Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on R d
aa r X i v : . [ m a t h . SP ] J un ABSENCE OF EIGENVALUES OF ANALYTIC QUASI-PERIODICSCHR ¨ODINGER OPERATORS ON R d YUNFENG SHI
Abstract.
In this paper we study on L ( R d ) the quasi-periodic Schr¨odingeroperator H = − ∆ + λV ( x ) , where V is a real analytic quasi-periodic functionand λ >
0. We first show that H has no eigenvalues in low energy region . Wealso provide in low energy region the new phase transition parameter, i.e. thecompetition between the strength of coupling and the length for frequencies. Introduction and main results
Let H = − ∆ + V ( x ) be the Schr¨odinger operator defined on L ( R d ). Thequestion of determining for which potential V ( x ) such H has no positive eigenvaluesattracted a great deal of attention over years, see e.g., [Kat59, Sim69, Agm70, FH83,JK85, IJ03, Mar19]. In general, those works require V ( x ) to obey certain decayinglaw at ∞ . This may exclude a large class of potentials without decaying, such asthe quasi-periodic one. The aim of the present paper tries to establish absenceof positive eigenvalues for some quasi-periodic Schr¨odinger operators on R d for arbitrary d ≥ R . In 1 D and periodic potentials case, the Floquet theory workswell, and as a result the spectrum consists of intervals (bands) and is purely abso-lutely continuous ( ac ). When the periodic potential is replaced by a quasi-periodicone, the spectral properties change dramatically resulting from the so-called small-divisors effect. Dinaburg-Sinai [DS75] first proved the existence of ac spectrum bya using a KAM reducibility argument. This result was further developed later byR¨ussmann [R¨us80] and Moser-P¨oschel [MP84]. Surace [Sur90] showed for potentialsof the form V ( x ) = λ (cos x + cos( θ + ωx )) , (1.1)the whole spectrum contains no eigenvalues if λ is small and the frequency ω is Dio-phantine. He extended the multi-scale analysis (MSA) of Fr¨ohlich-Spencer [FS83] towork in momentum space. The breakthrough came from Eliasson [Eli92], in whichhe proved for any nonconstant analytic quasi-periodic potential and Diophantinefrequency, the spectrum is purely ac in high energy region. His result also appliedto the whole spectrum for small quasi-periodic potentials. Moreover, he showed Date : June 30, 2020.
Key words and phrases.
Absence of eigenvalues, Multi-dimensional continuous Schr¨odingeroperators, Quasi-periodic potentials, Phase transition. the spectrum is a Cantor set for a generic set of analytic quasi-periodic potentials,which exhibits completely different spectral features compared with periodic poten-tials. For quasi-periodic potentials λV ( x ) with large λ >
0, Anderson localization(AL, i.e., pure point spectrum with exponentially decaying eigenfunctions) and de-localization transition [YZ14] may be expected. Sorets-Spencer [SS91] improvedHerman’s subharmonic trick to obtain the positivity of Lyapunov exponent at lowenergy for potentials (1.1) if λ ≫
1. Fr¨ohlich-Spencer-Wittwer [FSW90] proved thefirst AL for potentials of the form (1.1) at low energy if λ ≫
1. They performed aMSA for Green’s functions in x ∈ R space directly. It has been proven by You-Zhou[YZ14] that there exists the phase transition from singular to purely ac spectrumfor 1 D and two frequencies quasi-periodic operators if the coupling is large. Veryrecently, Binder-Kinzebulatov-Voda [BKV17] proved a non-perturbative AL (i.e.AL under only positive Lyapunov exponents assumption) for general analytic po-tentials in finite energy intervals by applying methods of [BG00, GS08]. For moreresults in 1 D quasi-periodic setting, we refer to [Bje06, DG14, Liu18].If one increases the space dimension to d ≥
2, the situation becomes significantlymore complicated and much less results were obtained in this setting. Unlike the1 D case, it was conjectured (i.e. the Bethe-Sommerfeld conjecture [Par08, PS10])that the spectrum of any periodic Schr¨odinger operator in higher dimensions con-tains finitely many gaps. By contrast, Damanik-Fillman-Gorodetski [DFG19] hadconstructed dD almost-periodic Schr¨odinger operators whose spectrum is a general-ized Cantor set of zero Lebesgue measure. The first result concerning ac spectrumof a 2 D quasi-periodic Schr¨odinger operator was due to Karpeshina-Shterenberg[KS19], where they obtained the existence of ac spectrum at high energy, andthe spectrum contains a semi-axis. The proof of this elegant result is based ontheir new MSA in momentum space. Before this work, they [KS13] got a similarresult for ( − ∆) l + V ( x ) with l ≥
