Absence of point spectrum for the self-dual extended Harper's model
aa r X i v : . [ m a t h . SP ] S e p ABSENCE OF POINT SPECTRUM FOR THE SELF-DUAL EXTENDEDHARPER’S MODEL
RUI HAN
Abstract.
We give a simple proof of absence of point spectrum for the self-dual extended Harper’smodel. We get a sharp result which improves that of [1] in the isotropic self-dual regime. Introduction
We study the extended Harper’s model on l ( Z ):( H λ,α,θ u ) n = c λ ( θ + nα ) u n +1 + ˜ c λ ( θ + ( n − α ) u n − + v ( θ + nα ) u n , (1.1)where c λ ( θ ) = λ e − πi ( θ + α ) + λ + λ e πi ( θ + α ) and v ( θ ) = 2 cos 2 πθ . ˜ c λ ( θ ) = c λ ( θ ) for θ ∈ T andits analytic extension when θ / ∈ T . We refer to λ = ( λ , λ , λ ) as coupling constants, θ ∈ T = [0 , α as the frequency.In [2] the authors partitioned the parameter space into the following three regions. λ λ + λ λ + λ = λ
11 Region I Region IIRegion III L II L I L III
Region I: ≤ λ + λ ≤ , < λ ≤ Region II: ≤ λ + λ ≤ λ , ≤ λ , Region III: max { , λ } ≤ λ + λ , λ > duality transformation σ : λ = ( λ , λ , λ ) → ˆ λ = ( λ λ , λ , λ λ ), wehave the following observation [2]: Observation . σ is a bijective map on 0 ≤ λ + λ , < λ .(i) σ (I ◦ ) = II ◦ , σ (III ◦ ) = σ (III ◦ ) (ii) Letting L I := { λ + λ = 1 , < λ ≤ } , L II := { ≤ λ + λ ≤ , λ = 1 } , and L III := { ≤ λ + λ = λ } , σ (L I ) = L III and σ (L II ) = L II .As σ bijectively maps III ∪ L II onto itself, the literature refers to III ∪ L II as the self-dual regime .We further divide III into III λ = λ ( isotropic self-dual regime ) and III λ = λ ( anisotropic self-dualregime ).A complete understanding of the spectral properties of the extended Harper’s model for a.e. θ has been established: Theorem 1.2. [1]
The following Lebesgue decomposition of the spectrum of H λ,α,θ holds for a.e. θ . • For all Diophantine α , for Region I , H λ,α,θ has pure point spectrum. • For all irrational α , for Regions II , III λ = λ , H λ,α,θ has purely absolutely continuous spec-trum. • For all irrational α , for Region III λ = λ , H λ,α,θ has purely singular continuous spectrum. As pointed out in [1], the main missing link between [2, 3] and Theorem 1.2 is the followingtheorem, excluding eigenvalues in the self-dual regime. We say θ is α -rational if 2 θ ∈ Z α + Z ,otherwise we say θ is α -irrational. Theorem 1.3. [1]
For all irrational α , • for λ ∈ III λ = λ ∪ L II , H λ,α,θ has empty point spectrum for all α -irrational θ . • for λ ∈ III λ = λ , H λ,α,θ has empty point spectrum for a.e. θ . In [1] the authors had to exclude more phases than α -rational θ in the isotropic self-dual regime.In this paper we give a simple proof of the following theorem. Theorem 1.4.
For all irrational α , for λ ∈ III , H λ,α,θ has empty point spectrum for all α -irrational θ .Remark . Our result for the isotropic self-dual regime III λ = λ is sharp. Indeed, according toProposition 5.1 in [1], for α -rational θ , H λ,α,θ has point spectrum.We organize this paper in the following way: in Section 2 we include some preliminaries, inSection 3 we present two lemmas that will be used in Section 5, then we deal with III λ = λ andIII λ = λ ∩ { λ + λ = 1 } in Section 4 and III λ = λ ∩ { λ + λ > } in Section 5.2. Preliminaries
Rational approximation.
Let { p m q m } be the continued fraction approximants of α , then12 q m +1 ≤ k q m α k T ≤ q m +1 . (2.1)The exponent β ( α ) is defined as follows β ( α ) = lim sup m →∞ ln q m +1 q m . (2.2)It describes how well is α approximated by rationals.2.2. Self-dual extended Harper’s model.
