, for instance, encode thechemical species of atomic nuclei if A is finite. In addition, recent developments [30] show thatalphabets that are compact metric spaces (such as compact intervals) can be used in practice asa tool. The family of all atomic configurations is represented by the infinite Cartesian product A L := Q x ∈L A = { ξ : L → A } . Since A is compact, this space is compact.Define the deformed cube Q r := { M x ∈ R d : | x | max ≤ r } of side length r >
0. The configurationspace A L becomes a compact metric space if equipped with the metric d A L ( ξ, η ) := min (cid:26) inf (cid:26) r : r ∈ (0 , ∞ ) , d A (cid:0) ξ ( x ) , η ( x ) (cid:1) ≤ r for all x ∈ Q r ∩ L (cid:27) , (cid:27) , (1.1)c.f. Lemma 2.1. The infimum over the empty set equals + ∞ by convention. If A is finite we choose d A to be the discrete metric. In this case, if d A L ( ξ, η ) ≤ r for some r >
1, then ξ | Q r = η | Q r . The translation group L acts naturally on this space, as a full shift denoted by t , namely ∀ ξ ∈ A L , ξ = (cid:0) ξ ( x ) (cid:1) x ∈L , (cid:0) t h ξ (cid:1) ( x ) := ξ ( x − h ) , h ∈ L . A quantum particle moving on the lattice L , such as a valence electron, is modeled by:(i) a Hilbert space of states taken to be H := ℓ ( L ) ⊗ C N , where N represents the number ofinternal degrees of freedom of the particle, and(ii) a bounded self-adjoint Hamiltonian H ξ , acting on H , where ξ ∈ A L represents the atomicconfiguration with which the particle interacts.For the sake of notation, M N ( C ) denotes the set of N × N matrices with complex coefficientsequipped with the operator norm k M k op := sup | x |≤ | M x | defined by the Euclidean norm. Inaddition, we use k M k max := sup | x | max ≤ | M x | max induced by the max-norm.Let z ∈ L be a translation vector. Denote by U z the unitary operator acting on H , which isinduced by the translation with z ∈ L . Then translating the origin of coordinates in the latticeis equivalent to shifting the atomic configuration in the background, leading to the following covariance condition U z H ξ U − z = H t z ξ . In addition, it is natural to consider the situation in which the map ξ ∈ A L H ξ ∈ B ( H )is strongly continuous [9]. It is worth remarking that thanks to the covariance condition, thespectra of H ξ and H t z ξ are identical [9, 4]. Hence the spectrum does not change along the orbit O rb ( ξ ) := { t z ξ : z ∈ L} of ξ . In addition, thanks to the strong continuity condition, if η ∈ O rb ( ξ ), then H η has its spectrum contained in the spectrum of H ξ , i.e., σ ( H η ) ⊆ σ ( H ξ ). Thesebasic conditions are satisfied in particular if the Hamiltonian obeys two additional properties: • There exists a finite subset
R ⊆ L , which is called range , and some N × N -matrix valuedcomplex coefficients t h,x ( ξ ) such that H ξ ψ ( x ) = X h ∈R t h,x ( ξ ) ψ ( x − h ) . In this case we say that the Hamiltonian has finite range R . The covariance conditionimposes that t h,x ( ξ ) = t h, ( t − x ξ ), hence from now on we will simply write t h instead of t h, . The strong continuity imposes that t h is continuous w.r.t. ξ ∈ Ω. The self-adjointnessof H ξ imposes that h ∈ R if and only if − h ∈ R and that t − h ( ξ ) = t h ( t h ξ ) ∗ . Here A ∗ denotes the adjoint of the matrix A ∈ M N ( C ). This gives H ξ ψ ( x ) = X h ∈R t h ( t − x ξ ) ψ ( x − h ) , (1.2)where t h : A L → M N ( C ) is continuous, R = −R is finite and t − h ( ξ ) = t h ( t h ξ ) ∗ . • If the alphabet A is finite, a function t : A L → M N ( C ) is called strongly pattern equivariant [32] (and correspondingly the family H = ( H ξ ) ξ ∈ A L is called strongly pattern equivariantHamiltonian ) if there exists a radius R t ≥ ξ and η coincideon Q R t , then t ( ξ ) = t ( η ). The radius 1 ≤ R t < ∞ is called the radius of influence of t .In the present framework ( A , d A ) is allowed to be any compact metric space, thus requiring t ( ξ ) = t ( η ) whenever ξ and η coincide on Q R t is too restrictive. Instead, we say that t : A L → M N ( C ) is strongly pattern equivariant with β -H¨older continuous coefficients ifthere exists a radius R t ≥ C t ≥ k t ( ξ ) − t ( η ) k op ≤ C t sup y ∈ Q Rt ∩L d A (cid:0) ξ ( y ) , η ( y ) (cid:1) β . (1.3)Therefore, the family H = ( H ξ ) ξ ∈ A L will be called strongly pattern equivariant with β -H¨older continuous coefficients whenever each of its coefficients t h requires the previousregularity constraints. In this case, the radius of influence R H of H is defined by R H := sup { R t h : h ∈ R} , ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 3 and we also introduce (see (1.3) for the definition of C t ) C hop := sup h ∈R C t h < ∞ . It is worth noticing that for A ⊆ R finite, the usual discrete Schr¨odinger operator H ξ ψ ( x ) = X h ∈ Z d , | h | =1 ψ ( x − h ) + ξ ( x ) ψ ( x ) , x ∈ Z d , (1.4)acting on ℓ ( Z d ) is strongly pattern equivariant [9] since the map v : A Z d → R where v ( ξ ) = ξ (0)induces a multiplicative real potential v ( t − x ξ ) = ξ ( x ). The first term is the discrete Laplaceoperator ∆, which models the kinetic energy. The range of ∆ is R ∆ = {± e , . . . , ± e d } where { e , . . . , e d } is the standard basis of Z d , while its radius of influence equals to 1. Also, v is ”local”and only sees one point at the time, i.e. the origin. Thus R v = { } and its radius of influence is R v = 1. It follows that the range of H ξ is R = R ∆ ∪ R v while its radius of influence R H = 1.1.2. A concise description of our main results.
We are mainly interested in the spectralproperties of an operator family H Ξ := ( H ξ ) ξ ∈ Ξ . A subset Ξ ⊆ A L is called invariant if given any ξ ∈ Ξ and h ∈ L , then t h ξ ∈ Ξ. An invariant, closed set Ξ is called a subshift and the set of allsubshifts is denoted by J . Then J may be equipped with the Hausdorff metric d H A L induced by d A L , see (2.1). Thus, ( J , d H A L ) becomes a compact metric space, c.f. [4, Proposition 4].For Ξ ∈ J , the spectrum σ ( H Ξ ) of the operator family H Ξ := ( H ξ ) ξ ∈ Ξ is defined by S ξ ∈ Ξ σ ( H ξ ).Since k H ξ k is uniformly bounded in ξ ∈ Ξ and H ξ is self-adjoint, σ ( H Ξ ) is a compact subset of R .Then the distance between two spectra σ ( H Ξ ) and σ ( H Θ ) for Ξ , Θ ∈ J is defined by the Hausdorffmetric d H on the set K ( R ) of compact subsets of R [24], see Section 2.2 for details.In recent works [3, 4] it has been shown that the mapΣ H : J → K ( R ) , Ξ σ ( H Ξ ) , is continuous in the corresponding Hausdorff topologies where H = ( H ξ ) ξ ∈ Ξ is a Hamiltonian asdefined in (1.2). We note that the continuity result holds under much more general conditionsthan we consider here and its proof relies on constructing a continuous field of C ∗ -algebras usingthe uniform continuity of the operator coefficients.Our current paper deals with the question regarding the connection between the regularity of thecoefficients (see (1.3)) and the regularity of the map Σ H .The decay of the off-diagonal influences additionally the regularity of the spectra. This is measuredby the Schur- β norm k · k β for 0 < β ≤
1. Specifically, this norm is defined for a finite rangeHamiltonian H = ( H ξ ) ξ ∈ A L given in (1.2) by k H k β := X h ∈R k t h k ∞ (1 + | h | ) β , where k t h k ∞ := sup ξ ∈ A L k t h ( ξ ) k op . (1.5)For simplification, let us first present our main result in the special case where A is finite. It assertsthat Σ : J → K ( R ) , Ξ σ ( H Ξ ) is Lipschitz continuous for every strongly pattern equivariantHamiltonian H . Proposition 1.1.
Let A be finite and consider a strongly pattern equivariant Hamiltonian H =( H ξ ) ξ ∈ A L with finite range R and strongly pattern equivariant coefficients t h , h ∈ R as defined in (1.2) . Then there exists a constant C d, L such that Σ H is Lipschitz continuous: d H (cid:0) σ ( H Ξ ) , σ ( H Θ ) (cid:1) ≤ C d, L C hop R H k H k d H A L (cid:0) Ξ , Θ (cid:1) , Ξ , Θ ∈ J . The latter result is an immediate consequence of the following main theorem where ( A , d A ) is acompact metric space only. Theorem 1.2.
Let H = ( H ξ ) ξ ∈ A L be a finite range strongly pattern equivariant Hamiltonian with β -H¨older continuous coefficients for some < β ≤ . Then there exists a constant C d, L onlydepending on the dimension d and the lattice L such that Σ H is β -H¨older continuous: d H (cid:0) σ ( H Ξ ) , σ ( H Θ ) (cid:1) ≤ C d, L C hop k H k β R βH d H A L (cid:0) Ξ , Θ (cid:1) β , Ξ , Θ ∈ J . SIEGFRIED BECKUS, JEAN BELLISSARD, HORIA CORNEAN
Proof of Proposition 1.1: If A is finite, every strongly pattern equivariant function t h : A L → C is Lipschitz continuous ( β = 1) with Lipschitz constant C t h and some R t h ≥
1. Thus, C hop =max { C t h : h ∈ R} and R H = max { R t h : h ∈ R} are finite. Hence, the statement follows fromTheorem 1.2. ✷ Example 1.3.
