Ballistic transport for one-dimensional quasiperiodic Schrödinger operators
aa r X i v : . [ m a t h - ph ] S e p BALLISTIC TRANSPORT FOR ONE-DIMENSIONAL QUASIPERIODICSCHR ¨ODINGER OPERATORS
LINGRUI GE AND ILYA KACHKOVSKIY
Abstract.
In this paper, we show that one-dimensional discrete multi-frequency quasiperi-odic Schr¨odinger operators with smooth potentials demonstrate ballistic motion on the setof energies on which the corresponding Schr¨odinger cocycles are smoothly reducible to con-stant rotations. The proof is performed by establishing a local version of strong ballistictransport on an exhausting sequence of subsets on which reducibility can be achieved bya conjugation uniformly bounded in the C ℓ -norm. We also establish global strong ballistictransport under an additional integral condition on the norms of conjugation matrices. Thelatter condition is quite mild and is satisfied in many known examples. Introduction
Types of ballistic motion.
Let H be a discrete Schr¨odinger operator on ℓ ( Z ):(1.1) ( Hψ )( n ) = ψ ( n −
1) + ψ ( n + 1) + V n ψ ( n ) , n ∈ Z . where { V n } n ∈ Z is a sequence of real numbers (the potential). The operator H is a Hamiltonianof a single quantum particle with wave function ψ : Z → C , whose time evolution is describedby the time-dependent Schr¨odinger equation :(1.2) i ∂ψ∂t = Hψ, ψ (0) ∈ ℓ ( Z ) . Using the spectral theorem, one may explicitly solve (1.2) via(1.3) ψ ( t ) = e − itH ψ (0) . Let B be a self-adjoint operator associated to an observable quantity. The Heisenberg evo-lution of B is described by B ( T ) = e iT H Be − iT H . In the present paper, we will be interested in spatial transport properties of a quantumparticle on the lattice Z . The relevant observable quantity is the position operator ( Xψ )( n ) := nψ ( n ) , n ∈ Z , which is an unbounded self-adjoint operator with the natural domain of definitionDom X = { ψ ∈ ℓ ( Z ) : X n ∈ Z | n | | ψ ( n ) | < + ∞} . One can check by direct calculation that the Heisenberg evolution of the position operatorcan be expressed in the following form:(1.4) X ( T ) := e iT H Xe − iT H = X + Z T e itH Ae − itH dt, T ∈ R , where(1.5) Aψ ( n ) = i ( ψ ( n + 1) − ψ ( n − current operator (a tight binding analogue of the gradient operator i ∇ ). Since A is bounded, (1.4) implies that X = X (0) and X ( T ) have the same domain.We will be interested in the phenomenon of ballistic motion , which states, informally, thatthe position of the particle grows linearly with time (“ X ( T ) ≈ T ”). More precisely, we willaddress the following limits:(1.6) lim T → + ∞ T X ( T ) ψ , where, initially, ψ ∈ Dom X . One can consider the limit (1.6), if it exists, as the “asymptoticvelocity” of the state ψ at infinite time. The asymptotic velocity operator can therefore bedefined by(1.7) Q = s–lim T → + ∞ T X ( T ) = s–lim T → + ∞ T Z T e itH Ae − itH dt. The first limit can only be considered on Dom X , but, since the term T X (0) of (1.4) dis-appears as T → ∞ , it is natural to drop it from consideration. We say that a Schr¨odingeroperator H demonstrates strong ballistic transport , if the strong limit in the right hand sideof (1.7) exists, is defined on the whole ℓ ( Z ) and, moreover, ker Q = { } .An immediate consequence of (1.4) and (1.7) is that all of the moments of the positionoperator grow ballistically in time. More specifically, for any p ≥ = ψ ∈ Dom | X | p ,we have(1.8) lim T → + ∞ T − p h| X ( T ) | p ψ , ψ i = h| Q | p ψ , ψ i > . If we take p = 2, (1.8) immediately implies that H has ballistic motion . More precisely, wesay H has ballistic motion if(1.9) lim inf T → + ∞ T − h| X ( T ) | ψ , ψ i > , ψ ∈ Dom
X, ψ = 0 . Note that (1.9) is weaker than (1.8) with p = 2. One can also consider (1.9) for p = 2.Ballistic motion is one of the examples of wave packet spreading, which indicates absenceof localization. The fundamental work in this aspect is the RAGE Theorem [16] which statesfor a Schr¨odinger operator H , if ψ ∈ ℓ ( H ), then for any N > T → + ∞ T Z T X | n |≤ N |h δ n , e iT H ψ i| dt = 0 , where ℓ ( H ) = { ψ ∈ ℓ ( Z ) : µ ψ = µ ψ, c } = ℓ ( H ) ⊥ is the subspace corresponding to the continuous spectrum of H . In other words, a wavepacketin the continuous subspace of H will spend most of the time outside of any fixed compactsubset of Z . In the case of the absolutely continuous subspace, (1.10) can be further improvedto a version that does not involve time averaging:(1.11) lim T → + ∞ X | n |≤ N |h δ n , e iT H ψ i| dt = 0 , ψ ∈ ℓ ( H ) . Both (1.10) and (1.11) imply the following growth conditions on the moments:(1.12) lim T → + ∞ T Z T h| X ( t ) | p ψ , ψ i dt = + ∞ , = ψ ∈ ℓ ( H ) . (1.13) lim T → + ∞ h| X ( T ) | p ψ , ψ i = + ∞ , = ψ ∈ ℓ ( H ) . However, it is harder to estimate the exact rate of growth. In fact, this rate can be relatedto the Hausdorff dimension of the spectrum and spectral measures of H , see [46]. For theabsolutely continuous case, the Guarneri–Combes–Last Theorem [46] states that, for any p ≥ T → + ∞ T p +1 Z T h| X ( t ) | p ψ , ψ i dt > , ∀ ψ ∈ Dom | X | p , = ψ ∈ ℓ ( H ) . One can compare the above versions of transport as follows:existence of (1.7) with trivial kernel ⇒ (1.8) ⇒ (1.9) for all p ⇒ (1.14) ⇒ (1.12) ⇓ (1.13) . (1.15)Thus, strong ballistic transport (as defined in (1.7)) can be viewed as the strongest versionof ballistic motion. Note that, since the operator A is bounded, (1.4) implies an elementary ballistic upper bound on the wave packet spreading. In other words, no transport can bestronger than ballistic.In general, ballistic transport is not expected on any spectra other than purely absolutelycontinuous. In particular, it was shown in [49] that point spectrum cannot support anyballistic motion. However, one can still expect it after restricting the operator to a subspacethat supports purely absolutely continuous spectrum. In this regard, we will need a versionof the above definition that would be local in energy. Let K ⊂ R be a Borel subset. We willsay that H has strong ballistic transport on K if there exists a self-adjoint operator Q suchthat(1.16) s–lim T → + ∞ T Z T e itH K ( H ) A K ( H ) e − itH dt = K Q K and ker Q = Ran( K ) ⊥ . While we will be able to establish (1.16) in a range of situations,a significant gain in generality can be achieved by slightly relaxing the above definition.We will say that H has local ballistic transport on K if there exists a self-adjoint operator Q and a sequence of Borel subsets {K j } ∞ j =1 such that K = ∪ j K j and H satisfies (1.16) oneach K j . As a part of the definition, we require that Q is the same operator for all K j ,and ker Q ( K ) = Ran( K ) ⊥ . For the purpose of the diagram (1.15), local ballistic transportimplies lower bounds on wavepacket spreading just as good as strong ballistic transport.More precisely, if ψ ∈ Ran K ( H ), then, for large j , we have(1.17) 1 T Z T e itH Ae − itH ψ dt = K j Qψ + ψ ⊥ ( T ) + o (1) , where ψ ⊥ ( T ) is orthogonal to K j Qψ , and hence can only increase the norm. Note that weare using the right hand side of (1.6) instead of T X ( T ) ψ , since we cannot guarantee that theintersection Ran K ( H ) ∩ Dom X is large enough. However, if ψ ∈ Dom X is sufficiently closeto Ran K ( H ) (for example, k (1 − K ( H )) ψ k < k K Qψ k ), then (1.17) implies a ballistic LINGRUI GE AND I. KACHKOVSKIY lower bound on k X ( T ) ψ k . The set of such ψ is dense in Ran K ( H ). The difference between(1.16) and (1.17) is that the latter may have a non-trivial “tail” which stays within the rangeof K ( H ), but eventually escapes any Ran( K j ( H )) with finite j . However, this tail can onlystrengthen the ballistic lower bound. As a consequence, local ballistic transport still implies(1.8) and (1.9).Unlike (1.14) and (1.13), we are not aware of any results of the form (1.7)–(1.9) for generalSchr¨odinger operators with absolutely continuous spectra. Instead, all known results onlyapply to potentials of special structure. First results of this type were obtained in [9] forperiodic operators in the continuum. Later, a tight binding analogue was obtained in [20]for discrete periodic Jacobi matrices, motivated by applications to XY spin chains. See alsorelated paper [19] about anomalous (non-ballistic) transport for Fibonacci-type operatorswith singular continuous spectra. The limit-periodic case was studied in [23] where ananalogue of (1.6) was proved by periodic approximations.1.2.
Quasiperiodic operators.
