Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schrödinger Operators on the Real Line
aa r X i v : . [ m a t h . SP ] S e p ABSENCE OF ABSOLUTELY CONTINUOUS SPECTRUMFOR GENERIC QUASI-PERIODIC SCHR ¨ODINGEROPERATORS ON THE REAL LINE
DAVID DAMANIK AND DANIEL LENZ
Abstract.
We show that a generic quasi-periodic Schr¨odinger opera-tor in L ( R ) has purely singular spectrum. That is, for any minimaltranslation flow on a finite-dimensional torus, there is a residual set ofcontinuous sampling functions such that for each of these sampling func-tions, the Schr¨odinger operator with the resulting potential has emptyabsolutely continuous spectrum. Introduction
In this paper we consider Schr¨odinger operators(1.1) [ Hψ ]( x ) = − ψ ′′ ( x ) + V ( x ) ψ ( x )in L ( R ) with quasi-periodic potentials(1.2) V ( x ) = f ( ω + xα ) . Here, ω, α ∈ T d = R d / Z d for some d ∈ Z + , f ∈ C ( T d ) real-valued, and x ∈ R . The case where V is periodic is classical and well understood, andhence we will primarily focus on the aperiodic case. This necessarily meansthat d ≥ α and f . We will assumethat α is such that the translation flow in question is minimal (i.e., all orbitsare dense) to ensure that the torus dimension d is chosen appropriately, andmoreover f needs to be non-constant to avoid periodicity.The spectral properties of operator of the form (1.1) with potentials ofthe form (1.2) have been studied intensively since the 1980’s, with manymajor advances occurring in the past two decades. Much of this work hasbeen reviewed in several recent survey papers, including [5, 6, 11, 12, 16].We should point out, however, that some of these survey papers discuss thediscrete analogs of these operators, which act in ℓ ( Z ) as[ H ( d ) ψ ]( n ) = ψ ( n + 1) + ψ ( n −
1) + V ( d ) ( n ) ψ ( n )with V ( d ) ( n ) = f ( ω + nα ) , but many results exist in both settings. Date : September 4, 2019.D.D. was supported in part by NSF grant DMS–1700131.
There are, of course, some notable exceptions. One of the most importantexceptions is that Avila’s global theory for discrete one-frequency quasi-periodic Schr¨odinger operators with analytic sampling functions [1] does notyet have a continuum counterpart. In this paper we will address anotherresult, which is known in the discrete case, but whose continuum counterpartis desirable to have because of recent progress on the Deift conjecture, whichmakes a connection with continuum quasi-periodic Schr¨odinger operators.The Deift conjecture [7, 8] states that the KdV equation with almost pe-riodic initial data admits global solutions that are almost periodic in (spaceand) time. The conjecture has been proved under suitable assumptions[4, 9]. These results, and really their proofs, need that the initial data,when considered as potentials, give rise to Schr¨odinger operators with ab-solutely continuous spectrum. It was therefore pointed out in [4] that theassumptions will likely fail generically in a suitable sense.Concretely, the abstract sufficient conditions for the Deift conjecture tohold have been verified for suitable classes of quasi-periodic functions ofthe form (1.2); see [4]. On the other hand, for discrete quasi-periodicSchr¨odinger operators, it is known [2] that a generic quasi-periodic potentialwill give rise to a Schr¨odinger operator with empty absolutely continuousspectrum. One should therefore expect that also in the continuum case,which is the one relevant to the study of the KdV equation and the Deiftconjecture, the absolutely continuous spectrum will be empty for a genericquasi-periodic potential.The purpose of this paper is to prove this statement:
Theorem 1.1.