Consider the Schr¨odinger operator defined in (1.4) with nearest neighbor inter-action on L = Z d . Clearly, Proposition 1.1 implies the Lipschitz continuity of the spectra with R H = 1, see discussion right after (1.4). ✷ The previous spectral estimates can be extended to Hamiltonians with infinite range. Specifically,let H = ( H ξ ) ξ ∈ A L be given by (1.2) while the range R is infinite. For β >
0, we call H a stronglypattern equivariant Hamiltonian with β -H¨older continuous coefficients and infinite range if k H k β and C hop are both finite. Let H | r be the restriction of H to the range R ∩ Q r , see (2.2) for details.If the radius of influence R H | r satisfies R H | r ≤ C H r , r ≥ ≤ C H < ∞ independent of r , we say that H admits a radius of influence withlinear growth in r . Then we have the following result: Theorem 1.4.
Let H = ( H ξ ) ξ ∈ A L be a strongly pattern equivariant Hamiltonian (possible infiniterange) with β -H¨older continuous coefficients for < β ≤ . If the radius of influence of H has alinear growth with constant ≤ C H < ∞ , then d H (cid:0) σ ( H Ξ ) , σ ( H Θ ) (cid:1) ≤ C d, L k H k β ( C βH + C hop ) d H A L (cid:0) Ξ , Θ (cid:1) β , Ξ , Θ ∈ J , where C d, L > is the same constant as in Theorem 1.2. For ξ ∈ A L , consider Ξ ξ := O rb ( ξ ) ∈ J . Such subshifts are called topological transitive . Notethat the set of topological transitive subshifts { Ξ ξ : ξ ∈ A L } ⊆ J is not closed in the Hausdorfftopology [5, Example 2]. Corollary 1.5.
Let H = ( H ξ ) ξ ∈ A L be a strongly pattern equivariant Hamiltonian (possibly withinfinite range) having β -H¨older continuous coefficients where < β ≤ . If the radius of influenceof H has linear growth with constant ≤ C H < ∞ , then d H (cid:0) σ ( H ξ ) , σ ( H η ) (cid:1) ≤ C d, L k H k β ( C βH + C hop ) d H A L (cid:0) Ξ ξ , Ξ η (cid:1) β , ξ, η ∈ A L , where C d, L > is the same constant as in Theorem 1.2. Proof:
Since ξ H ξ is continuous in the strong operator topology and the operator familyis equivariant, the inclusion σ ( H η ) ⊆ σ ( H ξ ) follows for all η ∈ Ξ ξ . Hence, σ ( H Ξ ξ ) = σ ( H ξ ) isderived and Theorem 1.4 finishes the proof. ✷ In particular, if Ξ ξ = Ξ η holds for some ξ, η ∈ A L , then their spectra σ ( H ξ ) and σ ( H η ) coincide forany strongly pattern equivariant Hamiltonian. Note that Ξ ξ = Ξ η defines an equivalence relationon A L , which we call orbit closure equivalence .Clearly, Lipschitz/H¨older continuity of Σ H implies the continuity of this map. This provides adifferent proof (in a special case) of the much more general statement [4, Theorem 2], namely Corollary 1.6.
Let H = ( H ξ ) ξ ∈ A L ∈ C be a strongly pattern equivariant Hamiltonian (possibleinfinite range) with β -H¨older continuous coefficients. Then the map Σ : J → K ( R ) , Ξ σ ( H Ξ ) , is continuous in the corresponding Hausdorff topologies. Proof: If H has finite range, the statement follows from Theorem 1.2. If R is infinite, thestatement follows by the fact that H can be approximated in norm by H r with finite range. ✷ ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 5 Comments and Method.
Continuity of spectral gaps has been proven in some cases inthe past. The problem occurred, in particular, with the dependence of the spectrum as a functionof an external, uniform magnetic field [47, 13, 14]. It also occurred for the Schr¨odinger operatoron the line R , with almost periodic potentials in which the frequency module is varying [19].Lipschitz continuity of the spectral gaps was studied more thoroughly in [11] for a broad class ofmodels for 2 D -crystal electrons in a uniform magnetic field. This was extended in [36, 37, 38] aswell. The problem of continuity w.r.t. changing the underlying atomic configurations was onlyrecently systematically investigated in connection with the need to compute the spectrum of aHamiltonian describing the electron motion in an aperiodic environment [2, 3, 4, 5]. In addition,the convergence of the density of states measures is studied in [6]. For it has been remarkedfor a long time by physicists that periodic approximations provided the most efficient numericalmethod to achieve the result [26, 42, 12]. A folklore theorem was also showing that the accuracyof such methods was exponentially fast in the period of the approximation. This was proved insome way in [44, 45]. The existence and construction of periodic approximations in dimensionone was recently characterized in [5]. In several of the studies mentioned above, the concept ofcontinuous field of C ∗ -algebras was explicitly used. One of the advantages of this concept is thatthe spectrum of a self-adjoint continuous section of such a field is always continuous [29, 17, 18, 2].However the question of whether a continuous field of C ∗ -algebras is Lipschitz continuous or evendifferentiable, has not been addressed in a systematic way so far. Therefore, to the best of theirknowledge, the authors believe that the Lipschitz/H¨older continuity dependence of the spectrumin terms of the underlying atomic configuration, described in the present article, is new.A bit of explanation for the method used here is in order, a method which employs and adapts anumbers of ideas from [13, 14] as well as [4]. Consider the simplest case, the Schrdinger operatordefined in (1.4) by the discrete Laplacian plus a potential on Z d . It is well-known, that if twodifferent potentials are close in the uniform norm (sup norm), then their spectra is close as wellin the Hausdorff metric. This follows as then their operator norm difference is small as welland all considered operators are self-adjoint. In the present paper, these potentials are describedvia different configurations in A Z d . Clearly, the assumption that two potentials are close in theuniform norm is too restrictive in general. For instance, if the alphabet A is finite it yieldsthat both potentials are equal if they are close enough in the uniform norm. This is in particularproblematic if one seeks to approximate non-periodic configurations via periodic ones. To overcomethis difficulty, the first powerful concept is coming from using the Hausdorff metric d H A L on theset of subshifts [4]. Two configurations ξ, η ∈ Ω generate ”close” subshifts if there exists a largeradius r such that they share almost the same local pattern of size r [3, 5]. Specifically, locallyon large areas the two configurations are close modulo translation. The second method [13, 14]consists in showing that there is δ > C > z satisfiesdist( z , σ ( H η )) > Cd H A L (Ξ ξ , Ξ η ) δ , then z belongs to the resolvent set of H ξ as well. This is obtainedthrough localizing the resolvent operators with the help of a Lipschitz-partition of unity. Thistrick implements the fact that the potentials are only locally close but not uniformly as discussedbefore. Then for each local region selected this way, in which H ξ sees a local pattern, shared with η , there is a translation at finite distance bringing the same pattern for η in this local region.Hence up to a translation depending on the region of localization, H η and H ξ are close in thisregion, thanks to the strongly pattern equivariance condition. This closeness is translated into aproof of the existence of ( z − H ξ ) − , see Proposition 4.4.This result is actually coming as a surprise. Indeed, in view of [2], the best that could be expectedis, for a Hamiltonian with Lipschitz continuous coefficients, to be p /
2. Asdiscussed in [2], such a loss of regularity is usually due to gap closing, as observed, for instance, inthe Harper model [26]. In the present case, the pattern equivariance condition is actually a strongconstraint on the system. Theorem 1.2 suggest then that, even if there are gaps closing in somelimit, it cannot occur at a slower rate than the one imposed by this Lipschitz continuity. One modelhas been numerically investigated in the literature, which is called the Kohmoto model [42]. Itrepresents a paradigm for one-dimensional quasicrystals. It contains a real parameter α that labels SIEGFRIED BECKUS, JEAN BELLISSARD, HORIA CORNEAN the slope of the line implementing the physical space in a cut-and-project scheme. It will be shownin a forthcoming paper [7], that the present estimate applies to this parameter, in that α defines aspecific subshift Ξ α to which Theorem 1.2 applies. The combinatoric distance implements, on theset of slopes, a topology making the real line completely disconnected, but compatible with theencoding by a continuous fraction expansion. It is actually difficult from looking at the numericalresults in [42] to see this Lipschitz continuity, because the map α ∈ R → Ξ α ∈ J is actuallydiscontinuous if R is endowed with its usual Euclidean topology [10].In light of [4, 6], it is natural to ask for extensions for Hamiltonians on Delone sets. This settingincludes important geometric examples such as the Penrose tilings or the octagonal lattice, whichare not included here. The main technical difficulty is that the corresponding operators for differentDelone sets act on different Hilbert spaces that are not isomorphic, in general. This problem isstudied in a forthcoming work. Furthermore, the continuity result of the map Σ H in [4] requiresamenability. Since some arguments provided here extend to more general groups, it is interestingto ask if there is a connection between the amenability and the existence of a Lipschitz partitionof unity in the group.1.4. Organization of the paper.
More details about the configuration space and the Hausdorfftopology are provided in Section 2.1. We discuss several properties of the considered class ofHamiltonians in Section 2.2. Section 3 is devoted to the concept of Lipschitz-partitions of unityand superoperators that are crucial ingredients in the proof of the main theorem. In Section 4,the proofs of the main results are given.