The next natural class of operators with absolutely con-tinuous spectra, where one can expect ballistic motion/ballistic transport, is quasiperiodicSchr¨odinger operators , which will be the subject of the present paper. Let v ∈ C s ( T d ; R ) be asmooth function. We will identify Z d -periodic functions on R d with functions on T d . Let also α ∈ R d be a frequency vector . We will always assume that { , α , . . . , α d } are independentover Q . An smooth multi-frequency quasiperiodic Schr¨odinger operator is an operator of theform(1.18) ( H x ψ )( n ) = ψ ( n −
1) + ψ ( n + 1) + v ( x + nα ) ψ ( n ) , n ∈ Z . Here x ∈ T d is the quasiperiodic phase, and one usually considers the whole family { H x } x ∈ T d .Quasiperiodic operators (1.18) with small analytic potentials v are often known to havepurely absolutely continuous spectra, see [2, 7, 10, 11, 14, 22]. In [42], it was shown thata large class of such operators (in all cited regimes, except for the Liouville case in [10])satisfies x -averaged strong ballistic transport . In other words, instead of (1.6), one has thefollowing convergence statement in the direct integral space L ( T d × Z ):s–lim T → + ∞ (cid:18) T Z ⊕ T d X ( x, T ) dx (cid:19) = Z ⊕ T d Q ( x ) dx. where X ( x, T ) := e iT H x Xe − iT H x . The proof used the duality method based on [37]. Like [20],the work [42] was motivated by applications to the XY spin chains. The x -averaged versionof ballistic transport implies existence of the limit (1.7) on a subsequence of time scales foralmost every x and hence is sufficient for the conclusion on the XY spin chain. However, itdoes not imply any of the claims (1.7)–(1.9) in full. In the same year, a KAM-type approachwas developed in [54] in order to obtain bounds of type (1.9) in the perturbative setting.The advantage is that it works for all x and does not require to take a subsequence of timescales. However, it falls short of establishing existence of (1.7). The KAM method of [54]was later expanded in [52] to treat the one-frequency Liouvillean case, by further weakening(1.9) to a lower bound on some transport exponents. Except for potentials decaying on infinity, where one can obtain these bounds using scattering theory.In general, ballistic transport is expected to be stable under decaying perturbations. We do not go into thedetails in the present paper.
Outline of the approach.
The goal of the present paper is to obtain a result whichhas the advantages of both [42] and [54]. One can see it as a refinement of either of thepapers, however, the general line of the argument is closer to [42]. In the quasiperiodic case,one of the results of [42] is the calculation of the asymptotic velocity operator Q ( x ), but,since it is only obtained on a sequence of time scales, one cannot exclude the possibility oflarge oscillations around the limiting value. Moreover, [42] predicts a possible mechanismof convergence: after applying duality, it becomes a procedure of diagonal truncation of anoperator dual to (1.5) in the basis of the eigenvectors of the dual Hamiltonian with purelypoint spectrum. The convergence of the truncation is only obtained in the Fourier dual directintegral space L ( T × Z d ), which is not enough to guarantee pointwise strong convergencein the original direct integral space. A natural question arises: can extra information onthe dual operator family improve the rate of convergence in (1.7) ? If yes, what kind ofinformation can be used?
In order to obtain a pointwise bound, we would like to replace L ( T d × Z ) by L ∞ ( T d ; ℓ ( Z ))or by C ( T d ; ℓ ( Z )). One can try to obtain that by improving convergence in the dual space:for example, to ℓ ( Z d ; L ( T )). On any finite box, ℓ and ℓ -norms are equivalent (with theconstants depending on the size of the box). Therefore, one possible way of obtaining ℓ -convergence would be to obtain a uniform ℓ bound on the tails. The latter can be achievedby investigating quantitative character of the localization for the dual model. For example,one can take advantage of exponential dynamical localization in expectation which has beenobtained in [31] and [27] under some assumptions. Along these lines, one can obtain thedesired control on the tails, which would imply strong ballistic transport for almost every x ∈ T d . This approach was partially implemented in the preprint [41], which is no longerintended for publication since the current paper supersedes it in several ways.The main results of the present paper are Theorem 2.1 (the local result) and Theorem2.2 (the global result). In Theorem 2.1, we state that the operators (1.18) have local bal-listic transport on the set of energies on which the corresponding Schr¨odinger cocycles areC s -reducible with s > d (see Section 2.1 for precise definitions). We do not require anyquantitative information on the conjugating matrices and do not care about Diophantineproperties of the frequency vector. While the result falls short of the complete strong ballis-tic transport, most of its conclusions (such as ballistic motion) also hold, as described above.In Theorem 2.2, we state that one can obtain strong ballistic transport under an additionalintegral condition on the norms of the conjugating matrices. Several known examples, in-cluding the settings of [54] and [27], satisfy this condition.The proof of the local result is based on the following observation: suppose that R is theset of energies under consideration, and K ⊂ K ⊂ . . . ⊂ R is a sequence of Borel subsets such that R r ( ∪ j K j ) has zero spectral measure with respect to H x . Then, it is sufficient to check that the limit (1.16) exists on each K j . The main problemin obtaining “nice” localization bounds for the dual model is the fact that regularity ofthe conjugation matrices (Bloch waves) is not uniform in the energy and depends on theDiophantine properties of the rotation number. Quantitative estimates, such as in [27], canbe quite delicate. On the other hand, if one is allowed to restrict to a subset of energies, wecan get, basically, as good control of the localization parameters as desired. In particular,we can get a ridiculously strong version of uniform localization, which is not even remotelyavailable on the whole spectrum. As expected, the constants will get worse as one increases LINGRUI GE AND I. KACHKOVSKIY the set of energies under consideration. Since we only need ℓ control of the tails, we alsodo not require Anderson (exponential) localization, and are satisfied with polynomial decayof eigenfunctions, which allows us to consider smooth potentials rather than analytic. Theidea of restricting to an exhausting subset of energies/rotation numbers while maintainingcontrol on the regularity is not unlike the argument in [26].The global result is somewhat more delicate. While we cannot expect any uniform re-ducibility bounds, the desired bound still contains an integral in θ , and hence, just as in theproofs of dynamical localization, one can hope for a quantitative result “in expectation”.Using a variant of the covariant representation for the eigenfunctions of the dual operatorby duality such as in [37], we reduce the integral(1.19) Z T |h δ m , e itL θ δ n i| dθ that appears in the proof of dynamical localization, to a convolution-type bound on theeigenfunctions which, in turn, can be controlled in terms of C s or Sobolev norms of theconjugating matrices, averaged over the rotation number. Unfortunately, in order to obtainbetter bounds, we would ideally want to estimate a different, smaller integral Z T |h δ m , e itL θ δ n i| r dθ, r > , and we were not able to find any way to take advantage of r >
1, which actually appearsin our desired bounds. Still, by taking some losses, we were able to obtain a bound by aseries of convolution-type estimates for expressions of the form (1.19). As a result, in theglobal theorem, the smoothness requirement is of the form C s with s > d/
2, rather than s > d as in the local result. Still, our integral condition is satisfied by a large margin in themodels where exponential dynamical localization is obtained such as [27]. It is also easy toreformulate our global result as a conditional one: for example, strong ballistic transport willhold on K if we assume C -reducibility on K (without any quantitative control) and powerlaw dynamical localization on K : Z T |h δ m , K ( L θ ) e itL θ δ n i| dθ ≤ C (1 + | m − n | ) s , s > d, which is weaker than, say, exponential dynamical localization in expectation.In both cases, the stated arguments would only imply the corresponding version of ballistictransport for almost every x ∈ T d , since the duality ignores measure zero subsets of phases.In the case of the dual operator, this can be a real issue: for example, one cannot expectlocalization for all θ ∈ T [40]. However, quantities related to the absolutely continuousspectrum are known to be more phase stable. We were able to recover continuity in x bycomparing the pre-limit expressions in the definition of Q ( x ) and the alternative definition of Q ( x ) and showing that they are both uniformly continuous in the strong operator topology.In the latter case, we had to use quantitative continuity of the absolutely continuous spectralmeasures discussed in Section 5. As stated, one can only obtain it in the setting of localballistic transport, since one has to restrict the operator to one of the subsets K j . However,that particular part survives after passing to the union of K j , and thus is also applicable tothe global case. Acknowledgments.
Ge is partially supported by the NSF grant DMS–1901462, andKachkovskiy is currently supported by NSF DMS–1846114.Both authors would like to thank S. Jitomirskaya for facilitating their collaboration andfor comments on the manuscript.2.
Preliminaries and statements of the results
Schr¨odinger cocycles and reducibility.
Let A ∈ C s ( T d ; SL(2 , R )), and consider afrequency vector α ∈ R d such that { , α , . . . , α d } are independent over Q . By definition, a quasiperiodic C s -smooth SL(2 , R ) -cocycle is a map( α, A ) : ( T d × C → T d × C ;( x, v ) ( x + α, A ( x ) v ) . The iterates of ( α, A ) are of the form ( α, A ) n = ( nα, A n ), where A n ( x ) := ( A ( x + ( n − α ) · · · A ( x + α ) A ( x ) , n ≥ A − ( x + nα ) A − ( x + ( n + 1) α ) · · · A − ( x − α ) , n < . We will usually simply call the above maps cocycles. Similarly, one can talk about SL(2 , C )-cocycles. The Lyapunov exponent of the cocycle ( α, A ) is defined by L ( α, A ) := lim n →∞ n Z T d ln k A n ( x ) k dx. A cocycle ( α, A ) is called uniformly hyperbolic if, for every x ∈ T d , there exists a continuoussplitting C = E s ( x ) ⊕ E u ( x ) such that for every n ≥ | A n ( x ) v | ≤ Ce − cn | v | , v ∈ E s ( x ) , | A n ( x ) − v | ≤ Ce − cn | v | , v ∈ E u ( x + nα ) , for some constants C, c >
0. This splitting is invariant under the dynamics, i.e., A ( x ) E s ( x ) = E s ( x + α ) , A ( x ) E u ( x ) = E u ( x + α ) , ∀ x ∈ T d . Assume that A ∈ C ( T d ; SL(2 , R )) is homotopic to the identity. It induces the projectiveskew-product F A : T d × S → T d × S with F A ( x, w ) := (cid:18) x + α, A ( x ) · w | A ( x ) · w | (cid:19) . In other words, F A : T d × T → T d × T can be expressed as ( x, y ) ( x + α, y + ϕ x ( y )), where ϕ x : R → R is a 1-periodic continuous function (defined modulo translations by integers onboth copies of R ). Let µ be any probability measure on T d × T invariant under F A and whoseprojection onto the coordinate x is given by the Lebesgue measure. The number(2.1) ρ ( α, A ) := Z T d × T ϕ x ( y ) dµ ( x, y ) mod Z depends neither on the lift ϕ nor on the measure µ , and is called the fibered rotation number of ( α, A ) (see [29, 39] for more details; see also [11, Appendix A] for a detailed exposition).Let R θ denote the rotation matrix(2.2) R θ := (cid:18) cos 2 πθ − sin 2 πθ sin 2 πθ cos 2 πθ (cid:19) , θ ∈ T . LINGRUI GE AND I. KACHKOVSKIY
Any continuous map A : T d → SL(2 , R ) is homotopic to x R n · x for a unique n ∈ Z d . Wecall n the degree of A and denote it by deg A . The fibered rotation number is invariant underreal conjugacies which are homotopic to the identity. More generally, if ( α, A ) is conjugatedto ( α, A ), i.e., B ( x + α ) − A ( x ) B ( x ) = A ( x ), for some B : T d → PSL(2 , R ) with deg B = n ,then(2.3) ρ ( α, A ) = ρ ( α, A ) + n · α. A typical example of a quasiperiodic cocycle is a
Schr¨odinger cocycle ( α, S E − v ), where S E − v ( x ) := (cid:18) E − v ( x ) −
11 0 (cid:19) , E ∈ R . Any formal solution ψ = { ψ ( n ) } n ∈ Z of the eigenvalue equation H x ψ = Eψ , where H x is thequasiperiodic Schr¨odinger operator (1.18)( H x ψ )( n ) = ψ ( n −
1) + ψ ( n + 1) + v ( x + nα ) ψ ( n ) , n ∈ Z , x ∈ T d , satisfies the following relation with S E − v ( x ): (cid:18) ψ n +1 ψ n (cid:19) = S E − v ( x + nα ) (cid:18) ψ n ψ n − (cid:19) , ∀ n ∈ Z . It is well known that the spectrum σ ( H x ), denoted by Σ α,v , is a compact subset of R ,independent of x if { , α , . . . , α d } are rationally independent. The spectral properties of H x and the dynamics of ( α, S E − v ) are related by the Johnson’s theorem [38]: E ∈ Σ α,v ifand only if ( α, S E − v ) is not uniformly hyperbolic. Throughout the paper, we will use thenotation L ( E ) = L ( α, S E − v ) and ρ ( E ) = ρ ( α, S E − v ) for brevity.2.2. Reducibility of quasiperiodic cocycles.