Let | c | λ ( θ ) = p c λ ( θ )˜ c λ ( θ ) be the analytic functionthat coincides with | c λ ( θ ) | when θ ∈ T .The presence of singularities of c λ ( θ ) is explicit: Observation . (e.g. [1]) The function c λ ( θ ) has at most two zeros. Necessary conditions for realroots are λ ∈ III λ = λ or λ ∈ III λ = λ ∩ { λ + λ = λ } . Moreover, • for λ ∈ III λ = λ , c λ ( θ ) has real roots determined by2 λ cos 2 π ( θ + α − λ , (2.3) and giving rise to a double root at θ = − α if λ ∈ III λ = λ ∩ { λ + λ = λ } . • for λ ∈ III λ = λ ∩ { λ + λ = λ } , c λ ( θ ) has only one simple real root at θ = − α . Remark . By the definition of the duality transformation σ , Observation 2.1 implies that c ˆ λ ( θ )has singular point if and only if λ ∈ III λ = λ or λ ∈ III λ = λ ∩ { λ + λ = 1 } .It will be clear in Section 4 that presence of singularities of c ˆ λ ( θ ) indeed simplifies the proof ofempty point spectrum of H λ,α,θ . 3. Lemmas
Lemma 3.1.
For λ ∈ III λ = λ ∩ { λ + λ > } , when λ > λ , we have c ˆ λ ( θ ) | c | ˆ λ ( θ ) = e − πi ( θ + α )+ if ( θ ) and ˜ c ˆ λ ( θ ) | c | ˆ λ ( θ ) = e πi ( θ + α ) − if ( θ ) , for a real analytic function f ( θ ) on T with R T f ( θ )d θ = 0 . Proof.
By the definition of c ˆ λ ( θ ) we have c ˆ λ ( θ ) = λ λ e − πi ( θ + α ) + 1 λ + λ λ e πi ( θ + α ) (3.1) = λ λ e − πi ( θ + α ) ( e πi ( θ + α ) − y + )( e πi ( θ + α ) − y − ) , (3.2)where y ± = − ±√ − λ λ λ . Note that y + = y − with | y + | = | y − | = r λ λ > , when 1 ≤ p λ λ , (3.3) y + , y − ∈ R with | y + | > | y − | = 2 λ λ + √ − λ λ > , when λ + λ > > p λ λ . (3.4)Note that c ˆ λ ( θ ) | c | ˆ λ ( θ ) = s c ˆ λ ( θ )˜ c ˆ λ ( θ ) = e − πi ( θ + α ) s ( e πi ( θ + α ) − y + )( e πi ( θ + α ) − y − )( e − πi ( θ + α ) − y + )( e − πi ( θ + α ) − y − ) . (3.5)By (3.3), we have Z T arg ( e πi ( θ + α ) − y + )( e πi ( θ + α ) − y − )( e − πi ( θ + α ) − y + )( e − πi ( θ + α ) − y − ) d θ = 0 , (3.6)and | ( e πi ( θ + α ) − y + )( e πi ( θ + α ) − y − )( e − πi ( θ + α ) − y + )( e − πi ( θ + α ) − y − ) | ≡ . (3.7)Thus there exists a real analytic function g ( θ ) on T such that( e πi ( θ + α ) − y + )( e πi ( θ + α ) − y − )( e − πi ( θ + α ) − y + )( e − πi ( θ + α ) − y − ) = e ig ( θ ) , (3.8)with R T g ( θ )d θ = 0. Taking f ( θ ) = g ( θ ) / (cid:3) RUI HAN
Lemma 3.2.
There is a subsequence { p ml q ml } of the continued fraction approximants of α so that forany analytic function f on T with R T f ( θ )d θ = 0 , we have lim l →∞ f ( x ) + f ( x + α ) + · · · + f ( x + q m l α − α ) = 0 uniformly in x ∈ T . Proof.