Acknowledgements.
This research was supported through the program “Research in Pairs” bythe Mathematisches Forschungsinstitut Oberwolfach in 2018. This research has been supportedby grant 8021–00084B
Mathematical Analysis of Effective Models and Critical Phenomena inQuantum Transport from The Danish Council for Independent Research | Natural Sciences.2.
Framework
In this section, an introduction is provided of the used concepts and notations of the configurationspace A L and the Hausdorff metric d H A L on the set J of closed, invariant subsets of A L . In thesecond part, a detailed elaboration of the studied Hamiltonians is presented. In addition, someauxiliary statements are provided that are used in the proof of the main theorem.2.1. The configuration space and local patterns.
Throughout this work ( A , d A ) is a compactmetric space and the configuration space A L = { ξ : L → A } is endowed with the product topology. Lemma 2.1.
Let ( A , d A ) be a compact metric space. Then ( A L , d A L ) is a compact metric spaceinducing the product topology on A L where d A L is defined in (1.1) Proof:
Since A is compact, A L is compact as well. Thus it suffices to show that d A L defines ametric as it is immediate to see that d A L generates the product topology on A L . Since d A is ametric, it is straightforward to show d A L ( ξ, η ) = d A L ( η, ξ ) and that d A L ( ξ, η ) = 0 implies ξ = η .In order to prove the triangle inequality, let ε >
0. Then there is an r ε > e r ε > d A L ( ξ, η ) ≤ r ε ≤ d A L ( ξ, η ) + ε , d A L ( η, ζ ) ≤ e r ε ≤ d A L ( η, ζ ) + ε , and d A (cid:0) ξ ( x ) , η ( x ) (cid:1) ≤ r ε for all x ∈ Q r ε ∩ L , d A (cid:0) η ( y ) , ζ ( y ) (cid:1) ≤ e r ε for all y ∈ Q e r ε ∩ L . Define 1 /r := 1 /r ε + 1 / e r ε . Thus, r < r ε and r < e r ε holds implying Q r ⊆ Q r ε and Q r ⊆ Q e r ε . Thenthe triangle inequality for d A gives d A (cid:0) ξ ( x ) , ζ ( x ) (cid:1) ≤ d A (cid:0) ξ ( x ) , η ( x ) (cid:1) + d A (cid:0) η ( x ) , ζ ( x ) (cid:1) ≤ r ε + 1 e r ε , x ∈ Q r ∩ L . Hence, d A L (cid:0) ξ, ζ (cid:1) ≤ r ε + 1 e r ε ≤ d A L (cid:0) ξ, η (cid:1) + d A L (cid:0) η, ζ (cid:1) + 2 ε ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 7 follows for ε > d A L . ✷ Since A L is compact, J is contained in K (cid:0) A L (cid:1) which denotes the set of compact subsets of A L . Theset K (cid:0) A L (cid:1) gets a compact metrizable space if equipped with the Hausdorff topology [15], also calledChabauty-Fell topology [16, 22]. More precisely, the Hausdorff metric d H A L : K (cid:0) A L (cid:1) × K (cid:0) A L (cid:1) → [0 , ∞ ) induced by d A L is defined as follows d H A L (Ξ , Ξ ) := max ( sup ξ ∈ Ξ inf ξ ∈ Ξ d A L ( ξ , ξ ) , sup ξ ∈ Ξ inf ξ ∈ Ξ d A L ( ξ , ξ ) ) (2.1)and (cid:0) K ( A L ) , d H A L (cid:1) is a compact metric space. Since J ⊆ K (cid:0) A L (cid:1) is a closed subset [3, 4], ( J , d H A L ) isa compact metric space. If A is finite, the topology on J can also be described by its local patterntopology defined in [3, Section 3.3]. It asserts that the convergence of subshifts in the Hausdorfftopology is equivalent to the convergence of the local patterns [3, Theorem 3.3.22]. This fact isimplicitly used in this work (as the operators are localized on patches by the Lipschitz-partitionof unity), which is implemented via the following lemma. Lemma 2.2.
The following assertions hold. (a)
Let Ξ , Θ ∈ J . For every ξ ∈ Ξ there exists an η := η ( ξ ) ∈ Θ such that d A L (cid:0) ξ, η ( ξ ) (cid:1) ≤ d H A L (Ξ , Θ) . (b) If ξ, η ∈ A L satisfy d A L ( ξ, η ) ≤ r with r > , then d A (cid:0) ξ ( x ) , η ( x ) (cid:1) ≤ r − , x ∈ Q r − ∩ L . Proof: (a) Let ξ ∈ Ξ be arbitrary. Because Θ is compact, there exists η ( ξ ) ∈ Θ such that d A L ( ξ, η ( ξ )) = inf η ∈ Θ d A L ( ξ, η ). Then (2.1) implies d A L ( ξ, η ( ξ )) ≤ d H A L (Ξ , Θ).(b) Since r − > d A L ( ξ, η ) < r − are strict, the estimate follows immediately by thedefinition of the infimum in (1.1). ✷ Remark 2.3. (i) Clearly, the role of Ξ and Θ can be interchanged. Specifically, for each η ∈ Θ,there is an ξ ( η ) ∈ Ξ such that d A L (cid:0) ξ ( η ) , η (cid:1) ≤ d H A L (Ξ , Θ) .(ii) If A is finite, d A is the discrete metric and Lemma 2.2 can be reformulated as follows: LetΞ , Θ ∈ J be such that r := d H A L (Ξ , Θ) − >
1. Then for every ξ ∈ Ξ there is an η := η ( ξ ) ∈ Θ suchthat ξ | Q r = η | Q r . The reader is referred for a purely topological discussion of this observation in[3, 5]. ✷ Hamiltonians.
As described in the introduction, the Hamiltonians are self-adjoint, boundedoperators on the Hilbert space H := ℓ ( L ) ⊗ C N . Denote by B ( H ) the C ∗ -algebra of bounded,linear operators on the Hilbert space H equipped with the operator norm k · k . The group L isrepresented by its left regular representation defined by U z ϕ ( x ) := ϕ ( x − z ) , ϕ ∈ H , x, z ∈ L . Let
R ⊆ L be a finite subset and t h : A L → M N ( C ) be continuous for h ∈ R . Define the hoppingfunction t h,ξ : L → M N ( C ) by t h,ξ ( x ) := t h ( t − x ξ ) for ξ ∈ A L . With this at hand, the Hamiltonian H ξ : H → H defined in Equation (1.2) is represented by H ξ := X h ∈R b t h,ξ U h where b f : H → H denotes the multiplication operator ( b f ϕ )( x ) := f ( x ) ϕ ( x ) by the function f : L → M N ( C ). The operator H ξ is linear and uniformly bounded in ξ ∈ A L , namely k H k := sup ξ ∈ A L k H ξ k ≤ X h ∈R k t h k ∞ < ∞ . SIEGFRIED BECKUS, JEAN BELLISSARD, HORIA CORNEAN
Since t h : A L → M N ( C ) and t : A L → A L are continuous, ξ H ξ is continuous with respect tothe strong operator topology. Furthermore, an elementary computation leads to the covariancecondition U z H ξ U − z = H t z ξ for z ∈ L and ξ ∈ A L .Let A be the ∗ -algebra generated by the operator families ( H ξ ) ξ ∈ A L defined in Equation (1.2).The involutive and algebraic structure on A is defined pointwise in ξ ∈ A L . Equipped withthe norm k H k := sup ξ ∈ A L k H ξ k , A is a normed ∗ -algebra and its completion C is a C ∗ -algebra.Then C is a sub- C ∗ -algebra of the reduced groupoid C ∗ -algebra defined by the transformationgroupoid A L ⋊ t L , see e.g. [3, 4]. If Ξ ∈ J is a subshift, a C ∗ -algebra C (Ξ) is similarly defined.Specifically, it is the closure of the ∗ -algebra of operators H Ξ := ( H ξ ) ξ ∈ Ξ . This C ∗ -algebra is againa sub- C ∗ -algebra of the reduced groupoid C ∗ -algebra defined by Ξ ⋊ t L .The main focus in this work is on the study of self-adjoint H ∈ C with strongly pattern equivarianthopping functions. Before providing the precise definition, the following auxiliary statement isproven. Lemma 2.4.