We will only consider cocycles ( α, A ) withdeg A = 0. A quasiperiodic C s -cocycle ( α, A ) with { , α , . . . , α d } rationally independent iscalled C s -rotations reducible if there exists B ∈ C s ( T d ; SL(2 , R )) and θ ∈ C s ( T d ; R ) such that(2.4) B ( x + α ) − A ( x ) B ( x ) = R θ ( x ) . We will call a cocycle reducible if it is rotations reducible to a constant rotation. In this case,one can choose θ ≡ ρ ( α, A ). For reducible cocycles, it will be more convenient diagonalizethe rotation matrix and consider B ∈ C s ( T d ; SL(2 , C )) satisfying(2.5) B ( x + α ) − A ( x ) B ( x ) = (cid:18) e πiρ ( α,A ) e − πiρ ( α,A ) (cid:19) . Note that our use of the definition is more narrow than usual. More accurately, we shouldhave used the wording “reducible to a constant rotation”. Usually, one considers reducibilityto a general constant matrix in the right hand side of (2.4).Let { H x } x ∈ T d be a quasiperiodic operator family, and ( α, S E − v ) be the correspondingSchr¨odinger cocycle. Define the following subset: R sα,v = { E ∈ R : ( α, S E − v ) is C s -reducible }⊂ RR sα,v = { E ∈ R : ( α, S E − v ) is C s -rotations reducible } . We will sometimes drop the indices and simply use R or RR , if the values of the indices areclear from the context. From Shnol’s theorem [13, 28, 50, 51], it follows that RR sα,v ⊂ Σ α,v . Moreover, subordinacytheory [32, 33, 43] implies that the restriction of the spectral measure of H x into RR sα,v ispurely absolutely continuous for any s ≥
0. The same also holds for R sα,v .We will also need some conventions about normalizations of the cocycles in L ( T d ). Letus rewrite the reducibility equation (3.1) as (cid:18) ( E − v ( x )) b ( x ) − b ( x ) ( E − v ( x )) b ( x ) − b ( x ) b ( x ) b ( x ) (cid:19) = (cid:18) e πiθ b ( x + α ) e − πiθ b ( x + α ) e πiθ b ( x + α ) e − πiθ b ( x + α ) (cid:19) . One can see that the columns of B ( x ) are not intertwined, and one can multiply onecolumn and divide another by the same constant without affecting the determinant. Notealso that k b k L ( T d ) = k b k L ( T d ) , k b k L ( T d ) = k b k L ( T d ) . As a consequence, we can choosea constant so that the columns are “balanced”: k b k L ( T d ) = k b k L ( T d ) = k b k L ( T d ) = k b k L ( T d ) , without affecting the regularity of the matrix B in the variable x . So, we would have(2.6) k B k ( T d ) = 4 k b ij k ( T d ) , ∀ i, j ∈ { , } , where in the left hand side we are using the Hilbert–Schmidt matrix norm. In the statementsof the main results, we will always assume that the conjugation matrix B is balanced in theabove sense. Also, we will not always require det B ( x ) = 1, but sometimes instead choose B to be L -normalized (and balanced).2.3. Statements of the results.
In order to formulate the main result, we will need thedefinition of the density of states measure of the operator family { H x } x ∈ T d : for a Borel subset B ⊂ R , define(2.7) N ( B ) = Z T d h B ( H x ) δ , δ i dx. In other words, the density of states measure is the expectation value of the spectral measures.We also introduce the integrated density of states (denoted by the same symbol with a slightabuse of notation):(2.8) N ( E ) := N (( −∞ , E ]) = N (( −∞ , E )) , the cumulative distribution function of the density of states measure. It is well knownthat N is a non-decreasing continuous function of E . Clearly, if the spectral measuresare absolutely continuous, then the IDS is also absolutely continuous (with respect to theLebesgue measure). The IDS is related to the fibered rotation number defined above in (2.1)in the following way [21]: N ( E ) = 1 − ρ ( E ) . Let
K ⊂ R be a Borel subset. The following function will be important:(2.9) g K ( E ) = ( πN ′ ( E ) , E ∈ K , E ∈ R \K . Note that g K ( E ) is well defined (Lebesgue) almost everywhere on K∩ Σ α,v . As a consequence,the operator g K ( H x ) is well defined as long as H x has purely absolutely continuous spectrumon K .For a (Borel) subset K ⊂ R , denote by K ( x ) the indicator function of K . If H isa self-adjoint operator on ℓ ( Z ), denote by H ( K ) the restriction of H into the subspace Ran K ( H ) ⊂ ℓ ( Z ). Here, K ( H ) is considered in the standard sense of functional calculusfor self-adjoint operators. For the current operator A defined in (1.5), let A ( x, K ) := K ( H x ) A K ( H x ) . In the case of A ( x, K ), it is convenient not to restrict it into Ran K ( H x ) and instead let ithave a zero block.We are ready to formulate the first (local) main result of the paper. Theorem 2.1.
Let { H x } x ∈ T d be a quasiperiodic operator family with v ∈ C s ( T d ; R ) , s > d .Denote by R the set of energies on which the corresponding Schr¨odinger cocycle is C s -reducible. Then H x has local ballistic transport on R . In other words, there exists a repre-sentation R = ∪ j K j such that the following limit exist for all x ∈ T and all K j : Q ( x, K j ) = s–lim T → + ∞ T Z T e itH x A ( x, K j ) e − itH x dt = g K j ( H x ) . As a consequence, ker Q ( x, K j ) = (Ran K j ( H x )) ⊥ . Theorem 2.1 is “soft” and requires very little regularity. As a consequence, we can only getlocal bounds. Still, as stated in the Introduction, even these bounds imply ballistic motionsuch as in [54]. If R hasIf one has some control over the dependence of k B k C s in the E variable, the result canbe improved to “true” strong ballistic transport. Unfortunately, there is no hope in gettingany kind of estimates that are uniform in energy, since regularity of the reducibility matrixdepends on Diophantine properties of the rotation number (see, for example, [27]). However,we can formulate a sufficient integral-type condition. We will say that ( α, S v − E ) is C s -reducible in expectation on K if it’s C s -reducible for every E ∈ K , and there exists a choiceof L ( T d )-normalized conjugations B ( E ; x ) such that(2.10) Z K k B ( E ; · ) k s ( T d ) dρ ( E ) < + ∞ . We can now formulate the second (global) main result.
Theorem 2.2.
Let { H x } x ∈ T d be a quasiperiodic operator family whose cocycles are is C s -reducible in expectation on K for some s > d/ . Then the family { H x } x ∈ T d has strongballistic transport on K . In other words, the following limit exists for all x ∈ T : Q ( x, K ) = s–lim T → + ∞ T Z T e itH x A ( x, K ) e − itH x dt = g K ( H x ) . As a consequence, ker Q ( x, K ) = (Ran K ( H x )) ⊥ . We will also state a version of Theorem 2.2 in terms of the localization property of thedual operator ( L θ ψ )( n ) = X m ∈ Z d ˆ v n − m ψ ( m ) + 2 cos 2 π ( n · α + θ ) ψ ( n ) , n ∈ Z d . We will say that the family { L θ } θ ∈ T has s -power law dynamical localization (sPDL) on K , ifthe spectra of L θ ( K ) are purely point for almost every θ ∈ T , and there are C > s > Z T |h δ m , K ( L θ ) e itL θ δ n i| dθ ≤ C (1 + | m − n | ) s . The following is a corollary of the proof of Theorem 2.2.
Corollary 2.3.