Suppose f is analytic on | Im θ | ≤ δ , then | ˆ f ( n ) | ≤ ce − πδ | n | for some constant c > β ( α ) = 0, then by solving the coholomogical equation we get f ( x ) = h ( x + α ) − h ( x ) forsome analytic h ( x ). Then lim m →∞ ( f ( x ) + f ( x + α ) + · · · + f ( x + q m α − α ))= lim m →∞ ( h ( x + q m α ) − h ( x )) = 0uniformly in x .Case 2. If β ( α ) >
0, choose a sequence m l such that q m l +1 ≥ e β q ml . Then | f ( x ) + f ( x + α ) + · · · + f ( x + q m l α − α ) | = | X | n |≥ ˆ f ( n )(1 + e πinα + · · · + e πin ( q ml − α ) e πinx | = | X | n |≥ ˆ f ( n ) 1 − e πinq ml α − e πinα e πinx |≤ X ≤| n |≤ q ml − c (cid:12)(cid:12)(cid:12)(cid:12) − e πinq ml α − e πinα (cid:12)(cid:12)(cid:12)(cid:12) + X | n |≥ q ml ce − πδ | n | q m l ≤ c q m l q m l +1 + cq m l e − πδ q ml → l → ∞ uniformly in x . (cid:3) Consequence of point spectrum
This part follows from [1]. We present the material here for completeness and readers’ convenience.Suppose { u n } is an l ( Z ) solution to H λ,α,θ u = Eu , where λ = ( λ , λ , λ ). This means(4.1) c λ ( θ + nα ) u n +1 + ˜ c λ ( θ + ( n − α ) u n − + 2 cos(2 π ( θ + nα )) u n = Eu n . Let u ( x ) = P n ∈ Z u n e πinx ∈ L ( T ). Multiplying (4.1) by e πinx and then summing over n , we get(4.2) e πiθ c ˆ λ ( x ) u ( x + α ) + e − πiθ ˜ c ˆ λ ( x − α ) u ( x − α ) + 2 cos 2 πx u ( x ) = Eλ u ( x ) , where ˆ λ = ( λ λ , , λ λ ). If we multiply (4.1) by e − πinx and sum over n , we get(4.3) e − πiθ c ˆ λ ( x ) u ( − x − α ) + e πiθ ˜ c ˆ λ ( x − α ) u ( − x + α ) + 2 cos 2 πx u ( − x ) = Eλ u ( − x ) . Thus writing (4.2), (4.3) in terms of matrices, we get1 c ˆ λ ( x ) (cid:18) Eλ − πx − ˜ c ˆ λ ( x − α ) c ˆ λ ( x ) 0 (cid:19) (cid:18) u ( x ) u ( − x ) e − πiθ u ( x − α ) e πiθ u ( − ( x − α )) (cid:19) = (cid:18) u ( x + α ) u ( − ( x + α )) e − πiθ u ( x ) e πiθ u ( − x ) (cid:19) (cid:18) e πiθ e − πiθ (cid:19) (4.4) Let M θ ( x ) ∈ L ( T ) be defined by M θ ( x ) = (cid:18) u ( x ) u ( − x ) e − πiθ u ( x − α ) e πiθ u ( − ( x − α )) (cid:19) . Let A ˆ λ,E/λ ( x ) = 1 c ˆ λ ( x ) (cid:18) Eλ − πx − ˜ c ˆ λ ( x − α ) c ˆ λ ( x ) 0 (cid:19) be the transfer matrix associated to H ˆ λ,α,θ and R θ = (cid:18) e πiθ e − πiθ (cid:19) be the constant rotation matrix. Then (4.4) becomes A ˆ λ,E ( x ) M θ ( x ) = M θ ( x + α ) R θ . (4.5)Taking determinant, we have the following proposition. Proposition 4.1. [1] If θ is α -irrational, then | det M θ ( x ) | = b | c | ˆ λ ( x − α )(4.6) for some constant b > and a.e. x ∈ T . Regions
III λ = λ and III λ = λ ∩ { λ + λ = 1 } We will show the following lemma.
Lemma 5.1. If θ is α -irrational, then for λ ∈ III λ = λ or λ ∈ III λ = λ ∩ { λ + λ = 1 } , H λ,α,θ hasno point spectrum. Proof.
According to Remark 2.1, we have c ˆ λ ( x ) = 0 for some x ∈ T . Note that presenceof singularity implies c ˆ λ ( x ) / ∈ L ( T ). Thus by (4.6), det M θ ( x ) / ∈ L ( T ). This contradicts with M θ ( x ) ∈ L ( T ). (cid:3) Regions
III λ = λ ∩ { λ + λ > } Without loss of generality, we assume λ > λ . Fix θ . Denote det M θ ( x ) = g ( x ) for simplicity. Lemma 6.1. If θ is α -irrational, then H λ,α,θ has no point spectrum in the anisotropic self-dualregion. Proof.