Let h ∈ L and t : A L → M N ( C ) be continuous. (a) Then k t k ∞ = k t ◦ t − h k ∞ and U − h b t ξ = b t t − h ξ U − h hold for every ξ ∈ A L . (b) If, additionally, t : A L → M N ( C ) is a strongly pattern equivariant function with β -H¨oldercontinuous coefficients with constant C t and radius of influence R t , then t ◦ t − h : A L → C is strongly pattern equivariant with β -H¨older continuous coefficients with radius of influ-ence R t + | M − h | max and constant C t ◦ t − h = C t . Proof: (a) The equality k t k ∞ = k t ◦ t − h k ∞ follows immediately by definition of the uniformnorm, c.f. Equation (1.5). Let ξ ∈ A L . Then the identities (cid:0) U − h b t ξ ϕ (cid:1) ( x ) = t (cid:0) t − x − h ξ (cid:1) · ϕ ( x + h ) = (cid:0)b t t − h ξ U − h ϕ (cid:1) ( x )hold for every ϕ ∈ H and x ∈ L .(b) Since the inclusion Q R t + h = M (cid:8) ˜ y + M − h ∈ R d : | ˜ y | max ≤ R t (cid:9) ⊆ Q R t + | M − h | max holds, the estimate k t ( t − h ξ ) − t ( t − h η ) k op ≤ C t max x ∈ Q Rt + h ∩L d A (cid:0) ξ ( x ) , η ( x ) (cid:1) β ≤ C t max x ∈ Q Rt + | M − h | max ∩L d A (cid:0) ξ ( x ) , η ( x ) (cid:1) β follows. Hence, assertion (b) is proven. ✷ Consider the operator family H = ( H ξ ) ξ ∈ A L ∈ C of the form given in Equation (1.2) where t h : A L → M N ( C ) is strongly pattern equivariant with β -H¨older continuous coefficients for each h ∈ R . Then assertion (a) and (b) of Lemma 2.4 imply that H is self-adjoint if (R1) R = −R ; (R2) the function t h satisfies t − h ( ξ ) = t ∗ h ( t − h ξ ) where t ∗ h ( t − h ξ ) denotes the adjoint of thematrix t h ( t − h ξ ).Lemma 2.4 (b) implies that the function t − h in (R2) is still strongly pattern equivariant with β -H¨older continuous coefficients with the same constant C t h but different radius of influence.For β ≥ Schur β -norm of H was already defined in (1.5) and denoted by k H k β . It isstraightforward to check that k H k ≤ k H k β holds for all β ≥
0. Thus, if H = ( H ξ ) ξ ∈ A L is givenby Equation (1.2) with infinite range R and k H k β < ∞ for some β ≥
0, then H ∈ C follows. Inthis case H is approximated in the C ∗ -norm by the restriction H | r := (cid:0) H ξ | r (cid:1) ξ ∈ A L defined by H ξ | r := X h ∈R∩ Q r b t h,ξ U h . (2.2) ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 9 More precisely, k H − H | r k → r → ∞ and k H k β < ∞ . It is worth mentioning that − Q r = Q r holds implying − ( R ∩ Q r ) = R ∩ Q r . Thus, the range set R r := R ∩ Q r of H | r satisfies (R1) if R does so.The spectrum of a Hamiltonian is studied in this work. Let H = ( H ξ ) ξ ∈ A L ∈ C be self-adjoint.Each operator H ξ : H → H has spectrum σ ( H ξ ) ⊆ R being compact and non-empty. For Ξ ∈ J ,define the spectrum of the operator family H Ξ := ( H ξ ) ξ ∈ Ξ by σ ( H Ξ ) := [ ξ ∈ Ξ σ ( H ξ ) ⊆ R . Since sup ξ ∈ Ξ k H ξ k ≤ k H k is finite, σ ( H Ξ ) is a non-empty compact subset of R . Furthermore, ρ ( H Ξ ) := C \ σ ( H Ξ ) is called the resolvent set of the operator family H Ξ := ( H ξ ) ξ ∈ Ξ , which is anopen subset of C . If h is the element in the C ∗ -algebra induced by the transformation groupoidΞ ⋊ t L that defines H Ξ , then σ ( h ) = σ ( H Ξ ), see e.g. [35, 3, 4]. It is worth mentioning that theclosure in the definition of σ ( H Ξ ) is not necessary since the union of the spectra is already closedby the amenability of the group L ⊆ R d [20, 40]. Then the distance of two spectra σ ( H Ξ ) and σ ( H Θ ) for Ξ , Θ ∈ J is measured by the Hausdorff distance on the compact subset of R , namely d H (cid:0) σ ( H Ξ ) , σ ( H Θ ) (cid:1) := max ( sup E ∈ σ ( H Ξ ) inf E ∈ σ ( H Θ ) | E − E | , sup E ∈ σ ( H Θ ) inf E ∈ σ ( H Ξ ) | E − E | ) . The subsection is finished with two auxiliary statements that are used in Proposition 4.4 whichis the heart of the proof of the main theorem. Recall that U h : H → H , U h ψ = ψ ( · − h ) , is thetranslation operator acting on the Hilbert space H := ℓ ( L ) ⊗ C N . For the sake of simplicity, thesymbol U h is also used for the translation operator acting on the Hilbert space ℓ ( L ).For ϕ ∈ ℓ ( L ), we write ϕ ≥ ϕ ( z ) ≥ z ∈ L . Then a linear bounded operator A : ℓ ( L ) → ℓ ( L ) is called positivity preserving , if ϕ ≥ Aϕ ≥ Lemma 2.5.
Let < β ≤ and H := (cid:0) H ξ : H → H (cid:1) ξ ∈ A L be a strongly pattern equivariantHamiltonian with β -H¨older continuous coefficients. Let t h,β : A L → C be defined as t h,β ( ξ ) :=(1 + | h | ) β/ k t h ( ξ ) k op and b t h,β,ξ : L → C , x t h,β ( t − x ξ ) . Define H βξ : ℓ ( L ) → ℓ ( L ) by: H βξ := X h ∈R b t h,β,ξ U h . Then H β := (cid:0) H βξ (cid:1) ξ ∈ A L is a positivity preserving self-adjoint operator family satisfying (R1) , (R2) and its operator norm k H β k := sup ξ ∈ A L k H βξ k is bounded by k H k β . Proof:
By definition, H βξ : ℓ ( L ) → ℓ ( L ) is a linear operator. Then the estimate (see (1.5) forthe definition of k t k ∞ ) k H βξ k ≤ X h ∈R (cid:13)(cid:13)b t h,β,ξ (cid:13)(cid:13) k U h k ≤ X h ∈R k t h k ∞ (1 + h ) β = k H k β follows. The range R of H (and so of H β ) satisfies (R1) . Furthermore, t − h,β ( ξ ) = (1 + | h | ) β/ k t − h ( ξ ) k op = (1 + | h | ) β/ k t ∗ h ( t − h ξ ) k op = t ∗ h,β ( t − h ξ )is derived as t h , h ∈ R , satisfy (R2) . Thus, the functions t h,β : A L → [0 , ∞ ) , h ∈ R , alsosatisfy (R2) implying that H β is self-adjoint by Lemma 2.4. Clearly, the translation operator U h is positivity preserving. Furthermore, the composition and sum of two positivity preservingoperators is positivity preserving. Thus, the operator H βξ is positivity preserving for each ξ ∈ A L since t h,β ≥ ✷ Lemma 2.6.
Let < β ≤ and H := (cid:0) H ξ : H → H (cid:1) ξ ∈ A L be a strongly pattern equivariantHamiltonian with β -H¨older continuous coefficients. Define H ∞ : ℓ ( L ) → ℓ ( L ) by H ∞ := X h ∈R k t h k ∞ U h . Then H ∞ is a positivity preserving, self-adjoint operator such that k H ∞ k ≤ k H k β . Proof:
The statement follows similarly as the previous one. ✷ Technical tools
In this section some technical tools are introduced and auxiliary statements are proven. The firstpart deals with Lipschitz-partitions of unity that are used to localize the Hamiltonians as describedin Section 1.3. Secondly, the so called superoperators are introduced, which are Lipschitz contin-uous maps on the Hilbert space H onto ℓ ( L ). It is important to notice that these superoperatorsare not linear. However, for the purpose of this paper, it suffices that they are bounded.3.1. Partition of unity.
A function Ψ : R d → R is called Lipschitz continuous if there is aconstant
C > | Ψ( x ) − Ψ( y ) | ≤ C | x − y | for all x, y ∈ R d . The smallest constantsatisfying the previous estimate is called Lipschitz constant , which is denoted by C L . A familyof functions (Ψ i ) i ∈ I is called uniformly Lipschitz continuous if C L := sup i C L ( i ) < ∞ where C L ( i ) > i . The set of Lipschitz continuous functions is denoted byLip( R d ).The notion of Lipschitz-partition of unity will play a crucial role in this work. Such a partitioncan be chosen to be subordinate to any given covering ( V i ) i ∈ I of R d , in general. Throughout thiswork the index set I will be the lattice L and a specific covering is chosen. This cover is assumedto be uniformly locally finite, which reflects in condition (P2) below. Specifically, the covering( V i ) i ∈ I is called uniformly locally finite if there is an N ∈ N such that for each x ∈ R d there are atmost N sets V i that contain x . Definition 3.1.
A family of functions (Ψ z ) z ∈L ⊆ Lip( R d ) with ≤ Ψ z ≤ is called a Lipschitz-partition of unity if the family (Ψ z ) z ∈L is uniformly Lipschitz continuous and (P1) P z ∈L Ψ z ( x ) = 1 for all x ∈ R d ; (P2) The set V z := (cid:8) z ′ ∈ L : supp (cid:0) Ψ z ′ (cid:1) ∩ supp (cid:0) Ψ z (cid:1) = ∅ (cid:9) is finite uniformly in z ∈ L , namely, N := N (Ψ) := sup z ∈L ♯V z < ∞ . For the sake of clarity, we will construct a concrete example of such a partition.
Example 3.2.
For r >
0, recall the notion of the deformed cube Q r := (cid:8) M x ∈ R d : | x | max ≤ r (cid:9) . Let 0 ≤ ψ ≤ ψ ( x ) = 1 for x ∈ K := (cid:8) x ∈ R d : | x | max ≤ (cid:9) and supp( ψ ) ⊆ U := (cid:8) x ∈ R d : | x | max < (cid:9) ⊆ R d . Since ( x + U ) x ∈ Z d is a uniformly locally finiteopen cover of R d , (cid:0) z + M U (cid:1) z ∈L is so as well as M is invertible where M U := { M x : x ∈ U } .Since the cover is uniformly locally finite, there is an N ∈ N satisfying ♯ { z ∈ L : x ∈ z + M U } ≤ N uniformly in x ∈ R d . The constraint ψ ( x ) = 1 for x ∈ K implies P z ′ ∈L ψ (cid:0) M − ( x − z ′ ) (cid:1) ≥ R d → [0 ,
1] by Ψ( x ) := ψ ( M − x ) P z ′∈L ψ ( M − ( x − z ′ )) . Its support supp(Ψ) is contained in M U ⊆ Q . It is also straightforward to check that the family (Ψ z ) z ∈L defined by Ψ z ( x ) := Ψ( x − z )satisfies all the conditions in Definition 3.1. ✷ Remark 3.3.