Let { H x } x ∈ T d be a quasiperiodic operator family whose cocycles are C -reducible on K , and the dual family { L θ } θ ∈ T satisfies s -power law dynamical localization on K with some s > d . Then the family { H x } x ∈ T d has strong ballistic transport on K . The assumptions of Theorems 2.1 and/or 2.2 are satisfied for several different classes ofoperators. In order to formulate some of them, recall that a frequency vector α ∈ R d iscalled Diophantine (denoted α ∈ DC d ( γ, τ ) for some γ > , τ > d −
1) if(2.11) dist( k · α, Z ) ≥ γ | k | − τ , ∀ k ∈ Z d \{ } . We will use the notation DC d = [ γ> τ>d − DC( γ, τ ) . In the one-frequency case α ∈ R \ Q , denote also β ( α ) = lim sup k →∞ ln q k +1 q k , where p k q k → α are the continued fraction approximants. Note that α ∈ DC implies β ( α ) = 0,but not vice versa. Remark 2.4.
Condition (2.10) is formulated in terms of C s -norms in order to avoid over-loading this section with terminology. In fact, in the proof we will use (weaker) SobolevH s -norms, since they behave better under some convolution-type operations appearing inthe process.2.4. Applications of Theorems 2.1 and 2.2.
As stated earlier, Theorem 2.1 falls in themiddle between ballistic motion and strong ballistic transport. Its advantage is that it isapplicable in a wide range of situations.(1) Let v ∈ C ω ( T ; R ) be an analytic one-frequency potential, and β ( α ) = 0. Thenthere exists a Borel subset Σ ⊂ R such that Σ supports the absolutely continuouscomponents of the spectral measures of H x for all x ∈ T , and the correspondingSchr¨odinger cocycle S E − v is analytically rotations reducible for all E ∈ Σ due to[10, Theorem 1.2]. Since β ( α ) = 0, by solving the cohomological equation, one canimprove rotations reducibility to reducibility for all E ∈ Σ. Thus, Theorem 2.1applies. One can state its conclusion in the following way: if { H x } x ∈ T is an analyticone-frequency quasiperiodic operator family with β ( α ) = 0 and Σ does not supportsingular spectral measures of H x , then H x has local ballistic transport and, as aconsequence, has ballistic motion on Σ.(2) In [53], some of the results [10, 11] were extended to the case of finitely smoothcocycles. As a consequence, the results from the previous case also extend to finitelydifferentiable potentials. (3) In [12], it was shown that almost Mathieu operators with potentials v ( x ) = 2 λ cos(2 πx )with log λ < − β ( α ) satisfy full measure analytic reducibility. As a consequence, theyalso satisfy local ballistic transport (and hence ballistic motion) on the whole spec-trum. The corresponding quantitative localization results for the dual operator exist[34–36] exist, but are very delicate.Let us now discuss applications of the more precise Theorem 2.2 and Corollary 2.3.(1) In [2, 22] it was shown that if α ∈ DC d , v ∈ C ω ( T d ; R ), and 0 < ε < ε ( α, v ), then theoperators H x with the potential εv have purely absolutely continuous spectra (see also[54]), and their Schrodinger cocycles are reducible on a set of energies of full spectralmeasure. In [27], exponential dynamical localization in expectation (which is strongerthat sPDL for all s ) was established for the corresponding dual operators. Therefore,Corollary 2.3 applies. The proof of [27] is based on quantitative reducibility estimates.It can be checked that these estimates, actually, guarantee convergence of the integral(2.10) (within a significant margin), and therefore one can also apply Theorem 2.2directly. Therefore, in the setting of [54], we actually have strong ballistic transporton the whole spectrum, rather than just ballistic motion.(2) A combination of [10] and [7] implies that, if v ∈ C ω ( T ; R ), β ( α ) < + ∞ , and 0 < ε <ε ( v, β ( α )), then the operators { H x } x ∈ T with potentials εv have purely absolutelycontinuous spectrum, and the corresponding Schr¨odinger cocycles are reducible foralmost every energy. Exponential dynamical localization for the dual operators hasbeen established in [31] (as stated, only for the almost Mathieu operator, but theargument easily extends to the general long range case, since it relies on [7, Theorem5.1] which is established for the long range case; see also [27] for the Diophantinecase). Therefore, again, Corollary 2.3 implies strong ballistic transport on the whole ℓ ( Z ). Note that, for β = 0, it gives a non-perturbative version of the result of [54],also with strong ballistic transport.3. On reducibility and localization
In this section, we will refine some of the results from [37] in order to extend them to alocal quantitative setting. For a function f ∈ L ( T d ), denote the Sobolev norm by k f k s ( T d ) = X m ∈ Z d (1 + | m | ) s | ˆ f ( m ) | , where ˆ f ( m ) are the Fourier coefficients: f ( x ) = X m ∈ Z d ˆ f ( m ) e πm · x . We will only consider s > d/
2, in which case H s is embedded into C ( T d ) and its elementsare ordinary continuous functions, rather than equivalence classes. Coincidentally, the samecondition is sufficient for H s being an algebra with respect to the pointwise multiplication,which will also be important; see, for example [1, Theorem 4.39]. Proposition 3.1.
Let s > d/ and f, g ∈ H s ( T d ) . Then f g ∈ H s ( T d ) , and k f g k H s ( T d ) ≤ C ( d, s ) k f k H s ( T d ) k g k H s ( T d ) . Let v ∈ C( T d , R ), and α ∈ R d such that { , α , . . . , α d } are independent over Q . Considerthe following quasiperiodic Schr¨odinger cocycle ( α, S E − v ), where S E − v ( x ) = (cid:18) E − v ( x ) −
11 0 (cid:19) . Adapting the definition from Section 1, we will say that ( α, S E − v ) is H s -reducible if thereexists B ∈ H s ( T d ; GL(2 , C )) such that(3.1) B ( x + α ) − S E − v ( x ) B ( x ) = (cid:18) e πiρ ( E ) e − πiρ ( E ) (cid:19) , ∀ x ∈ T d , where ρ ( E ) is the fibered rotation number of ( α, S E − v ), as defined in Section 1. As aconsequence, deg B = 0. Definition 3.2.
Let
K ⊂ R be a Borel subset. We will say that ( α, S E − v ) is H s -reducibleon K if there exists and an L -normalized balanced family of conjugating matrix functions { B ( E ) } E ∈K , satisfying (3.1) for all E ∈ K , and the following bound:(3.2) Z K k B ( E, · ) k s ( T d ) dρ ( E ) < + ∞ At this moment, we also do not assume any regularity of B E in the variable E . Forexample, B ( E ) itself may not be measurable in E , as long as there is an upper norm boundby a measurable function satisfying (3.2). However, one can obtain the following:(1) Assuming that B ( E ) with above properties exist on K , one can pick a measurableparametrization of B ( E ) in E .(2) As in Section 1, let R be the set of energies such that ( α, S E − v ) is C s -reducible. Then,for a given c >
0, the set of E ∈ R such there exists B ( E ) satisfying (3.1) with, say, k B ( E ) k C s ( T d ;SL(2 , R )) ≤ c , is measurable.Claims (1) and (2) can be obtained from the following fact: selecting a B satisfying (3.1)is the same as selecting two linearly independent Bloch wave solutions of the Schr¨odingerequation which, in turn, are completely determined by their initial data. These solutionsdetermine the values of B ( x ) on a dense subset of x , and therefore contain all informationon the regularity of the corresponding Bloch functions, as well as the matrix elements of B (as long as the latter are continuous). One can also independently obtain measurabilityfor almost every E (which is equally good in our case) from the duality arguments below,similarly to [37].Recall that the rotation number of the Schr¨odinger cocycle ( α, S E − v ) is a continuous non-increasing map ρ : R → [0 , / α,v onto [0 , / N ( E ) = 1 − ρ ( E )implies that the pre-image of the Lebesgue measure on [0 , /
2] under ρ is half of the density ofstates measure. For θ ∈ [0 , / r ( Z + α · Z d ), denote by E ( θ ) the unique value E ∈ Σ α,v suchthat ρ ( E ) = θ (note that the uniqueness is violated at the endpoints of spectral gaps, whichcorrespond to the removed values of θ ). Extend it as an even function into [ − / , R by 1-periodicity. Denote the resulting function, defined on R r ( Z + α · Z d ),by the same symbol E ( θ ). LetΘ = ( ρ ( K ) ∪ ( − ρ ( K ))) r ( Z / α · Z d / . Then E : Θ → Σ α,v is a measurable map which takes each of its values twice, and whoserange is equal to K except, at most, for a countable subset. Note that we only needed toremove Z + α · Z d for the above argument. However, the further construction will requireremoval of half- α -rational frequencies.Let us recall the definition of the dual operator family.(3.3) ( L θ ψ )( n ) = X m ∈ Z d ˆ v n − m ψ ( m ) + 2 cos 2 π ( n · α + θ ) ψ ( n ) , θ ∈ T . In order to formulate the main result of this section, introduce the translation operator: T a : ℓ ( Z d ) → ℓ ( Z d ) , ( T a ψ )( n ) := ψ ( n + a ) . An important property of the eigenvectors of the operators (3.3) is the following covariancerelation. Suppose that L θ ψ = Eψ , ψ ∈ ℓ ( Z d ) Then(3.4) L θ + ℓ · α T ℓ ψ ( θ ) = E ( θ ) T ℓ ψ ( θ ) , ∀ ℓ ∈ Z d . As a consequence, if one wants to study localization properties of the family { L θ } θ ∈ T , itmay be beneficial to pick only one representative from each “equivalence class” defined by(3.4). There are obvious difficulties with this approach, as it dangerously resembles theprocedure of constructing a non-measurable subset of the circle. However, in our setting it ispossible and is discussed, for example, in [37]. The main result of this section is the followingrefinement of [37]. Theorem 3.3.