Given d ≥ and a minimal translation flow on T d , R ∋ x ω + xα ∈ T d , there is a dense G δ -set S ⊆ C ( T d ) such that for every f ∈ S , the Schr¨odinger operator in L ( R ) with potential V ( x ) = f ( ω + xα ) has purely singular spectrum. Remarks 1.2. (a) The minimality of the flow is a property of α ∈ T d , andthe result holds for any such fixed α . The set S will then depend on thechoice of α .(b) There is no quantifier on ω ∈ T d in the statement of the result, eventhough the potential V depends on it. This is due to the constancy of theabsolutely continuous spectrum in ω , which is a result of Last and Simon[17, Theorem 1.5].(c) This result shows that there is a generic obstruction to an extension ofthe BDGL approach [4] or the EVY approach [9] to the Deift conjecture[7, 8].One can also consider one-parameter families of potentials and operatorsby varying the coupling constant: Theorem 1.3.
Given d ≥ and a minimal translation flow on T d , R ∋ x ω + xα ∈ T d , there is a dense G δ -set S ⊆ C ( T d ) such that for every UASI-PERIODIC SCHR ¨ODINGER OPERATORS AND SINGULAR SPECTRUM 3 f ∈ S and Lebesgue almost every λ > , the Schr¨odinger operator in L ( R ) with potential V ( x ) = λf ( ω + xα ) has purely singular spectrum. Preliminaries
Discontinuous Periodic Functions Having Limit-Periodic Lim-its.
Recall that a bounded uniformly continuous function on R is called almost periodic if for any ε > t ∈ R with k f − f ( · − t ) k ∞ < ε is relatively dense. A bounded uniformly continuous function on R is called limit-periodic if it is a uniform limit of continuous periodic functions. Butwhat if we have a uniformly convergent sequence of discontinuous periodicfunctions? Can the limit be limit-periodic? Clearly, we need to assume atleast the continuity of the limit, but what else is needed?The following statement is likely well known, but since it will play a rolein the proof of our main result, we include its short proof for the convenienceof the reader. Proposition 2.1.
Suppose f ∈ C ( R ) is uniformly continuous and, for n ≥ , f n ∈ L ∞ ( R ) is periodic. If k f n − f k ∞ → as n → ∞ , then f is limit-periodic.Proof. The issue is that the f n may be discontinuous and hence the remedywill be to make them continuous via mollification and then to observe thatthe continuous mollified functions still converge uniformly to f . Compare[10, Section C.4] for the definitions and general results below.Explicitly, define η ∈ C ∞ ( R ) by η ( x ) = ( C exp (cid:16) | x | − (cid:17) if | x | < , | x | ≥ , where C > R R η ( x ) dx = 1. Then, for ε >
0, set η ε ( x ) = 1 ε η (cid:16) xε (cid:17) and, for n ≥ f εn = η ε ∗ f , that is, f εn ( x ) = Z R η ε ( x − y ) f n ( y ) dy. By the uniform continuity of f , Theorem 6 in [10, Section C.4] and itsproof (especially the proof of part (iii)), imply that for each n ≥ ε > f εn is smooth (and in particular continuous) and k f εn − f n k ∞ → ε →
0. Thus, the statement follows by diagonalization, that is, for asuitable sequence ε n →
0, the functions f ε n n are continuous, periodic (byconstruction) and converge uniformly to f , showing that f is indeed limit-periodic. (cid:3) A function q on R is called eventually periodic if there exists a periodicfunction p with p ( x ) = q ( x ) for all sufficiently large x ∈ R . D. DAMANIK AND D. LENZ
Corollary 2.2.
Suppose f ∈ C ( R ) is almost periodic and, for n ≥ , f n ∈ L ∞ ( R ) is eventually periodic. If k f n − f k ∞ → as n → ∞ , then f islimit-periodic.Proof. Let ε > p ∈ L ∞ ( R ) with k f − p k < ε .By assumption there exists an eventually periodic q ∈ L ∞ ( R ) (viz q = f m for sufficiently large m ) with k f − q k < ε. As q is eventually periodic, there exists a periodic p ∈ L ∞ ( R ) with p ( x ) = q ( x ) for all sufficiently large x . Let P > p ( x ) = p ( x + P ) for all x ∈ R .As f is almost periodic, there exists a sequence ( t n ) in R with k f t n − f k ∞ → n → ∞ . Here, we set g t := g ( · − t ). There exist then unique k n ∈ N and 0 ≤ s n < P with t n = k n P + s n . Restricting attention to asubsequence if necessary, we can then assume without loss of generality that s n → s . As f is uniformly continuous, we can even assume without loss ofgenerality s n = s for all n . To simplify notation we will assume s = 0.Hence, f − p is the pointwise limit of f − q t n for n → ∞ . This gives k f − p k ∞ ≤ lim sup n k f − q t n k ∞ ≤ lim sup n ( k f − f t n k ∞ + k f t n − q t n k ∞ )= (lim n k f − f t n k ∞ ) + lim sup n k f t n − q t n k ∞ = k f − q k ∞ < ε. Here, we used the invariance of k · k ∞ under translation in the penultimatestep. (cid:3) Transfer Matrices, Lyapunov Exponents, and Weyl-Titch-marsh Functions.