It is worth mentioning that N ≥ R .Thus, N ≥ R d . For indeed, if N ≤ R by restricting the partition to the first component in R d , a contradiction. ✷ Definition 3.4.
Let r > and Ψ ∈ Lip( R d ) be such that Ψ z := Ψ( · − z ) , z ∈ L , defines aLipschitz-partition of unity with N := N (Ψ) , Lipschitz constant C L > and supp(Ψ) ⊆ Q .Define the family of functions (cid:0) Ψ ( r ) z (cid:1) z ∈L by Ψ ( r ) z : R d → R , Ψ ( r ) z ( x ) := Ψ (cid:16) xr − z (cid:17) . Furthermore, χ ( r ) z : R d → { , } denotes the characteristic function of the support supp (cid:0) Ψ ( r ) z (cid:1) ⊆ R d . ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 11 Example 3.2 shows that Ψ ∈ Lip( R d ) exists such that (Ψ z ) z ∈L is a Lipschitz-partition of unity.The latter defined family (cid:0) Ψ ( r ) z (cid:1) z ∈L of functions turns out to be also a Lipschitz-partition of unitywith same bound in (P2) : Lemma 3.5.
Let r > and Ψ ∈ Lip( R d ) with Lipschitz constant C L > be such that (Ψ z ) z ∈L isa Lipschitz-partition of unity. Then the family (cid:0) Ψ ( r ) z (cid:1) z ∈L defined in Definition 3.4 is a Lipschitz-partition of unity with Lipschitz constant C L r satisfying N (Ψ) = N (cid:0) Ψ ( r ) (cid:1) . If supp(Ψ) ⊆ Q s forsome s > , then the support of Ψ ( r ) z is contained in rz + Q sr . Proof:
It is immediate to see the estimate (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:12)(cid:12) ≤ C L r (cid:12)(cid:12) x − y (cid:12)(cid:12) , x, y ∈ R d . (3.1)Condition (P1) follows by a short computation while (P2) and N (Ψ) = N (cid:0) Ψ ( r ) (cid:1) are derived fromthe identity V z = n z ′ ∈ L | supp (cid:0) Ψ ( r ) z ′ (cid:1) ∩ supp (cid:0) Ψ ( r ) z (cid:1) = ∅ o . Finally, it is straightforward to show supp (cid:0) Ψ ( r ) z (cid:1) ⊆ rz + Q sr . ✷ Lemma 3.6.
Let r > and (Ψ ( r ) z ) z ∈L be the Lipschitz-partition of unity defined in Definition 3.4.For ≤ β ≤ and z ∈ L , the estimate (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:1)(cid:12)(cid:12) ≤ (cid:18) | x − y | r (cid:19) β − β C βL (cid:16) χ ( r ) z ( x ) + χ ( r ) z ( y ) (cid:17) holds for all x, y ∈ R d . Proof:
Combining (3.1) with the inequality (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:12)(cid:12) ≤ (cid:18) C L | y − x | r (cid:19) β − β , ≤ β ≤ . The last ingredient is the identity (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) Ψ ( r ) z ( x ) − Ψ ( r ) z ( y ) (cid:12)(cid:12) (cid:0) χ ( r ) z ( x ) + χ ( r ) z ( y ) (cid:1) , finishing the proof. ✷ Superoperators.
Recall that H denotes the Hilbert space ℓ ( L ) ⊗ C N on which the Hamil-tonians act. Furthermore, χ ( r ) z : R d → { , } denotes the characteristic function of the supportsupp(Ψ ( r ) z ). The corresponding multiplication operator by the function χ ( r ) z on H (acting as theidentity on C N ) is denoted by the symbol b χ ( r ) z , with k b χ ( r ) z k = 1 for each z ∈ L . In the following C c ( L ) denotes the set of functions ϕ : L → C with finite support in L . Note in the followingthat ( Bϕ )( x ) ∈ C N holds if B : H → H , ϕ ∈ H and x ∈ L . In this case | ( Bϕ )( x ) | denotes theEuclidean length of the vector Bϕ ( x ). Lemma 3.7.
Consider an operator family ( A z ) z ∈L ⊆ B (cid:0) H (cid:1) with k A k := sup z ∈L k A z k < ∞ . Thenthe map O ( A ) : C c ( L ) ⊗ C N → ℓ ( L ) defined by (cid:0) O ( A ) ϕ (cid:1) ( x ) := X z ∈L b χ ( r ) z ( x ) (cid:12)(cid:12)(cid:0) A z b χ ( r ) z ϕ (cid:1) ( x ) (cid:12)(cid:12) , ϕ ∈ C c ( L ) ⊗ C N , x ∈ L , satisfies (cid:13)(cid:13) O ( A ) ϕ − O ( A ) ϕ (cid:13)(cid:13) H ≤ N k A k k ϕ − ϕ k H , ϕ , ϕ ∈ C c ( L ) ⊗ C N . Furthermore, O ( A ) uniquely extends to a continuous bounded map on H to ℓ ( L ) such that sup (cid:8) k O ( A ) ϕ k H : ϕ ∈ H with k ϕ k H ≤ (cid:9) ≤ N k A k . Proof:
As introduced in Definition 3.1, V ( r ) z denotes the set of all z ′ ∈ L such that supp(Ψ ( r ) z ) ∩ supp(Ψ ( r ) z ) = ∅ . Let ϕ ∈ C c ( L ) ⊗ C N . First note that ϕ z ( x ) := (cid:12)(cid:12)(cid:0) A z b χ ( r ) z ϕ (cid:1) ( x ) (cid:12)(cid:12) = N X k =1 (cid:12)(cid:12)(cid:0) A z b χ ( r ) z ϕ (cid:1) k ( x ) (cid:12)(cid:12) ! , x, z ∈ L , defines an element in ℓ ( L ). Thus, the Cauchy-Schwarz inequality on ℓ ( L ) yields (cid:12)(cid:12)(cid:10)b χ ( r ) z ϕ z , b χ ( r ) z ′ ϕ z ′ (cid:11) ℓ ( L ) (cid:12)(cid:12) ≤ (cid:13)(cid:13)b χ ( r ) z ϕ z (cid:13)(cid:13) ℓ ( L ) (cid:13)(cid:13)b χ ( r ) z ′ ϕ z ′ (cid:13)(cid:13) ℓ ( L ) ≤ k ϕ z k ℓ ( L ) k ϕ z ′ k ℓ ( L ) . Note that the latter inner product vanishes if z ∈ L and z ′ V ( r ) z . Furthermore, a short compu-tation leads to k ϕ z k ℓ ( L ) = (cid:13)(cid:13) A z b χ ( r ) z ϕ (cid:13)(cid:13) H ≤ k A k k b χ ( r ) z ϕ k H . Since 2 ab ≤ a + b for a, b ≥
0, the previous considerations yield k O ( A ) ϕ k ≤ X z ∈L X z ′ ∈ V ( r ) z (cid:12)(cid:12)(cid:10)b χ ( r ) z | (cid:0) A z b χ ( r ) z ϕ (cid:1) | , b χ ( r ) z ′ | (cid:0) A z ′ b χ ( r ) z ′ ϕ (cid:1) | (cid:11) ℓ ( L ) (cid:12)(cid:12) ≤ k A k X z ∈L X z ′ ∈ V ( r ) z (cid:0) k b χ ( r ) z ϕ k H + k b χ ( r ) z ′ ϕ k H (cid:1) . Let l x be the number of z ′ ∈ L satisfying that χ ( r ) z ′ ( x ) = 0 for fixed x ∈ L . Thus, l x ≤ N is derivedfor all x ∈ L by (P2) and Lemma 3.5. Hence, X z ∈L X z ′ ∈ V ( r ) z χ ( r ) z ′ ( x ) ≤ N ♯ (cid:8) z ′ ∈ V ( r ) z : χ ( r ) z ′ ( x ) = 0 (cid:9) ≤ N follows. Using the previous considerations, the estimate X z ∈L X z ′ ∈ V ( r ) z k b χ ( r ) z ′ ϕ k H = X x ∈L N X k =1 | ϕ k ( x ) | X z ∈L X z ′ ∈ V ( r ) z χ ( r ) z ′ ( x ) ≤ k ϕ k H N is proven. Similarly, X z ∈L X z ′ ∈ V ( r ) z k b χ ( r ) z ϕ k H = X x ∈L N X k =1 | ϕ k ( x ) | N ♯ n z ∈ L (cid:12)(cid:12) χ ( r ) z ( x ) = 0 o ≤ k ϕ k H N is deduced. These observations imply k O ( A ) ϕ k H ≤ N k A k k ϕ k H for all ϕ ∈ C c ( L ). Let ϕ , ϕ ∈C c ( L ) ⊗ C N . Since we have: (cid:12)(cid:12)(cid:0) O ( A ) ϕ − O ( A ) ϕ (cid:1) k ( x ) (cid:12)(cid:12) ≤ X z ∈L b χ ( r ) z (cid:12)(cid:12) | (cid:0) A z b χ ( r ) z ϕ (cid:1) k ( x ) | − | (cid:0) A z b χ ( r ) z ϕ (cid:1) k ( x ) | (cid:12)(cid:12) ≤ X z ∈L b χ ( r ) z (cid:12)(cid:12)(cid:0) A z b χ ( r ) z ( ϕ − ϕ ) (cid:1) k ( x ) (cid:12)(cid:12) , the previous considerations lead to (cid:13)(cid:13) O ( A ) ϕ − O ( A ) ϕ (cid:13)(cid:13) H ≤ (cid:13)(cid:13) O ( A )( ϕ − ϕ ) (cid:13)(cid:13) H ≤ N k A k k ϕ − ϕ k H , namely O ( A ) is Lipschitz continuous. Thus, there is a unique continuous extension O ( A ) : H → ℓ ( L ) satisfying sup (cid:8) k O ( A ) ϕ k H : ϕ ∈ H with k ϕ k H ≤ (cid:9) ≤ N k A k as C c ( L ) ⊗ C N ⊆ H is dense. ✷ ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 13 Lemma 3.8.