Suppose that the family of Schr¨odinger cocycles ( α, S E − v ) is H s -reducible on K ⊂ R with sone s > d/ . Construct the subset Θ ⊂ [0 , and the function E ( · ) as above.Then there exists a measurable function f : Θ × T d → C , such that the following claims hold. (1) For each θ ∈ Θ , k f ( θ, · ) k L ( T d ) = 1 , and Z Θ k f ( θ, · ) k s ( T d ) dθ < + ∞ . (2) For each θ ∈ Θ , the vector ψ ( θ ; m ) = ˆ f ( θ, m ) ( the Fourier transform in the secondvariable ) is an eigenvector of the dual operator: L θ ψ ( θ ) = E ( θ ) ψ ( θ ) . (3) For ℓ ∈ Z d such that θ − ℓ · α ∈ Θ , construct additional eigenvectors of L θ by ψ ℓ ( θ ) = T ℓ ψ ( θ − ℓ · α ) , so that, using (3.4) , we have L θ + ℓ · α ψ ℓ ( θ ) = E ( θ ) ψ ℓ ( θ ) . Then, for almost every θ ∈ T , the spectrum of L θ ( K ) is purely point, and the con-structed eigenfunctions (3.5) { ψ ℓ ( θ ) : θ − ℓ · α ∈ Θ } , form a complete system for L θ ( K ) . (4) Denote by ψ ∗ ( θ ) the following convolution vector: ψ ∗ ( θ ; p ) = X m ∈ Z d | ψ ( θ ; m ) ψ ( θ ; m + p ) | . Then the following Sobolev localization property holds: (3.6) X p (1 + | p | ) s Z Θ | ψ ⋆ ( θ ; p ) | dθ < + ∞ . Proof.
Most of the the argument is very close to [37]. See also a similar argument in [26,Section 3]. Let E ( θ ) be constructed as above. The arguments of [37] imply that one cantake f ( x, θ ) = B ( x, E ( θ )) k B ( x, E ( θ )) k L ( T d ) for θ ∈ Θ ∩ [0 , /
2] and extend it by the relation f ( x, − θ ) = f ( x, θ ) into Θ. Then, for each θ ∈ Θ, ψ ( θ ; n ) = ˆ f ( θ ; n ) would be an ℓ ( Z d )-normalized eigenfunction of L θ :(3.7) L θ ψ ( θ ) = E ( θ ) ψ ( θ ) , which implies the first two claims. Let us establish completeness. Again, the argument issimilar to [37]: we calculate the “partial density of states measure”, using the expression(2.7) with the spectral projection of L θ replaced by the projection onto the subspace spannedby eigenfunctions (3.5). If that measure coincides with the complete IDS, this would indicatecompleteness of the eigenfunctions (for almost every θ ). The calculation is straightforwardif we assume ψ ( θ ) to be measurable. One can recover measurability from that of B ( E ), butthere is also a more direct argument as follows.Let P ℓ ( θ ) be the spectral projection of L θ onto the eigenspace corresponding to the eigen-value E ( θ − ℓ · α ), for θ − ℓ · α ∈ Θ. The above construction implies P ℓ ( θ ) = 0 for θ ∈ Θ + ℓ · α .Let ϕ ( θ ) be a measurable branch of an element from P ( θ ), k ϕ ( θ ) k = 1. Eventually, we willshow that ϕ ( θ ) = c ( θ ) ψ ( θ ) for almost every θ . However, at this point we cannot state thatthe spectrum of L θ is simple. Fortunately, for the following calculations ϕ ( θ ) is just as goodas ψ ( θ ). Denote ϕ ℓ ( θ ) := T ℓ ϕ ( θ − ℓ · α ) . Then, by covariance, we have the following eigenvalue equation similar to (3.7). L θ ϕ ℓ ( θ ) = E ( θ − ℓα ) ϕ ℓ ( θ ) , if θ − ℓ · α ∈ Θ . As a consequence, we have ϕ ℓ ( θ ) ∈ Ran P ℓ ( θ ), and X ℓ Z T h P ℓ ( θ ) δ , δ i Θ ( θ − ℓ · α ) dθ ≥ X ℓ Z T |h ϕ ℓ ( θ ) , δ i| Θ ( θ − ℓ · α ) dθ = X ℓ Z T |h ϕ ( θ − ℓ · α ) , δ − ℓ i| Θ ( θ − ℓ · α ) dθ = X ℓ Z T |h ϕ ( θ ) , δ − ℓ i| Θ ( θ ) dθ = | Θ | = N ( K ) . Since the left hand side cannot be larger than | N ( K ) | , all inequalities are actually equal-ities, which also implies simplicity of the spectrum for almost every θ . Since measurableparametrization of eigenvectors was obtained independently of measurability of B ( E ) andthat the eigenvalues of L θ are simple on Θ, this gives us measurability of B ( E ) in retrospec-tive.It remains to establish Claim (4). We will obtain it as a consequence of Claim 1. Let f ( θ, x ) = X m ∈ Z d | ψ ( θ ; − m ) | e πim · x , f ( θ, x ) = X m ∈ Z d | ψ ( θ ; m ) | e πim · x . Clearly, we have k f ( θ, · ) k H s ( T d ) = k f ( θ, · ) k H s ( T d ) = k f ( θ, · ) k H s ( T d ) . Then one can express ψ ∗ as a convolution: ψ ∗ ( θ ; p ) = ( ˆ f ( θ, · ) ∗ ˆ f ( θ, · ))( p ) , and hence, by definition of the Sobolev norm and Proposition 3.1, we have X p (1 + | p | ) s | ψ ∗ ( θ ; p ) | = k f ( θ, · ) f ( θ, · ) k s ≤ k f ( θ, · ) k s . One can now get Claim (4) by integrating in θ . (cid:3) We will also need a Sobolev version of the dynamical localization.
Theorem 3.4.
Under the assumptions of Theorem . , there exists h ∈ H s ( T d ) such that Z T |h δ p , K ( L θ ) e − itL θ δ q i| dθ = ˆ h ( q − p ) . Proof.
We have, using the notation of the previous theorem,(3.8) h pq := Z T |h δ p , K ( L θ ) e − itL θ δ q i| dθ ≤ X ℓ ∈ Z d Z Θ+ ℓ · α | ψ ℓ ( θ, p ) ψ ℓ ( θ, q ) | dθ = X ℓ ∈ Z d Z Θ+ ℓ · α | ψ ( θ − ℓ · α, p + ℓ ) ψ ( θ − ℓ · α, q + ℓ ) | dθ = X ℓ ∈ Z d Z Θ | ψ ( θ, p + ℓ ) ψ ( θ, q + ℓ ) | dθ ≤ (cid:18)Z Θ | ψ ∗ ( θ, q − p ) | dθ (cid:19) / . Due to covariance, h pq only depends on q − p . We have the following thanks to the last claimin Theorem 3.3: X p ∈ Z d (1 + | p | ) s | h p | ≤ Z Θ X p ∈ Z d (1 + | p | ) s | ψ ∗ ( θ ; p ) | dθ ≤ Z Θ k f ( θ ; · ) k s . (cid:3) Suppose that, instead of a Sobolev bound, we have a uniform boundsup θ ∈ Θ k f ( θ ; · ) k C s ( T d ) < + ∞ . In this case, the dual operator family demonstrates an extremely strong form of uniformlocalization, which would allow us, ultimately, to relax regularity requirements on the re-ducibility.
Lemma 3.5.
Suppose that, in the notation of Theorem . , we have sup θ ∈ Θ k f ( θ ; · ) k C s ( T d ) =: M < + ∞ . Then, for almost every θ ∈ T , we have the following uniform dynamical localization bound: ess–sup θ ∈ T |h δ p , K e itL θ δ q i| < C ( s, M )(1 + | p − q | ) s − d . Proof.
Using the representation from Theorem 3.3, we have | ψ ℓ ( θ ; q ) | = | ˆ f ( θ − ℓ · α ; q + ℓ ) | ≤ C ( M )(1 + | q + ℓ | ) s . The rest follows from Lemma 6.1. (cid:3) From localization to strong ballistic transport
In this section, we will prove Theorem 2.2 by studying the consequences of the resultsfrom the previous section to the operator (1.18):( H x ψ )( n ) = ψ ( n −
1) + ψ ( n + 1) + v ( x + nα ) ψ ( n ) , n ∈ Z . In order to formulate the main result, we will need to introduce the dual operator family.Define the Fourier coefficients of v by ˆ v n , where v ( x ) = X n ∈ Z d ˆ v n e πin · x . Let { L θ } θ ∈ T be the dual family on ℓ ( Z d ):(4.1) ( L θ ψ )( n ) = X m ∈ Z d ˆ v n − m ψ ( m ) + 2 cos 2 π ( n · α + θ ) ψ ( n ) . As stated in the introduction, denote by A the current operator on ℓ ( Z ):( Aψ )( n ) = i ( ψ ( n + 1) − ψ ( n − . For a Borel subset
K ⊂ R , we defined A ( x, K ) = K ( H x ) A K ( H x ) . Recall also that, by definition, H x ( K ) = K ( H x ) | Ran K ( H x ) . It is convenient to assume that A ( x, K ) acts on the whole ℓ ( Z ) and H x ( K ) is restricted toRan K ( H x ), since, in the latter case, the wording “ σ ( H x ( K )) is purely absolutely continuous”has intended meaning and does not need to account for the large kernel of Ran K ( H x ) ⊥ .Recall the definition of the function g K ( E ) : g K ( E ) = ( πN ′ ( E ) , E ∈ K , E ∈ R \K . Definition 4.1.
An analytic quasiperiodic operator family { H x } x ∈ T d will called K - regular ifthe following properties are satisfied:(1) The spectra of H x ( K ) are purely absolutely continuous.(2) The families K ( H x ) and g K ( H x ) are strongly continuous in the parameter x ∈ T d .The results of [17] imply that, under the above assumptions, k g K ( H x ) k ≤ Theorem 4.2.
Let { H x ( K ) } x ∈ T d be a K -regular family such that the dual operator family { L θ } θ ∈ T satisfies H s -localization in expectation on K for some s > d/ , or s -uniform power law localization in expectation on K for some s > d . Then the conclusion of Theorem . holds. In other words, for every x ∈ T d the limit Q ( x, K ) = s–lim T → + ∞ T Z T e itH x K ( H x ) A K ( H x ) e − itH x dt, exists and ker Q ( x, K ) = (Ran K ( H x )) ⊥ . Remark 4.3.