This subsection recalls important and well-known con-cepts, mainly to fix notation.Fixing d ≥ T d , R ∋ x ω + xα ∈ T d ,as well as a real-valued sampling function f ∈ C ( T d ), the transfer matrices are defined via ddx M f ( x, E, ω ) = A f ( E, ω + xα ) M f ( x, E, ω ) M f (0 , E, ω ) = I for x ∈ R , E ∈ C , ω ∈ T d , where A f ( E, ω ) = (cid:18) f ( ω ) − E (cid:19) . UASI-PERIODIC SCHR ¨ODINGER OPERATORS AND SINGULAR SPECTRUM 5
These transfer matrices are defined in such a way that u solves the differentialequation(2.1) − u ′′ ( x ) + f ( ω + xα ) u ( x ) = Eu ( x )if and only if it solves (cid:18) u ( x ) u ′ ( x ) (cid:19) = M f ( x, E, ω ) (cid:18) u (0) u ′ (0) (cid:19) . By the subadditive ergodic theorem, there is a number L ( E ) ≥
0, calledthe
Lyapunov exponent , so that L f ( E ) = lim | x |→∞ | x | log k M f ( x, E, ω ) k for almost every ω ∈ T d .The map E L f ( E ) is real-symmetric and subharmonic. Moreover, wehave (see [15, Lemma 3.2 and (49)–(50)])(2.2) L f ( E ) = − Z T d Re m + ,f,ω ( E ) dω for E ∈ C + , the upper half-plane, where m + ,f,ω is the Weyl-Titchmarsh m - function on the right half-line associated with the potential V ( x ) = f ( ω + xα ), defined by m + ,f,ω ( E ) = u ′ + ,f,ω (0) u + ,f,ω (0) , where u + ,f,ω is a solution of (2.1) that is square-integrable at + ∞ . A Semi-Continuity Result
In this section we discuss the continuum analog of the Avila-Damaniksemi-continuity result [2, Lemma 1]. The general structure of the proof willbe the same, and hence we will focus mostly on the aspects that are differentbetween the discrete case and the continuum case.Set M R ( f ) = Leb ( { E ∈ R ∩ [ − R, R ] : L f ( E ) = 0 } ) . Remark 3.1.
By the Ishii-Kotani-Pastur Theorem [15, Theorem 4.7] andthe Last-Simon Theorem [17, Theorem 1.5], we have that M R ( f ) = 0 if andonly if the Schr¨odinger operator in L ( R ) with potential V ( x ) = f ( ω + xα )has purely singular spectrum in the energy interval [ − R, R ] for every ω ∈ T d .Here is the continuum analog of [2, Lemma 1]: To see that such a solution exists, observe that E σ ( H ) by self-adjointness, andhence ˜ u + ,f,ω := ( H − E ) χ ( − , ∈ L ( R ). But by definition ˜ u + ,f,ω solves (2.1) on (0 , ∞ ).Thus, keeping it unchanged on the right half-line and extending it to a solution on R bysolving (2.1), we obtain u + ,f,ω . D. DAMANIK AND D. LENZ
Lemma 3.2.