Let B : ℓ ( L ) → ℓ ( L ) be a positivity preserving, linear bounded operator. Consideran operator family ( A z ) z ∈L ⊆ B ( H ) such that k A k := sup z ∈L k A z k < ∞ . Then the map O B ( A ) : C c ( L ) ⊗ C N → ℓ ( L ) defined by (cid:0) O B ( A ) ϕ (cid:1) ( x ) := X z ∈L b χ ( r ) z ( x ) (cid:16) B (cid:12)(cid:12) A z b χ ( r ) z ϕ (cid:12)(cid:12)(cid:17) ( x ) , ϕ ∈ C c ( L ) ⊗ C N , x ∈ L , satisfies (cid:13)(cid:13) O B ( A ) ϕ − O B ( A ) ϕ (cid:13)(cid:13) H ≤ N k A k k B k k ϕ − ϕ k H , ϕ , ϕ ∈ C c ( L ) ⊗ C N . Furthermore, O B ( A ) extends to a continuous bounded map on H to ℓ ( L ) such that sup (cid:8) k O B ( A ) ϕ k H : ϕ ∈ H with k ϕ k H ≤ (cid:9) ≤ N k A k k B k . Proof:
Recall that (cid:12)(cid:12) A z b χ ( r ) z ϕ (cid:12)(cid:12) is an element of ℓ ( L ) and so B (cid:12)(cid:12) A z b χ ( r ) z ϕ (cid:12)(cid:12) is well-defined. Followingthe lines of the proof Lemma 3.7 we obtain k O B ( A ) ϕ k H ≤ N k B k k A k k ϕ k H . Since B is a positivity preserving linear operator we have0 ≤ B (cid:0) | ϕ − ψ | − | ϕ | + | ψ | (cid:1) = B | ϕ − ψ | − (cid:0) B | ϕ | − B | ψ | (cid:1) . This implies a pointwise estimate: (cid:12)(cid:12)(cid:12) B (cid:12)(cid:12) A z b χ ( r ) z ϕ (cid:12)(cid:12) − B (cid:12)(cid:12) A z b χ ( r ) z ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ B (cid:12)(cid:12) A z b χ ( r ) z ( ϕ − ϕ ) (cid:12)(cid:12) . The latter yields the pointwise estimate: (cid:12)(cid:12) O B ( A ) ϕ − O B ( A ) ϕ (cid:12)(cid:12) ≤ X z ∈L b χ ( r ) z B (cid:12)(cid:12) A z b χ ( r ) z ( ϕ − ϕ ) (cid:12)(cid:12) = O B ( A )( ϕ − ϕ )which leads to (cid:13)(cid:13) O B ( A ) ϕ − O B ( A ) ϕ (cid:13)(cid:13) H ≤ (cid:13)(cid:13) O B ( ϕ − ϕ ) (cid:13)(cid:13) H ≤ N k B k k A k k ϕ − ϕ k H . Then the density of C c ( L ) ⊗ C N ⊆ H finishes the proof. ✷ Proofs of the main results
The following two auxiliary lemmas provide estimates on the potential and the kinetic term ifcommuted with the Lipschitz partition of unity.
Lemma 4.1.
For each r > , let (Ψ ( r ) z ) z ∈L be the Lipschitz-partition of unity defined in Defini-tion 3.4. Let < β ≤ and t : A L → M N ( C ) be a strongly pattern equivariant function with β -H¨older continuous coefficients, constant C t ≥ and radius of influence R t ≥ . Let Ξ , Θ ∈ J besuch that r := d H A L (Ξ , Θ) − > e R := R t + k M − k max k M k max + 1 Then for each ξ ∈ Ξ and z ∈ L , there exists an η ξ := η ( ξ, z, r, e R, L ) ∈ Θ such that: (cid:12)(cid:12)(cid:12)(cid:16)b t ξ b Ψ ( r − e R ) z − b Ψ ( r − e R ) z b t η z (cid:17) ϕ (cid:12)(cid:12)(cid:12) ( x ) ≤ r β C t χ ( r − e R ) z ( x ) (cid:12)(cid:12) ϕ ( x ) (cid:12)(cid:12) , ϕ ∈ H , x ∈ L , holds. Proof:
Let z ∈ L . For each x ∈ R d we have:min y ∈L | x − y | max ≤ k M k max min m ∈ Z d | M − x − m | max ≤ k M k max Thus there exists an z r ∈ L such that (cid:12)(cid:12) z r − ( r − e R ) z (cid:12)(cid:12) max ≤ k M k max . Since Ξ is invariant, we have t − z r ξ ∈ Ξ and due to Lemma 2.2 we may find ˜ η ξ ∈ Θ such that d A L (cid:0) t − z r ξ, ˜ η ξ (cid:1) ≤ d H A L (Ξ , Θ) . By definition, R t ≥ r >
2. Since Θ is invariant, η ξ := t z r ˜ η ξ ∈ Θ. ThenLemma 2.2 (b) implies d A (cid:0) t − z r ξ ( y ) , t − z r η ξ ( y ) (cid:1) = d A (cid:0) t − z r ξ ( y ) , ˜ η ξ ( y ) (cid:1) ≤ r − ≤ r , y ∈ Q r − ∩ L . (4.1)Consider some ϕ ∈ H and x ∈ L . Then a short computation leads to (cid:16)b t ξ b Ψ ( r − e R ) z ϕ − b Ψ ( r − e R ) z b t η ξ ϕ (cid:17) ( x ) = Ψ ( r − e R ) z ( x ) (cid:16) t (cid:0) t − x ξ (cid:1) ϕ − t (cid:0) t − x η ξ (cid:1) ϕ (cid:17) ( x ) . Clearly, the term vanishes if x supp (cid:0) Ψ ( r − e R ) z (cid:1) and 0 ≤ Ψ ( r − e R ) z ( x ) ≤ χ ( r − e R ) z ( x ). Thus, it sufficesto show k t (cid:0) t − x ξ (cid:1) − t (cid:0) t − x η ξ (cid:1) k op ≤ C t r − β , x ∈ supp (cid:0) Ψ ( r − e R ) z (cid:1) ∩ L . Let x ∈ supp (cid:0) Ψ ( r − e R ) z (cid:1) ∩ L . According to Definition 3.4, the inclusion supp(Ψ) ⊆ Q holds. Thus,Lemma 3.5 asserts x ∈ supp (cid:0) Ψ ( r − e R ) z (cid:1) ⊆ ( r − e R ) z + Q r − e R .Using the definition of e R , we have | x + x | max ≤ r − k M − k max k M k max − | x | max ≤ r − e R and | x | max ≤ R t . Thus, Q r − e R + Q R t is contained in Q r −k M − k max k M k max − implying x + Q R t ⊆ ( r − e R ) z + Q r −k M − k max k M k max − = z r + (cid:16) ( r − e R ) z − z r (cid:17) + Q r −k M − k max k M k max − . In addition, (cid:0) ( r − e R ) z − z r (cid:1) + Q r −k M − k max k M k max − ⊆ Q r − is an immediate consequence of theestimates (cid:12)(cid:12)(cid:12) M − (cid:16) ( r − e R ) z − z r (cid:17) + y (cid:12)(cid:12)(cid:12) max ≤ | y | max + k M − k max (cid:12)(cid:12)(cid:12) ( r − e R ) z − z r (cid:12)(cid:12)(cid:12) max ≤ r − y ∈ R d satisfying M y ∈ Q r −k M − k max k M k max − . Consequently we have: x + Q R t ⊆ z r + Q r − , x ∈ supp (cid:0) Ψ ( r − e R ) z (cid:1) ∩ L . Since t satisfies (1.3), x ∈ supp (cid:0) Ψ ( r − e R ) z (cid:1) ∩ L ⊆ ( r − e R ) z + Q r − e R ∩ L yields k t (cid:0) t − x ξ (cid:1) − t (cid:0) t − x η ξ (cid:1) k op ≤ C t max n d A (cid:0) ξ ( y ) , η ξ ( y ) (cid:1) β : y ∈ x + Q R t ∩ L o ≤ C t max n d A (cid:0) t − z r ξ ( y ) , t − z r η ξ ( y ) (cid:1) β : y ∈ Q r − ∩ L o . Then (4.1) yields the desired estimate. ✷ For two operators
A, B ∈ B ( H ) we denote their commutator AB − BA by [ A, B ]. The followingestimate is based on Lemma 3.6.
Lemma 4.2.