In Section 6, we obtain K -regularity as a consequence of local C rotationsreducibility for the corresponding Schr¨odinger cocycles. Therefore, it holds in all consideredcases.In order to prove Theorem 4.2, we will need several additional calculations with dualityinvolving direct integrals. Each of the families { H x } x ∈ T d and { L θ } θ ∈ T can be considered asa single operator in the appropriate direct integral space: H := Z ⊕ T d ℓ ( Z ) dx, e H = Z ⊕ T ℓ ( Z d ) dθ. Denote the unitary duality operator U : H → e H on functions Ψ = Ψ( x, n ) by(4.2) ( U Ψ)( θ, m ) = e Ψ( θ + m · α, m ) , where e Ψ denotes the Fourier transform in both discrete and continuous variables:(4.3) e Ψ( θ, m ) = X n ∈ Z Z T d e πinθ − πim · x Ψ( x, n ) dx. In the notation, we will always write the continuous variables before discrete variables inthe arguments of functions, even when they roles are switched under duality. As mentionedabove, the operator families { H x } x ∈ T d and { L θ } θ ∈ T can be represented by direct integrals H := Z ⊕ T d H x dx, L := Z ⊕ T L θ dθ. Aubry duality (see, for example, [24]) can be formulated as the unitary equivalence of theabove direct integrals:(4.4)
U HU − = L . One can apply duality to other operators and operator families on ℓ ( Z ). For example,the operator family corresponding to the operator A (constant in x ) has the following dualfamily: U (cid:18)Z ⊕ T d A dx (cid:19) U − = Z ⊕ T e A ( θ ) dθ, where(4.5) ( e A ( θ ) ψ )( m ) = 2 sin 2 π ( m · α + θ ) ψ ( m ) , m ∈ Z d . Note that an x -independent family may become θ -dependent after the duality transforma-tion, and vice versa. For any (Borel) function f , we have(4.6) U f ( H ) U − = U f (cid:18)Z ⊕ T H x dx (cid:19) U − = U (cid:18)Z ⊕ T f ( H x ) dx (cid:19) U − = Z ⊕ T d f ( L θ ) dθ = f ( L ) . For a Borel subset K , denote e A ( θ, K ) = K ( L θ ) e A ( θ ) K ( L θ ) . Then, one can check that e A ( θ, K ) is dual to A ( x, K ): U T Z T (cid:18)Z ⊕ T d e iH x t A ( x, K ) e − iH x t dx (cid:19) dt U − = 1 T Z T (cid:18)Z ⊕ T e iL θ t e A ( θ, K ) e − iL θ t dθ (cid:19) dt. The following proposition is, essentially, established in [42] for the case K = σ ( H x ) in aslightly different form. We include most of the proof for the convenience of the reader. Proposition 4.4.
Under the assumptions of Theorem . , denote by E k ( θ ) , ψ k ( θ ) the eigen-values and eigenfunctions of L θ ( K ) ( the exact choice of parametrization does not matter ) .Then, for almost every θ ∈ T , the following limit e Q ( θ, K ) := s–lim T → + ∞ T Z T e itL θ e A ( θ, K ) e − itL θ dt exists and is a diagonal operator in the representation of eigenvectors of L θ ( K ) . More pre-cisely, (4.7) e Q ( θ, K ) ψ k ( θ ) = 1 πN ′ ( E k ( θ )) ψ k ( θ ) . As a consequence, for almost every θ ∈ T we have e Q ( θ, K ) = g K ( L θ ) . Proof.
We only sketch the main ideas, since most of the argument is contained in [42]. Theexistence of the limit and the fact that it is diagonal in the basis of eigenvectors of L θ followsfrom the following standard calculation:1 T Z T D e itL θ e A ( θ, K ) e − itL θ ψ k ( θ ) , ψ ℓ ( θ ) E dt = (cid:18) T Z T e it ( E ℓ ( θ ) − E k ( θ )) dt (cid:19) h e A ( θ ) ψ k ( θ ) , ψ ℓ ( θ ) i and the fact that (cid:12)(cid:12)(cid:12)(cid:12) T Z T e it ( E ℓ ( θ ) − E k ( θ )) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤
1; lim T → + ∞ T Z T e it ( E ℓ ( θ ) − E k ( θ )) dt = ( , E k ( θ ) = E ℓ ( θ );0 , E k ( θ ) = E ℓ ( θ ) . As a consequence, we obtain(4.8) h e Q ( θ, K ) ψ k ( θ ) , ψ k ( θ ) i = h e A ( θ ) ψ k ( θ ) , ψ k ( θ ) i = X m ∈ Z d π ( θ + m · α ) | ψ k ( θ, m ) | . In order to establish (4.7), consider the Fourier transforms of the eigenvectors of L θ ( K ): f k ( x, θ ) = X n ∈ Z d e πin · x ψ k ( θ, n ) , where ψ k ( θ, n ) is the n th component of ψ k ( θ ) (the latter is considered as a vector from ℓ ( Z )).If θ / ∈ Z + α · Z d , then (see Appendix C of [8], and also Remark 5.1 in [37])(4.9) d k ( θ ) := e πiθ f k ( x, θ ) f k ( x − α, θ ) − e − πiθ f k ( x, θ ) f k ( x − α, θ ) = 0 . By direct calculation and (4.8), we also have(4.10) d k ( θ ) = h e Q ( θ, K ) ψ k ( θ ) , ψ k ( θ ) i , which implies that ker e Q = { } . Let S E − v ( x ) := (cid:18) E − v ( x ) −
11 0 (cid:19) be the Schr¨odinger cocycle, and consider a matrix function B ( x, θ ) defined by B ( x, θ ) := 1 | d k ( θ ) | / (cid:18) f k ( x, θ ) f k ( x, θ ) e − πiθ f k ( x − α, θ ) e πiθ f k ( x − α, θ ) (cid:19) ;note that the matrix is invertible since d k ( θ ) = 0. Then B ( x + α, θ ) − S E − v ( x ) B ( x, θ ) = (cid:18) e πiθ e − πiθ (cid:19) . Kotani’s theory (see the argument in [42] with additional references) implies that there existsa subset
E ⊂ K of full Lebesgue measure (as a consequence, full spectral measure for each H x ( K )) such that, if E k ( θ ) is constructed above and E k ( θ ) ∈ E , then d k ( θ ) = 1 πN ′ ( E k ( θ )) . Comparing the last equality with (4.10), both of which hold for almost every θ ∈ T , wecomplete the proof. Note that the function g is only defined Lebesgue almost everywhere on K . However, for almost every θ ∈ T , all eigenvalues E k ( θ ) will be at differentiability pointsof N , and hence g ( L θ ) will be well defined. As a consequence, using (4.6), we have Q ( K ) := Z ⊕ T d Q ( x, K ) dx = Z ⊕ T d g K ( H x ) dx (4.11) = U − (cid:18)Z ⊕ T e Q ( θ, K ) dθ (cid:19) U = U − (cid:18)Z ⊕ T g K ( L θ ) dx (cid:19) U := U − e Q ( K ) U . (cid:3) Denote by Q ( x, T, K ) the pre-limit expression:(4.12) Q ( x, T, K ) := 1 T Z T e itH x A ( x, K ) e − itH x dt, Q ( T, K ) = Z ⊕ T d Q ( x, T, K ) dx. We would like to show the following, for all p ∈ Z :(4.13) Q ( x, T, K ) δ p → Q ( x, K ) δ p , where { δ p } p ∈ Z denote the standard basis vectors in ℓ ( Z ). Let f p,T ( x ) = Q ( x, T, K ) δ p , f p ( x ) = Q ( x, K ) δ p . Denote by f p,T ( x, n ) and f p ( x, n ) the n -th components of f p,T ( x ), f p ( x ) respectively, where n ∈ Z . One can treat f p,T and f p as elements of L ( T d × Z ). Denote also by e f p,T ( θ, m ), e f p ( θ, m ) the Fourier transforms of f p,T , f p in both variables defined as in (4.3): e f p,T ( θ, m ) = X n ∈ Z Z T d e πinθ − πim · x f p,T ( x, n ) dx. Lemma 4.5.
For any p ∈ Z , x ∈ T d , and T > , we have k f p,T ( x ) − f p ( x ) k ℓ ( Z ) ≤ Z T X m ∈ Z d | e f p,T ( θ, m ) − e f p ( θ, m ) | ! dθ. (4.14) Proof.
First, let us note that both x f p,T ( x ) and x f p ( x ) are continuous as maps from T d to ℓ ( Z ). In particular, they are continuous component-wise. Denote by ˆ f p,T ( x, θ ) theFourier transform only in the variable n , and same for ˆ f p ( x, θ ). Using the Parseval’s identity,continuity in x , and ℓ bound for the Fourier transform, we have the following:(4.15)sup x k f p,T ( x ) − f p ( x ) k ℓ ( Z ) = sup x X n ∈ Z | f p,T ( x, n ) − f p ( x, n ) | = sup x Z T | ˆ f p,T ( x, θ ) − ˆ f p ( x, θ ) | dθ ≤ Z T (cid:18) ess–sup x | ˆ f p,T ( x, θ ) − ˆ f p ( x, θ ) | (cid:19) dθ ≤ Z T X m ∈ Z d | e f p,T ( θ, m ) − e f p ( θ, m ) | ! dθ. (cid:3) Remark 4.6.
It is crucial that the left hand side of (4.14) is continuous in x , otherwise wewould not have been able to obtain convergence for all x ∈ T d , as the right hand side of(4.14) does not allow to recover any data about measure zero subsets of T d in the variable x . The said continuity, ultimately, reduces to the assumption of K -regularity of the family { H x } x ∈ T . Remark 4.7.