For all choices of r, R, Λ > , the maps (3.1) ( B r ( L ∞ ( T d )) , k · k ) → [0 , ∞ ) , f M R ( f ) and (3.2) ( B r ( L ∞ ( T d )) , k · k ) → [0 , ∞ ) , f Z Λ0 M R ( λf ) dλ are upper semi-continuous. Here, B r denotes the closed ball with radius r in the essential-supremum norm.Proof. It is enough to show that (3.1) is upper semi-continuous, the uppersemi-continuity of (3.2) then follows from that via Fatou’s lemma.The proof of the upper semi-continuity of (3.1) proceeds in the same wayas in [2]. Assuming that the upper semi-continuity of (3.1) fails for somechoice of r, R, Λ >
0, there must be f n , f ∈ L ∞ ( T d ) such that(i) f n → f in L and pointwise as n → ∞ ,(ii) k f n k ∞ ≤ r for every n ≥ k f k ∞ ≤ r ,(iii) lim inf M R ( f n ) ≥ M R ( f ) + ε for some ε > m -functions m + ,f,ω in C + for almost every ω ∈ T d (this follows from a modification of theargument given in [13]). Thus, by (2.2), (ii), and dominated convergence,the associated Lyapunov exponents L f n converge pointwise in C + to L f .Next, consider the region U in C + bounded by the equilateral triangle T with sides I, J, K , where I = [ − R, R ] ⊂ R . From here the proof proceedsverbatim as in [2], using the Schwarz-Christoffel formula, as well as thefact that the Lyapunov exponent is harmonic in C + and subharmonic (andin particular upper semi-continuous) globally, to derive a contradiction to(iii). (cid:3) Small Perturbations That Destroy the AbsolutelyContinuous Spectrum
In this section we discuss how arbitrarily small perturbations can destroythe absolutely continuous spectrum. Here, we use results of [14].We first recall some basic concepts from [14]. A piece is a pair (
W, I )consisting of an interval I ⊆ R with length | I | > | I | = ∞ allowed)and a locally bounded function W on R supported on I . We abbreviatepieces by W I . Without restriction, we may assume that min I = 0. A finitepiece is a piece of finite length. The concatenation W I = W I | W I | . . . ofa finite or countable family ( W I j j ) j ∈ N , with N = { , , . . . , N } (for N finite)or N = N (for N infinite), of finite pieces is defined by I = , X j ∈ N | I j | , UASI-PERIODIC SCHR ¨ODINGER OPERATORS AND SINGULAR SPECTRUM 7 W = W + X j ∈ N, j ≥ W j (cid:16) · − (cid:16) j − X k =1 | I k | (cid:17)(cid:17) . In this case we say that W I is decomposed by ( W I j j ) j ∈ N .Let now V be a locally bounded function on R . We say that V hasthe finite decomposition property if there exist a finite set P of finite piecesand x ∈ R such that (1 [ x , ∞ ) V ) is a translate of a concatenation W I | W I | . . . with W I j j ∈ P for all j ∈ N . We say that V has the simple finitedecomposition property if it has the f.d.p. with a decomposition such thatthere is ℓ > W I − m − m | . . . | W I | W I | . . . | W I m m and W I − m − m | . . . | W I | U J | . . . | U J m m occur in the decomposition of V with a common first part W I − m − m | . . . | W I of length at least ℓ and such that1 [0 ,ℓ ) ( W I | . . . | W I m m ) = 1 [0 ,ℓ ) ( U J | . . . | U J m m ) , where W I j j , U J k k are pieces from the decomposition (in particular, all belongto P and start at 0) and the latter two concatenations are of lengths at least ℓ . Then W I = U J . The relevance of the simple finite decomposition property comes from thefollowing result from [18] (see [14] as well).
Lemma 4.1 (Theorem 7.1 of [18]) . Let W be a bounded measurable functionon R . Assume that both W and W ( −· ) have the simple finite decompositionproperty and are not eventually periodic. Then, the Schr¨odinger operator H W ψ = − ψ ′′ ( x ) + W ( x ) ψ ( x ) does not have any absolutely continuous spec-trum. Here is the main result of this section.
Proposition 4.2.