Let (Ψ ( r ) z ) z ∈L be the Lipschitz-partition of unity defined in Definition 3.4. For each z , h ∈ L , ≤ β ≤ and r ≥ , the estimate (cid:12)(cid:12)(cid:12)(cid:16)h U h , b Ψ ( r ) z i ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ | h | β r β − β C βL (cid:16) χ ( r ) z ( x − h ) + χ ( r ) z ( x ) (cid:17) (cid:12)(cid:12)(cid:0) U h ϕ (cid:1) ( x ) (cid:12)(cid:12) holds for all ϕ ∈ H and x ∈ L . Proof:
Let ϕ ∈ H and x ∈ L . A short computation leads to (cid:16)h U h , b Ψ ( r ) z i ϕ (cid:17) ( x ) = (cid:16) Ψ ( r ) z ( x − h ) − Ψ ( r ) z ( x ) (cid:17) (cid:0) U h ϕ (cid:1) ( x ) . In addition, Lemma 3.6 implies (cid:12)(cid:12)(cid:12) Ψ ( r ) z ( x − h ) − Ψ ( r ) z ( x ) (cid:12)(cid:12)(cid:12) ≤ | h | β r β − β C βL (cid:16) χ ( r ) z ( x − h ) + χ ( r ) z ( x ) (cid:17) . ✷ We are now interested in constructing an approximate inverse of H ξ − z using the resolvent of H η .Denote the distance of z ∈ C to a compact subset K ⊆ C bydist( z , K ) := inf (cid:8) | z − x | : x ∈ K (cid:9) . ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 15 Recall the notion of the spectrum σ ( H Θ ) and resolvent set ρ ( H Θ ) for an H ∈ C and a subshiftΘ ∈ J , which were defined in Section 2.2. Lemma 4.3.
Let r > , Θ ∈ J and (Ψ ( r ) z ) z ∈L be the Lipschitz-partition of unity defined inDefinition 3.4 with N := N (cid:0) Ψ ( r ) (cid:1) independent of r . Suppose H = ( H ξ ) ξ ∈ A L ∈ C is self-adjoint.For every z ∈ L , choose an arbitrary η z ∈ Θ . Then for every z ∈ ρ ( H Θ ) = C \ σ ( H Θ ) , the operator S ( z ) ∈ B ( H ) given by S ( z ) := X z ∈L b Ψ ( r ) z ( H η z − z ) − b χ ( r ) z . is well-defined and its operator norm is bounded by N dist ( z ,σ ( H Θ )) . Proof:
Note that H η − z is invertible for each η ∈ Θ as z ∈ ρ ( H Θ ). Consider the operator family A z := b Ψ ( r ) z ( H η z − z ) − for z ∈ L . Its operator norm is bounded by k A k = sup z ∈L k A z k ≤ sup η ∈ Θ k ( H η − z ) − k = sup η ∈ Θ (cid:0) z , σ ( H η ) (cid:1) = 1dist (cid:0) z , σ ( H Θ ) (cid:1) Let ϕ ∈ C c ( L ) ⊗ C N and x ∈ L . Since b Ψ ( r ) z = b χ ( r ) z b Ψ ( r ) z , the estimate (cid:12)(cid:12)(cid:0) S ( z ) ϕ (cid:1) ( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈L (cid:16)b χ ( r ) z A z b χ ( r ) z ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) O ( A ) ϕ (cid:1) ( x )follows where O ( A ) is the map defined in Lemma 3.7. Hence, Lemma 3.7 implies k (cid:0) S ( z ) (cid:1) k ≤ N k A k which coupled with the estimate on k A k ends the proof. ✷ Proposition 4.4.
Consider a strongly pattern equivariant Hamiltonian H := (cid:0) H ξ (cid:1) ξ ∈ A L with β -H¨older continuous coefficients of finite range with radius of influence R H . Let Ξ , Θ ∈ J be suchthat r := d H A L (Ξ , Θ) − > e R H := R H + k M − k max k M k max + 1 Let (Ψ ( r ) z ) z ∈L be a Lipschitz-partition of unity defined in Definition 3.4 with N := N (cid:0) Ψ ( r ) (cid:1) inde-pendent of r . Then if z ∈ ρ ( H Θ ) satisfies dist (cid:0) z , σ ( H Θ ) (cid:1) > N max (cid:8) C L , (cid:9) β ( r − e R H ) β C hop k H k β , (4.2) we have that z ∈ ρ ( H Ξ ) . Proof:
For ξ ′ ∈ A L , the Hamiltonian is defined by H ξ ′ := X h ∈R b t h,ξ ′ U h satisfying (R1) and (R2) where t h : A L → M N ( C ) , h ∈ R , are strongly pattern equivariant with β -H¨older continuous coefficients and R is finite. Furthermore, C hop = sup h ∈R C t h is finite and1 ≤ R t h ≤ R H holds for all h ∈ R .Let z ∈ ρ ( H Θ ) obeying (4.2). In particular, z ∈ ρ ( H η ) for all η ∈ Θ and dist (cid:0) z , σ ( H η ) (cid:1) ≥ dist (cid:0) z , σ ( H Θ ) (cid:1) . It suffices to prove that z ∈ ρ ( H ξ ) uniformly in ξ ∈ Ξ, i.e. that there exists δ > ξ ∈ Ξ such that dist (cid:0) z , σ ( H ξ ) (cid:1) > δ . This implies z ∈ ρ ( H Ξ ) = C \ S ξ ∈ Ξ σ ( H ξ ).In light of this, let ξ ∈ Ξ and we will prove that z ∈ ρ ( H Ξ ) going through the following steps:(i) An operator S ( z ) is constructed such that ( H ξ − z ) S ( z ) = I + T ( z ) + T ( z ) where the errorterms T ( z ) and T ( z ) come from the kinetic and the potential terms, respectively.(ii) It is shown that k T ( z ) k ≤ .(iii) It is shown that k T ( z ) k ≤ .(iv) Using (i)-(iii), z ∈ ρ ( H ξ ) is verified. (i): Recall that R t h ≥ < d H A L (Ξ , Θ) = r with r > e R H ≥ R t h + k M − k max k M k max + 1 ≥ C t h ≤ C hop for all h ∈ R , Lemma 4.1 applies with e R replaced by e R H and C t replaced by C hop . This implies that given any z ∈ L , there exists a η z = η ( ξ, z, r, e R H , L ) ∈ Θ satisfying (cid:12)(cid:12)(cid:12)(cid:16)(cid:16)b t h,ξ b Ψ ( r − e R H ) z − b Ψ ( r − e R H ) z b t h,η z (cid:17) ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ r β C hop χ ( r − e R H ) z ( x ) (cid:12)(cid:12) ϕ ( x ) (cid:12)(cid:12) (4.3)for all ϕ ∈ H and x ∈ L . It is worth noticing that η z is independent of h ∈ R by Lemma 4.1.With this chosen η z ∈ Θ for z ∈ L , define S ( z ) ∈ B ( H ) by S ( z ) := X z ∈L b Ψ ( r − e R H ) z ( H η z − z ) − b χ ( r − e R H ) z . According to Lemma 4.3, S ( z ) is a well-defined operator and k S ( z ) k ≤ N dist (cid:0) z , σ ( H Θ ) (cid:1) . (4.4)In the following we investigate the operator product H ξ S ( z ). In order to shorten notation, define: E h,z := b t h,ξ b Ψ ( r − e R H ) z − b Ψ ( r − e R H ) z b t h,η z , h ∈ R , z ∈ L . Then for each z ∈ L we have H ξ b Ψ ( r − e R H ) z = X h ∈R b t h,ξ h U h , b Ψ ( r − e R H ) z i + X h ∈R E h,z U h ! + b Ψ ( r − e R H ) z H η z . Define the operators T ( z ) , T ( z ) ∈ B ( H ) by T ( z ) := X z ∈L X h ∈R b t h,ξ h U h , b Ψ ( r − e R H ) z i A z ( z ) b χ ( r − e R H ) z , T ( z ) := X z ∈L X h ∈R E h,z U h A z ( z ) b χ ( r − e R H ) z , where A z ( z ) := ( H η z − z ) − ∈ B ( H ) for z ∈ L . Then the operator norm of this operator family( A z ( z )) z ∈L satisfies k A ( z ) k := sup z ∈L k A z ( z ) k ≤ k ( H η z − z ) − k = 1dist (cid:0) z , σ ( H η z ) (cid:1) ≤ (cid:0) z , σ ( H Θ ) (cid:1) . According to Lemma 3.5, (cid:0) Ψ ( r − e R H ) z (cid:1) z ∈L is a Lipschitz-partition of unity satisfying (P1) and (P2) . Since b Ψ ( r − e R H ) z = b Ψ ( r − e R H ) z b χ ( r − e R H ) z , (P1) implies that P z ∈L b Ψ ( r − e R H ) z b χ ( r − e R H ) z is equal tothe identity operator I. With this at hand, the previous considerations lead to( H ξ − z ) S ( z ) = X z ∈L b Ψ ( r − e R H ) z (cid:0) H η z − z (cid:1) ( H η z − z ) − b χ ( r − e R H ) z + T ( z ) + T ( z )= I + T ( z ) + T ( z ) . (ii): Let ϕ ∈ H and x ∈ L . Lemma 4.2 implies (cid:12)(cid:12) T ( z ) ϕ ( x ) (cid:12)(cid:12) ≤ X z ∈L X h ∈R (cid:13)(cid:13) t h (cid:0) t − x ξ (cid:1)(cid:13)(cid:13) op (cid:12)(cid:12)(cid:12) (cid:16)h U h , b Ψ ( r − e R H ) z i A z ( z ) b χ ( r − e R H ) z ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ − β C βL ( r − e R H ) β X z ∈L χ ( r − e R H ) z ( x ) X h ∈R | h | β (cid:13)(cid:13) t h (cid:0) t − x ξ (cid:1)(cid:13)(cid:13) op (cid:12)(cid:12)(cid:12)(cid:16) U h A z ( z ) b χ ( r − e R H ) z ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12) + X h ∈R | h | β (cid:13)(cid:13) t h (cid:0) t − x ξ (cid:1)(cid:13)(cid:13) op X z ∈L (cid:16)b χ ( r − e R H ) z (cid:12)(cid:12) A z ( z ) b χ ( r − e R H ) z ϕ (cid:12)(cid:12)(cid:17) ( x − h ) ! . Recall the notion of the Hamiltonian H β = ( H βξ ) ξ ∈ A Z d introduced in Lemma 2.5 where the hoppingterms are given t h,β,ξ : L → [0 , ∞ ) , x (1 + | h | ) β/ k t h ( t − x ξ ) k op . Also, O ( A ( z )) and O H βξ ( A ( z )) ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 17 are the superoperators defined in Section 3.2. Since | h | β ≤ (1 + h ) β , the previous estimate readsas follows (cid:12)(cid:12) T ( z ) ϕ ( x ) (cid:12)(cid:12) ≤ − β C βL ( r − e R H ) β (cid:16)(cid:0) O H βξ ( A ( z )) ϕ (cid:1) ( x ) + (cid:0) H βξ O ( A ( z )) ϕ (cid:1) ( x ) (cid:17) . Lemma 2.5 asserts that H βξ : ℓ ( L ) → ℓ ( L ) is positivity preserving, self-adjoint satisfying k H βξ k ≤k H k β . Thus, Lemma 3.7 and Lemma 3.8 imply k T ( z ) k ≤ C L r − e R H ) ! β N k H βξ k k A ( z ) k ≤ C L r − e R H ) ! β N k H k β (cid:0) z , σ ( H Θ ) (cid:1) . Invoking the lower bound (4.2) on the distance dist (cid:0) z , σ ( H Θ ) (cid:1) , the norm k T ( z ) k is smaller orequal than since C hop ≥
1, uniformly in ξ ∈ Ξ.(iii): Let ϕ ∈ H and x ∈ L . Estimate (4.3) at the beginning of the proof leads to (cid:12)(cid:12) T ( z ) ϕ ( x ) (cid:12)(cid:12) ≤ X z ∈L X h ∈R (cid:12)(cid:12)(cid:12)(cid:16) E h,z U h A z ( z ) b χ ( r − e R H ) z ϕ (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ r β C hop X z ∈L χ ( r − e R H ) z ( x ) X h ∈R k t h k ∞ (cid:12)(cid:12)(cid:12)(cid:0) U h A z ( z ) b χ ( r − e R H ) z ϕ (cid:1) ( x ) (cid:12)(cid:12)(cid:12) . As introduced in Lemma 2.6, H ∞ : ℓ ( L ) → ℓ ( L ) defined by H ∞ := P h ∈R k t h k ∞ U h is apositivity preserving, self-adjoint operator. Hence, the previous considerations imply (cid:12)(cid:12) T ( z ) ϕ ( x ) (cid:12)(cid:12) ≤ r β C hop (cid:0) O H ∞ ( A ( z )) ϕ (cid:1) ( x ) . Here O H ∞ ( A ( z )) : H → ℓ ( L ) is again the superoperator introduced in Lemma 3.8. Lemma 2.6additionally states that k H ∞ k ≤ k H k β . Hence, Lemma 3.8 leads to k T ( z ) k ≤ r β C hop N k H ∞ k k A ( z ) k ≤ r β C hop N k H k β (cid:0) z , σ ( H Θ ) (cid:1) ≤ k T ( z ) + T ( z ) k ≤ uniformly in ξ ∈ Ξ, for all z ∈ ρ ( H Θ )satisfying (4.2). Assuming for the moment that z has a non-zero imaginary part, we know that H ξ − z is invertible because H is self-adjoint and we can write:( H ξ − z ) − = S ( z ) (I + T ( z ) + T ( z )) − . The estimate (4.4) and k T ( z ) + T ( z ) k ≤ imply: k ( H ξ − z ) − k ≤ N dist (cid:0) z , σ ( H Θ ) (cid:1) uniformly both in ξ and in the imaginary part of z . By analytic continuation, the above estimateremains true for real elements of ρ ( H Θ ) satisfying (4.2). Moreover, the same estimate providesthe uniform lower bound we are looking for:dist (cid:0) z , σ ( H ξ ) (cid:1) ≥ dist (cid:0) z , σ ( H Θ ) (cid:1) N , ξ ∈ Ξ , which shows that z ∈ ρ ( H Ξ ). ✷ The previous proposition provides an estimate on the Hausdorff distance of the spectra whenever d H A L (Ξ , Θ) is small enough. The following (classical) statement delivers an estimate if d H A L (Ξ , Θ)is greater or equal than a certain constant. A slightly more general version can be found in [28,Chapter 5, Theorem 4.10].
Lemma 4.5.
Let
A, B ∈ L ( H ) be self-adjoint. Then d H (cid:0) σ ( A ) , σ ( B ) (cid:1) ≤ k A − B k ≤ (cid:8) k A k , k B k (cid:9) . Proof:
Let λ σ ( A ) such that d ( λ, σ ( A )) > k A − B k . Then the operator ( B − A )( A − λ ) − hasnorm strictly less than 1 and so I + (B − A)(A − λ ) − is invertible. Thus B − λ = (cid:0) I + (B − A)(A − λ ) − (cid:1) ( A − λ )is also invertible, which shows that λ σ ( B ). In other words, no element of σ ( B ) can be locatedat a distance larger than k A − B k from σ ( A ), which implies:sup λ ∈ σ ( B ) dist (cid:0) λ, σ ( A ) (cid:1) ≤ k A − B k . By interchanging A with B , the proof is over. ✷ Proof of Theorem 1.2.
Recall the notation e R H := R H + k M − k max k M k max + 1 and definethe constant C d, L := 16 N max (cid:8) k M − k max k M k max , C L , (cid:9) which only depends on the lattice L and the choice of the Lipschitz-partition of unity (and henceon the dimension), c.f. Section 3.1. If Ξ = Θ, then σ ( H Ξ ) = σ ( H Θ ). Now suppose Ξ = Θ, namely d H A L (Ξ , Θ) >
0. Set r := d H A L (Ξ , Θ) − . We analyze two cases: (i) 1 ≤ r ≤ e R H and (ii) 2 e R H < r .(i): From 1 ≤ r ≤ e R H we infer1 ≤ e R H r ! β ≤ {k M − k max k M k max , } R βH r β where the last inequality follows from the definition of e R H . Also, H is self-adjoint and k H ξ ′ k ≤k H k β for ξ ′ ∈ A L . Then Lemma 4.5 and the previous considerations imply d H (cid:0) σ ( H ξ ) , σ ( H η ) (cid:1) ≤ k H k β ≤
12 max {k M − k max k M k max , } R βH k H k β d H A L (Ξ , Θ) β for all ξ ∈ Ξ and η ∈ Θ. According to Remark 3.3, N ≥ σ ( H Ξ ) and σ ( H Θ ) is derived if 1 ≤ r ≤ e R H as C hop ≥ r > e R H ≥
2. For z ∈ σ ( H Ξ ), Proposition 4.4 leads todist (cid:0) z , σ ( H Θ ) (cid:1) ≤ N max (cid:8) C L , (cid:9) β ( r − e R H ) β C hop k H k β . By interchanging the role of Ξ and Θ we obtain: d H (cid:0) σ ( H Ξ ) , σ ( H Θ ) (cid:1) ≤ C d, L β C hop k H k β r − e R H ) β . In addition, the constraint r > e R H implies ( r − e R H ) − β ≤ (cid:0) r (cid:1) β which together with r := d H A L (Ξ , Θ) − and the previous estimate finishes the proof. ✷ Proof of Theorem 1.4.
If Ξ = Θ, then σ ( H Ξ ) = σ ( H Θ ) follows. Thus, without loss ofgenerality suppose d H A L (Ξ , Θ) >
0. Let r := d H A L (Ξ , Θ) − ≥
1. Recall that C d, L := 16 N max (cid:8) k M − k max k M k max , C L , (cid:9) Let us introduce a parameter s ≥ H | s := (cid:0) H ξ ′ | s (cid:1) ξ ′ ∈ A L with β -H¨older continuous coefficients, which is given by restrictingthe range to R ∩ Q s , see Section 2.2. Since the range of influence of H has linear growth, thereexists a C H ≥ R H | s (i.e. the radius of influence of H | s ) obeys R H | s ≤ C H s .First, assume that 1 ≤ r ≤ k M − k max k M k max + 4 C H . The second inequality in Lemma 4.5implies d H (cid:0) σ ( H ξ ) , σ ( H η ) (cid:1) ≤ k H k β ≤ k M − k max k M k max + 4 C H ) β k H k β d H A L (Ξ , Θ) β for all ξ ∈ Ξ and η ∈ Θ. According to Remark 3.3, N ≥ C hop ≥ ¨OLDER CONTINUITY OF THE SPECTRA FOR APERIODIC HAMILTONIANS 19 Second, assume that r > k M − k max k M k max + 4 C H . For every s ≥ s β k H ξ ′ − H ξ ′ | s k ≤ s β X | h | >s k t h k ∞ ≤ k H k β , where we used that s β ≤ (1 + h ) β for | h | > s . Hence, k H ξ ′ − H ξ ′ | s k ≤ s − β k H k β follows for all ξ ′ ∈ A L . Since the operators are self-adjoint, Lemma 4.5 implies d H (cid:0) σ ( H ξ ′ ) , σ ( H ξ ′ | s ) (cid:1) ≤ k H ξ ′ − H ξ ′ | s k ≤ k H k β s β , ξ ′ ∈ A L . (4.5)Define s := r − k M − k max k M k max C H >
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