Let U be the duality operator. Then e f p,T ( θ, m ) = ( U f p,T )( θ − m · α, m ) , e f p ( θ, m ) = ( U f p )( θ − m · α, m ) . Therefore, in order to show (4.13), we can apply Lemma 4.5 and reduce it to a convergencestatement about the images of f p,T under duality.We will use the following notation for the dual pre-limit expressions: e Q ( θ, T, K ) := 1 T Z T e itL θ e A ( θ, K ) e − itL θ dt, e Q ( T, K ) = Z ⊕ T e Q ( θ, T, K ) dθ. Finally, consider δ p as an element of L ( T d × Z ) that is a constant function in the x variable.Then, it’s Fourier transform in both variables is equal to( e δ p )( θ, m ) = ( U δ p )( θ − m · α, m ) = e πipθ δ ( m ) , which implies e f p,T ( θ, m ) = ( U Q ( T, K ) δ p )( θ − m · α, m ) = ( e Q ( T, K ) U δ p )( θ − m · α, m ) = e πipm · α e Q ( θ − m · α, T, K ) δ , and similarly e f p ( θ, m ) = ( U Q ( K ) δ p )( θ − m · α, m ) = ( e Q ( K ) U δ p )( θ − m · α, m ) = e πipm · α e Q ( θ − m · α, K ) δ . As a consequence, we can rewrite the conclusion of Lemma 4.5 as(4.16) sup x k f p,T ( x ) − f p ( x ) k ℓ ( Z ) ≤ Z T X m ∈ Z d (cid:12)(cid:12)(cid:12) e f p,T ( θ, m ) − e f p ( θ, m ) (cid:12)(cid:12)(cid:12)! dθ / = Z T X m ∈ Z d (cid:12)(cid:12)(cid:12)(cid:16) e Q ( θ − m · α, T, K ) δ − e Q ( θ − m · α, K ) δ (cid:17) ( m ) (cid:12)(cid:12)(cid:12)! dθ / ≤ X m ∈ Z d (cid:18)Z T (cid:12)(cid:12)(cid:12)(cid:16) e Q ( θ − m · α, T, K ) δ − e Q ( θ − m · α, K ) δ (cid:17) ( m ) (cid:12)(cid:12)(cid:12) dθ (cid:19) / = X m ∈ Z d (cid:13)(cid:13)(cid:13)D δ m , e Q ( · , T, K ) δ − e Q ( · , K ) δ E(cid:13)(cid:13)(cid:13) L ( T ) = (cid:13)(cid:13)(cid:13) e Q ( T, K ) δ − e Q ( K ) δ (cid:13)(cid:13)(cid:13) ℓ ( Z d ;L ( T )) . The factor e πipm · α was absorbed into the absolute value, and the second inequality is thetriangle inequality. Let ( P N Ψ)( θ, m ) = ( Ψ( θ, m ) , | m | ≤ N , | m | > N be the projection onto a neighborhood of the origin in discrete Z d variable. The following isthe main technical estimate of this section that uses the localization bounds. Lemma 4.8.
Suppose that the family { L θ } θ ∈ T satisfies Sobolev localization on K in the senseof Theorem . with s > d/ . Then the norms kQ ( T, K ) δ k ℓ ( Z d ; ℓ ( T )) are bounded uniformly in T . As a consequence, k (1 − P N ) e Q ( T, K ) δ k ℓ ( Z d ;L ( T )) ≤ c ( N ) , where c ( N ) → as N → + ∞ , uniformly in T .Proof. First, let us replace e Q ( θ, T, K ) by the non-averaged expression e itL θ e A ( θ, K ) e − itL θ (thusproving a stronger inequality). As a consequence, we would like to estimate(4.17) X n ∈ Z d (cid:18)Z T (cid:12)(cid:12)(cid:12) h δ n , e itL θ e A ( θ, K ) e − itL θ δ i (cid:12)(cid:12)(cid:12) dθ (cid:19) ≤ X n ∈ Z d (cid:18)Z T (cid:12)(cid:12)(cid:12) h δ n , e itL θ e A ( θ, K ) e − itL θ δ i (cid:12)(cid:12)(cid:12) dθ (cid:19) ≤ X n ∈ Z d X k ∈ Z d Z T (cid:12)(cid:12)(cid:12) h K ( L θ ) e − iL θ t δ n , δ k ih e A ( θ ) δ k , K ( L θ ) e − iL θ t δ i (cid:12)(cid:12)(cid:12) dθ ! ≤ X n ∈ Z d X k ∈ Z d Z T (cid:12)(cid:12) h K ( L θ ) e − iL θ t δ n , δ k ih δ k , K ( L θ ) e − iL θ t δ i (cid:12)(cid:12) dθ ! , where in the second inequality we used the fact that the integrand is bounded by 2 in absolutevalue to replace L norm by L norm, and then used the fact that e A ( θ ) is a diagonal operatoracting on δ k as a scalar (we also transferred K ( L θ ) to e − iL θ t , and hence there is no more K in e A ( θ )). In the case of sPDL, we can continue the chain of inequalities as follows, usingLemma 6.1:(4.17) ≤ X n ∈ Z d X k ∈ Z d Z T (cid:12)(cid:12) h K ( L θ ) e − iL θ t δ n , δ k ih δ k , K ( L θ ) e − iL θ t δ i (cid:12)(cid:12) dθ ! / ≤ X n ∈ Z d X k ∈ Z d Z T |h K ( L θ ) e − iL θ t δ n , δ k i| / |h δ k , K ( L θ ) e − iL θ t δ i| / dθ ! / ≤ X n ∈ Z d X k ∈ Z d (cid:18)Z T |h K ( L θ ) e − iL θ t δ n , δ k i| dθ Z T |h δ k , K ( L θ ) e − iL θ t δ i| dθ (cid:19) / ! / . Let h n = Z T |h K ( L θ ) e − iL θ t δ , δ n i| dθ, n ∈ Z d . Then, by covariance, we have the following bound (recall that ∗ denotes the standard con-volution for functions on Z d ):(4.17) ≤ X n ∈ Z d : | n | >N (cid:0) h / ∗ h / (cid:1) / ( n ) . Therefore, in order to obtain decay, we need to verify (cid:0) h / ∗ h / (cid:1) / ∈ ℓ ( Z d ). We will needto use some bounds on weighted ℓ spaces with the norms k u k ℓ s = X n ∈ Z d (1 + | n | ) s | u ( n ) | . Their properties are summarized in the Appendix. Since h ∈ ℓ s ( Z d ), we have the followinginclusions, see also Appendix (“+” means the number has to be strictly larger): h / ∈ ℓ s/ − d/ ; h / ∗ h / ∈ ℓ s − d + ; w := ( h / ∗ h / ) / ∈ ℓ s/ − d/ ; { (1 + | n | ) s/ − d/ w ( n ) } n ∈ Z d ∈ ℓ ( Z d ) . In order to get w into ℓ ( Z d ), we can use H¨older inequality, for which it would be sufficientto have { (1 + | n | ) − ( s/ − d/ } n ∈ Z d ∈ ℓ ( Z d ) . This, ultimately, gives us the requirement s/ − d/ > d/
2, which reduces to s > d/ (cid:3) Corollary 4.9.
The conclusion of Lemma . also holds for e Q ( K ) .Proof. Recall that, being a direct integral, e Q ( T, K ) converges to e Q ( K ) in the strong operatortopology on L ( T × Z d ). As a consequence, there is a subsequence of time scales T k such that e Q ( T k , K ) δ converges to e Q ( K ) δ almost everywhere on T × Z d as k → ∞ (here, as before, δ is considered as an element of L ( T × Z d ) constant in θ ). Hence, the result follows fromFatou’s lemma. (cid:3) Conclusion of the proof of Theorem 4.2.
Since the operator norms of Q ( x, T, K ) areuniformly bounded, it suffices to show that Q ( x, T, K ) δ p → Q ( x, K ) δ p for all p ∈ Z . In otherwords, it is sufficient to show that the right hand side of (4.16) converges to zero. Take N ≫
1. Using the triangle inequality, Lemma 4.5, and Corollary 4.9, we have (cid:13)(cid:13)(cid:13) e Q ( T, K ) δ − e Q ( K ) δ (cid:13)(cid:13)(cid:13) ℓ ( Z d ;L ( T )) ≤ (cid:13)(cid:13)(cid:13) P N (cid:16) e Q ( T, K ) δ − e Q ( K ) δ (cid:17)(cid:13)(cid:13)(cid:13) ℓ ( Z d ;L ( T )) + (cid:13)(cid:13)(cid:13) (1 − P N ) (cid:16) e Q ( T, K ) δ − e Q ( K ) δ (cid:17)(cid:13)(cid:13)(cid:13) ℓ ( Z d ;L ( T )) ≤ (2 N ) d/ (cid:13)(cid:13)(cid:13) e Q ( T, K ) δ − e Q ( K ) δ (cid:13)(cid:13)(cid:13) L ( T × Z d ) + 2 c ( N ) , where in the last inequality we use the fact that ℓ ( Z d ; L ( T )) norm is bounded by theL ( T × Z d ) norm on the range of P N (with an appropriate constant). Now, since k e Q ( T, K ) δ − e Q ( K ) δ k L ( T × Z d ) →
0, the proof can be completed using the standard ε/ (cid:3) . The proof of Theorem 4.2 in the uniform case.
Suppose, instead of Theorem 3.4, wehave the conclusion of Lemma 3.5 with s > d . By covariance, we have h n − m = ess–sup θ ∈ T |h δ n , K ( L θ ) e itL θ δ m i| for some h which satisfies h ( n ) ≤ M (1 + | n | ) − s . Similarly to (4.17), we can estimate (cid:12)(cid:12)(cid:12) h δ n , e itL θ e A ( θ, K ) e − itL θ δ i (cid:12)(cid:12)(cid:12) ≤ ( h ∗ h )( n ) ≤ C ( M )(1 + | n | ) s − d . Therefore, the conclusion reduces to { (1 + | n | ) − (2 s − d ) } n ∈ Z d ∈ ℓ ( Z d ), which is satisfied for s > d .4.1. On the proofs of the main results.