Given d ≥ , a minimal translation flow R ∋ x ω + xα ∈ T d , f ∈ C ( T d ) , and ε > , there exists ˜ f ∈ L ∞ ( T d ) such that k f − ˜ f k ∞ < ε and, for all ω ∈ T d , the potential ˜ V ( x ) = ˜ f ( ω + xα ) as well as ˜ V ( −· ) have the simple finite decomposition property and are not eventuallyperiodic. In particular, the Schr¨odinger operator in L ( R ) with potential ˜ V has purely singular spectrum.Proof. It suffices to show the first statement. The last statement then followsfrom the preceding lemma.Since the given flow is minimal, we can assume without loss of generalitythat the function f yields aperiodic potentials V ( x ) = f ( ω + xα ) (otherwiseuse a fraction of the given ε to perturb f within C ( T d ) in order to ensurethis property). D. DAMANIK AND D. LENZ
For the given ε >
0, let us now consider a sequence of partitions P ε,n of T d into finitely many boxes (parallelepipeds) of the following form: B γ,ℓ = γ + d − X j =1 t j e j + t d α : 0 ≤ t j < ℓ j for 1 ≤ j ≤ d , where γ ∈ T d and ℓ = ( ℓ , . . . , ℓ d ) with 0 < ℓ , . . . , ℓ d <
1. Here e j denotesthe vector that has a 1 as its j -th component and only 0’s otherwise.We require two properties from these partitions. These two propertiesmay be satisfied since f is uniformly continuous and the translation flow isminimal. First we ask that for every n and every box B γ,ℓ belonging to P ε,n ,the variation of f on B γ,ℓ is less than ε/
2, that is,(4.1) sup ω ∈ B γ,ℓ f ( ω ) − inf ω ∈ B γ,ℓ f ( ω ) < ε . Second, letting δ ε,n denote the maximum of k ℓ k ∞ taken over all boxes B γ,ℓ in the partition P ε,n , we require that δ ε,n → n → ∞ .Note that once the translation flow enters such a box B γ,ℓ , then it spendsexactly ℓ d time units in the box before it leaves it again. This is true foreach entry into the box, no matter where the entry happens.Let us now define a function f ε,n ∈ L ∞ ( T d ) as follows. On each box B γ,ℓ belonging to P ε,n , f ε,n takes values in the interval " inf ω ∈ B γ,ℓ f ( ω ) − min (cid:26) ε , n (cid:27) , sup ω ∈ B γ,ℓ f ( ω ) + min (cid:26) ε , n (cid:27) , and moreover the value of f ε,n at the point γ + P d − j =1 t j e j + t d α depends onlyon t d and is independent of t , . . . , t d − . Finally we require the dependenceof f ε,n on t d to be continuous and non-constant. Such a selection is clearlypossible since the interval of allowed values is non-degenerate. Moreover, byconstruction we have(4.2) k f − f ε,n k ∞ < ε. Now we claim that there is an n so that the statement of the proposi-tion holds for ˜ f := f ε,n . Assume this fails, and we have that in fact forevery n , the potential V ε,n ( x ) = f ε,n ( ω + xα ) or the potential V ε,n ( − x ) iseventually periodic or does not have the simple finite decomposition prop-erty. Now, clearly, these potentials have the finite decomposition propertyby construction, and the simplicity of the finite decomposition property ofthe potential follows by [14, Proposition 3.5] and the local non-constancyaspect of our construction. Thus, for each n the potential V ε,n or V ε,n ( −· )must be eventually periodic. Restricting attention to a subsequence we canassume without loss of generality that V ε,n ( x ) must be eventually periodic This will imply the finite decomposition property below. We can make it even more regular if needed, such as the function taking any value onlya finite number of times. This will then imply the simple finite decomposition property.
UASI-PERIODIC SCHR ¨ODINGER OPERATORS AND SINGULAR SPECTRUM 9 for every n . Note that the V ε,n are bounded and measurable, but in generaldiscontinuous. These eventually periodic functions converge (by construc-tion) uniformly to the function V ( x ) = f ( ω + xα ), which is clearly almostperiodic, and hence must be limit-periodic due to Corollary 2.2. But sinceit is manifestly quasi-periodic as well, it must therefore be periodic by [3,Corollary A.1.4]; contradiction (by our initial step). (cid:3) Remark 4.3.