Taking determinant in (4.5), we get:˜ c ˆ λ ( x − α ) c ˆ λ ( x ) g ( x ) = g ( x + α ) . This implies g ( x + kα ) = ˜ c ˆ λ ( x + kα − α ) · · · ˜ c ˆ λ ( x ) ˜ c ˆ λ ( x − α ) c ˆ λ ( x + kα − α ) · · · c ˆ λ ( x + α ) c ˆ λ ( x ) g ( x ) . (6.1)Taking k = q m l , as in Lemma 3.2, on one hand, since g ( x ) is an L function, as the determinant ofan L matrix, and lim l →∞ k q m l α k T = 0, we havelim l →∞ k g ( x + q m l α ) − g ( x ) k L = 0 . RUI HAN
By (6.1), this implies0 = lim l →∞ k g ( x + q m l α ) − g ( x ) k L = lim l →∞ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Q q ml − j = − ˜ c ˆ λ ( x + jα ) Q q ml − j =0 c ˆ λ ( x + jα ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · | g ( x ) | d x. (6.2)On the other hand, by Lemma 3.1lim l →∞ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Q q ml − j = − ˜ c ˆ λ ( x + jα ) Q q ml − j = − c ˆ λ ( x + jα ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · | g ( x ) | d x = lim l →∞ Z (cid:12)(cid:12)(cid:12)(cid:12) − | c | ˆ λ ( x − α ) | c | ˆ λ ( x + q m l α − α ) e − i (cid:16)P qml − j = − f ( x + jα )+ P qml − j =0 f ( x + jα ) (cid:17) e πiq ml x e πiq ml ( q ml − α (cid:12)(cid:12)(cid:12)(cid:12) · | g ( x ) | d x ≥ lim inf l →∞ (cid:18)Z | − e πiq ml x +2 πiq ml α || g ( x ) | d x − Z (cid:12)(cid:12)(cid:12)(cid:12) − | c | ˆ λ ( x − α ) | c | ˆ λ ( x + q m l α − α ) e − i (cid:16)P qml − j = − f ( x + jα )+ P qml − j =0 f ( x + jα ) (cid:17) e − πiq ml α (cid:12)(cid:12)(cid:12)(cid:12) · | g ( x ) | d x (cid:19) := lim inf l →∞ ( I − I ) . (6.3)Combining the fact k q m l α k T → | c | ˆ λ ( x − α ) | c | ˆ λ ( x + q m l α − α ) e − i (cid:16)P qml − j = − f ( x + jα )+ P qml − j =0 f ( x + jα ) (cid:17) e − πiq ml α → l → ∞ . Then by dominated convergence theorem, we get lim l →∞ I = 0. Then (6.3) implies that for anysmall constant δ >
0, lim l →∞ k g ( x + q m l α ) − g ( x ) k L ≥ lim inf l →∞ I ≥ lim inf l →∞ Z k q ml x + q ml α k T ≥ δ δ | g ( x ) | d x, where |{ x : k q m l x + q m l α k ≥ δ }| , | F m l ,δ | = 1 − δ . Thuslim l →∞ k g ( x + q m l α ) − g ( x ) k L ≥ lim inf l →∞ (4 δ k g k L − δ Z F cml,δ | g ( x ) | d x ) ≥ lim inf l →∞ (4 δ k g k L − δ k g k L ∞ ) . By (4.6) | g ( x ) | = b | c | ˆ λ ( x − α ) for some constant b >
0, thus k g k L , k g k L ∞ are positive finite numbers,so one can choose δ ∼ δ k g k L − δ k g k L ∞ is strictly positive. This contradicts with(6.2). (cid:3) Acknowledgement
This research was partially supported by the NSF DMS-1401204. I would like to thank SvetlanaJitomirskaya for useful discussions.
References
1. Avila, A., Jitomirskaya, S. and Marx, C.A., 2016. Spectral theory of extended Harper’s model and a question byErd˝os and Szekeres. arXiv preprint arXiv:1602.05111.2. Jitomirskaya, S. and Marx, C.A., 2012. Analytic quasi-perodic cocycles with singularities and the Lyapunovexponent of extended Harper’s model. Communications in mathematical physics, 316(1), pp.237-267.3. Jitomirskaya, S. and Marx, C.A., 2013. Erratum to: Analytic quasi-perodic cocycles with singularities and theLyapunov Exponent of Extended Harper’s Model. Communications in mathematical physics, 317, pp.269-271.
Department of Mathematics, University of California, Irvine CA, 92717
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