The proof of Theorem 2.2 is complete, modulo K -regularity which will be established in the next section. To finish the proof of Theorem2.1, consider K j = { E ∈ K : k B ( E ) k C s ( T d ) ≤ j } and apply the uniform result on each K j , together with K -regularity. To prove Corollary2.3, follow the same lines as in the proof of Theorem 4.2, using Lemma 6.1 instead of theSobolev bounds. We did not try to optimize the condition s > d in this case.5. Regularity of the absolutely continuous spectral measures
This section is mostly expository. Let { H x } x ∈ T d be a quasiperiodic operator family (1.18):( H x ψ )( n ) = ψ ( n −
1) + ψ ( n + 1) + v ( x + nα ) ψ ( n ) , n ∈ Z . For a Borel subset
K ⊂ R , we will say that the family of Schr¨odinger cocycles ( α, S E − v )is C s -uniformly rotations reducible on K , if there exists c > B ( E ; · ) ∈ C s ( T d ; SL(2 , R )), E ∈ K , such that(5.1) B ( x + α, E ) − S E − v ( x ) B ( x, E ) = R θ ( x ) = (cid:18) cos 2 πθ ( x ) − sin 2 πθ ( x )sin 2 πθ ( x ) cos 2 πθ ( x ) (cid:19) . where θ ∈ C s ( T ; R ) and k B ( · , E ) k C s ( T d ;SL(2 , R )) ≤ c, ∀ E ∈ K . For a Borel subset F ⊂ R , denote by µ xpq ( F ) = h δ p , F ( H x ) δ q i a spectral measure of H x . Clearly, µ xpq = µ x + nαp + n,q + n . Hence, one simplify the computations byassuming p = 0.Moreover, since δ and δ form a cyclic subspace for H x , one can easily check the following(say, by repeatedly applying H x to δ or δ and eliminating previous elements by induction): δ n = p ( n − x ( H x ) δ + q ( n − x ( H x ) δ , where p ( n − x , q ( n − x are polynomials of degree ≤ n −
1, whose coefficients are C s -smooth in x . As a consequence, in order to establish smoothness of spectral measures (see below), itwould sufficient to consider µ x and µ x ( x ). Proposition 5.1.
Suppose that a family of Schr¨odinger cocycles ( α, S E − v ) is C -uniformlyrotations reducible on a Borel subset K ⊂ R . Then, for all x ∈ T d , and any Borel subset F ⊂ K , we have (5.2) µ x ( F ) = 12 π Z F (cid:0) b ( x, E ) + b ( x, E ) (cid:1) dE. (5.3) µ x ( F ) = 12 π Z F ( b ( x, E ) b ( x + α, E ) + b ( x, E ) b ( x + α, E )) dE. Proof.
Both claims follow from some standard calculations from the Kotani theory. We will,essentially, use the notation from [18]. For Im
E > x ∈ T d , denote by u ± ( x, E ) theunique solutions of the eigenvalue equation Hu = Eu satisfying u ± ( x, E ; 0) = 1 , u ± ( x, E ; n ) → n → ±∞ , and the m -functions m ± ( x, E ) = − u ± ( x, E ; ± . Using the eigenvalue equation, one can also obtain u − ( x, E ; 1) = m − ( x, E ) + E − v ( x ) . The Green’s function can be expressed through the above Jost solutions: G nm ( x, E ) = h δ n , ( H x − E ) − δ m i = − u − ( x, E ; n ) u + ( x, E ; m ) m + ( x, E ) + m − ( x, E ) + E − v ( x ) . As a consequence, (5.4) G ( x, E ) = − m + ( x, E ) + m − ( x, E ) + E − v ( x ) , G ( x, E ) = m + ( x, E ) m + ( x, E ) + m − ( x, E ) + E − v ( x ) . We can also extend m ± ( E, x ) into E ∈ R by considering limits m ± ( E + iε, x ) as ε → E for which L ( E ) = 0. In particular, they will exist almosteverywhere on K . The values of m ± ( E, x ) for E ∈ R can be calculated as follows. Anymatrix B ∈ SL(2 , R ) defines the following action on the upper half plane C + : B ◦ z = B z + B B z + B , z ∈ C + . Suppose that B ( · , E ) satisfies (5.1). Then, for almost every pair ( E, x ) ∈ K × T d , we have(5.5) m + ( x, E + i
0) = B ( x, E ) ◦ i = − m − ( x, E + i . The continuity arguments similar to [6] (see also [18, Footnote on page 10]) imply that (5.5)actually holds for all x ∈ T d and almost every E ∈ K , where “almost every” depends on x .However, in the following considerations zero measure sets will not be important, and henceone can use (5.5) as an alternative definition of m ± ( x, E + i m + ( x, E + i
0) = b i + b b i + b = b b + b b b + b + i b + b , where b ij = b ij ( x, E ) are the matrix elements of B ( x, E ). Note that (5.5) implies that thedenominators in (5.4) are purely imaginary for E ∈ K + i
0. Therefore, one can calculatedensities of spectral measures µ x , µ x as follows: dµ x dE = 1 π Im G ( x, E + i
0) = 12 π Im m + ( x, E + i
0) = b + b π .dµ x dE = 1 π Im G ( x, E + i
0) = − Re m + ( x, E + i π Im m + ( x, E + i
0) = − b b + b b π . (cid:3) We immediately obtain the following regularity claim.
Corollary 5.2.
Under the assumptions of Proposition . , the spectral measures µ xpq areabsolutely continuous on K with respect to the Lebesgue measure. Moreover, their densitiesare Lipschitz continuous in x : (cid:12)(cid:12)(cid:12)(cid:12) dµ xpq dE − dµ ypq dE (cid:12)(cid:12)(cid:12)(cid:12) ≤ C p − q | x − y | , where C p − q depends on p − q and the constant c from the uniform rotations reducibilityassumption. Theorem 5.3.
Suppose that the family { H x } x ∈ T d is C -uniformly rotations reducible on aBorel subset K ⊂ R . Let g ∈ L ∞ ( R ) , supp g ⊂ K . Then, for any x ∈ T d , we have s–lim x → x g ( H x ) = g ( H x ) . Proof.
Since g ( H x ) are uniformly bounded, it would be sufficient to show g ( H x ) δ n → g ( H x ) δ n strongly. By shifting the x variable, one can assume n = 0. Since g ( H x ) δ are also uniformlybounded in ℓ ( Z ), it is sufficient to establish the following:(5.6) h δ n , g ( H x ) δ i → h δ n , g ( H x ) δ i , ∀ n ∈ Z (5.7) k g ( H x ) δ k → k g ( H x ) δ k . To establish (5.6), note |h δ n , g ( H x ) δ i − h δ n , g ( H x ) δ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z g ( E ) dµ xn ( E ) − Z g ( E ) dµ x n ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k L ∞ c n | x − x ||K| , where c n is the constant from Corollary 5.2. Similarly, (5.7) can be established using thefact k g ( H x ) δ k = h δ , | g ( H x ) | δ i , and then repeating the earlier argument applied to the function | g | . (cid:3) Corollary 5.4.
Suppose that the conclusion of Theorem is satisfied for a fixed function g ∈ L ∞ ( R ) and a sequence of Borel subsets K ⊂ K . . . . Then it also satisfied for K = ∪ j K j .Proof. The statement follows from the Banach — Steinhaus theorem: indeed, the familyof operators { g ( H x ) } x ∈ T d is uniformly bounded, and the convergence can be verified on thedense set ∪ j Ran( K j ( H x )), by applying the previous theorem with K = K j . (cid:3) Appendix
In this section, we will establish some elementary bounds which will happen to be usefullater. All functional spaces denoted by ℓ with some indices will be on Z d . Denote by ℓ s thespace of Fourier transform of functions from H s ( T d ) with the following norm: k u k ℓ s = X n ∈ Z d (1 + | n | ) s | u ( n ) | . Recall the H¨older inequality: for u ∈ ℓ p , v ∈ ℓ q , we have(6.1) k uv k ℓ r ≤ k u k ℓ p k v k ℓ q , r = 1 p + 1 q , ≤ p, q, r ≤ ∞ . The following lemma is elementary:
Lemma 6.1.
Let a ∈ Z d and s + s − d > . X n ∈ Z d | a − n | ) s (1 + | n | ) s ≤ c ( s , s , s , d )(1 + | a | ) s + s − d . Proof.
We have X n ∈ Z d | a − n | ) s (1 + | n | ) s ≤ X | n |≤ a/ + X | n − a |≤ a/ + X | n | >a/ , | n − a | >a/ | a − n | ) s (1 + | n | ) s ≤ | a/ | ) s X | n |≤ a/ | n | ) s + 1(1 + | a/ | ) s X | n |≤ a/ | n | ) s + X | n |≥ a/ | n | / s + s ≤ c ( s , s , a )(1 + | a | ) d − s − s . (cid:3) Finally, recall that u ( p ) = (1 + | n | ) − s belongs to ℓ r for rs > d . We will need the following“square root” bound. Lemma 6.2.
Suppose that u ∈ ℓ s , and let v ( n ) = | u ( n ) | / . Then k v k ℓ r ≤ C ( r, s, d ) k u k ℓ s , ≤ r < s − d . Proof.
The condition u ∈ ℓ s is equivalent to { (1 + | n | ) s u ( n ) } n ∈ Z d ∈ ℓ , which is in turnequivalent to { (1 + | n | ) s/ v ( n ) } n ∈ Z d ∈ ℓ . We can multiply it by an appropriate power of(1 + | n | ) − in order to get it back to ℓ : since { (1 + | n | ) s ′ } n ∈ Z d ∈ ℓ for s ′ > d/
4, we have byH¨older inequality { (1 + | n | ) − s ′ (1 + | n | ) s/ v ( n ) } n ∈ Z d ∈ ℓ , s ′ > d/ . which implies the statement of the lemma. (cid:3) Finally, the following is the dual version of the multiplicative Sobolev inequality [1, The-orem 4.39] on the language of convolutions. One can also prove it directly and use in theproof of multiplicative inequalities.
Lemma 6.3.
Let u ∈ ℓ s , v ∈ ℓ s . Denote their convolution by ( u ∗ v )( n ) = X m ∈ Z d u ( n − m ) v ( m ) . Then k u ∗ v k ℓ s ≤ C ( s, s , s ) k u k ℓ s k v k ℓ s , < s < s + s − d/ . References [1] R. Adams, J. Fournier, Sobolev Spaces, 2nd Edition, Academic Press, 2003.[2] S. Amor,
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