In the proposition above, once we know that the potential˜ V ( x ) = ˜ f ( ω + xα ) and ˜ V ( −· ) have the simple finite decomposition propertyand are not eventually periodic, these properties are inherited by any non-zero multiple of the potential. In particular it then also follows that, forevery λ >
0, the Schr¨odinger operator in L ( R ) with potential λ ˜ V haspurely singular spectrum.5. Closing the Jumps
We saw in Proposition 4.2 that by approximating a given continuous sam-pling function with a discontinuous sampling function, we can destroy theabsolutely continuous spectrum of the associated operator. The approxi-mation is with respect to the k · k ∞ norm. However, we wish to identifya continuous sampling function that is close to the original one, for whichthe absolutely continuous spectrum is empty. A second approximation istherefore necessary to close the jumps .Clearly, the discontinuous function (with the desired property) cannotbe approximated by a continuous function in the k · k ∞ norm. However,it is possible to approximate it in the k · k norm. This shows why thesemi-continuity result given by Lemma 3.2 is relevant. Moreover, since thelimit function has a zero value and the values are non-negative, the semi-continuity result becomes in effect a continuity result in the setting relevantto this discussion.The following lemma implements this two-step approximation: Lemma 5.1.
For f ∈ C ( T d ) and < ε, δ, R, Λ < ∞ , there exists g ∈ C ( T d ) such that k f − g k ∞ < ε , M R ( g ) < δ , and R Λ0 M R ( λg ) dλ < δ .Proof. Given f ∈ C ( T d ) and 0 < ε, δ, R, Λ < ∞ , Proposition 4.2 yields an˜ f ∈ L ∞ ( T d ) with k f − ˜ f k ∞ < ε and M R ( ˜ f ) = 0, as well as (cf. Remark 4.3) M ( λ ˜ f ) = 0 for every λ > f (via the mollifiers used in the proof of Proposition 2.1)to produce f n ∈ C ( T d ) with lim n →∞ k f n − ˜ f k = 0and sup n ∈ Z + k f n − f k ∞ < ε. By the non-negativity of the quantities in question, the vanishing limits, andthe semi-continuity properties from Lemma 3.2, it follows thatlim n →∞ M R ( f n ) = 0and lim n →∞ Z Λ0 M R ( λf n ) dλ = 0 . Thus, for n large enough, g = f n has the desired properties. (cid:3) Proof of the Main Results
In this section we prove the main results, Theorems 1.1 and 1.3. Theproofs are analogous to the corresponding proofs in [2]. Since they are veryshort, we give the details for the reader’s convenience.
Proof of Theorem 1.1.
For 0 < δ, R < ∞ , we define M R,δ = { f ∈ C ( T d ) : M R ( f ) < δ } . By Lemma 3.2, M R,δ is open, and by Lemma 5.1, M R,δ is dense. Thus, { f ∈ C ( T d ) : Σ ac ( f ) = ∅} = \ n ∈ Z + M n, n is a dense G δ set, as claimed. (cid:3) Proof of Theorem 1.3.
For 0 < δ, R, Λ < ∞ , we define M R,δ (Λ) = (cid:26) f ∈ C ( T d ) : Z Λ0 M R ( λf ) dλ < δ (cid:27) . By Lemma 3.2, M R,δ (Λ) is open, and by Lemma 5.1, M R,δ (Λ) is dense.Thus, { f ∈ C ( T d ) : Σ ac ( λf ) = ∅ for a.e. λ > } = \ n ∈ Z + M n, n ( n )is a dense G δ -set, as claimed. (cid:3) References [1] A. Avila, Global theory of one-frequency Schr¨odinger operators,
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Department of Mathematics, Rice University, Houston, TX 77005, USA
E-mail address : [email protected] Institute for Mathematics, Friedrich-Schiller University, Jena, 07743 Jena
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