2. We also mention the work of Karpeshina-Lee-Shterenberg-Stolz [KLSS17], in which the existence of ballistic transport for theSchr¨odinger operator with limit-periodic or quasi-periodic potential in dimensiontwo was established. There are also some results [PS16, PS12] for quasi-periodic(even almost periodic) operators on R d in dealing with asymptotic expansion of thespectral functions (such as the IDS).If d ≥
3, as far as we know, there are no localization or de-localization resultsavailable for a quasi-periodic Schr¨odinger operator on R d . This is the other moti-vation of the present work.In this paper we study on R d ( d ≥
1) the Schr¨odinger operators with analyticquasi-periodic potentials λV ( x ) and prove the absence of eigenvalues in low energyregion . Combined our main theorem and results of [Bje06, Bje07, YZ14, BKV17],we also provide the new phase transition parameter in low energy region (see (4)of Remark 1.1 in the following).Here is the set up of our main result:Let b i ∈ N (1 ≤ i ≤ d ) and b = P di =1 b i . Define T bd = Q di =1 ( R / π Z ) b i . Write ~θ = ( ~θ , · · · , ~θ d ) ∈ T bd for ~θ i = ( θ i , · · · , θ ib i ) ∈ T b i (1 ≤ i ≤ d ). Similarly, let ~ω =( ~ω , · · · , ~ω d ) ∈ [0 , π ] bd with ~ω i ∈ [0 , π ] b i (1 ≤ i ≤ d ). Let x = ( x , · · · , x d ) ∈ R d and define x~ω = ( x ~ω , · · · , x d ~ω d ) ∈ R bd . We consider on L ( R d ) the following Schr¨odinger operators with quasi-periodicpotentials H ( ~θ ) = − ∆ + λV ( ~θ + Kx~ω ) , (1.2) where the real function V constant is the potential, λ ≥ ~ω isthe frequency, ~θ is the phase and K > V ∈ C ω ( T bd ; R ) satisfying R T bd V ( ~θ )d ~θ = 0.Without loss of generality, we may assume for some ρ > ∀ k = ( k , · · · , k d ) ∈ Z bd ( k i = ( k i , · · · , k ib i ) ∈ Z b i , ≤ i ≤ d ) | b V k | ≤ e − ρ | k | , (1.3)where b V k = R T bd V ( ~θ ) e − ik · ~θ d ~θ (with k · ~θ = d P i =1 k i · ~θ i = d P i =1 b i P j =1 k ij θ ij ) is the Fouriercoefficient of V ( ~θ ) and | k | = max ≤ i ≤ d, ≤ j ≤ b i | k ij | .Denote by mes( · ) the Lebesgue measure. We have Theorem 1.1.
Let H ( ~θ ) be defined by (1.2) with V satisfying (1.3) . Then there is c ⋆ = c ⋆ ( b, d ) > (depending on b , · · · , b d , d ) such that, for δ > and < c < c ⋆ ,there exists ε = ε ( δ, c , ρ, b, d ) > such that the following holds: If < λK ≤ ε ,then there is some Ω = Ω(
K, λ ) ⊂ [0 , π ] bd satisfying mes([0 , π ] bd \ Ω) ≤ δ suchthat H ( ~θ ) has no eigenvalues in h − K (log K λ ) c , K (log K λ ) c i for every ~θ ∈ T bd and ~ω ∈ Ω . Remark 1.1. (1) Theorem 1.1 is non-vacuous, i.e., there exists a portion of thespectrum in I = h − K (log K λ ) c , K (log K λ ) c i . Indeed (see LemmaA.1 in the Appendix), one can show for any E ≥ M ≥ λ | V | max , σ ( H ( ~θ )) ∩ [ E − M, E + M ] = ∅ , where σ ( · ) denotes the spectrum. Inparticular, σ ( H ( ~θ )) ∩ [ − M, M ] = ∅ for any M > λ | V | max . Moreover, it isobvious that I ր R as λ → K → ∞ .(2) We make no smallness assumptions on λ >
0, and prove the first purelycontinuous spectrum result in low energy region for quasi-periodic opera-tors on R d for arbitrary d ≥
1. This can be regarded as a supplement ofresults of [KS19], where the existence of ac spectrum for a 2 D quasi-periodicSchr¨odinger operator was established in high energy region .(3) The theorem suggests that the length of frequencies is relevant for thespectral properties of continuous quasi-periodic Schr¨odinger operators.(4) If d = 1 , b > V satisfies some conditions, it was proved in [BKV17](combined with results of [Bje06]) that for λ ≥ λ ( K~ω, b ) > , λ / ] is pure point with exponential decay eigenfunctions (i.e. AL) fora.e. Diophantine ~ω . By contrast, our Theorem 1.1 implies that there areno eigenvalues in [0 , Cλ ] at all for any λ > K ≥ C √ λ . Those motivateus to provide the new phase transition parameter log K log λ from pure point to purely continuous spectrum in low energy region . Precisely, there may exista competition between λ and K : log K log λ > C > purely continuous spectrum; and log K log λ ≤ C may show pure point spectrum (even AL). Werefer to [YZ14] for the phase transition parameter involving the couplingand energy: log E log λ .(5) Our results can be extended to the case H = ( − ∆) l + λV ( ~θ + Kx~ω ), where l ∈ Z , l ≥ V ( · ) is Gevrey regular (see [Shi19b]). YUNFENG SHI (6) It should be true that the spectrum in I is purely absolutely continuousfor 0 < λK ≪ ~ω . However, we think Anderson localizationoccurs at low energy for λK ≫ ~ω .We outline the proof of Theorem 1.1:The main scheme is to perform MSA in momentum space Θ ∈ R d . Let E be aneigenvalue of H ( ~θ ), i.e. H ( ~θ )Ψ = E Ψ for some 0 = Ψ ∈ L ( R d ). Let k ~ω = ( k · ~ω , · · · , k d · ~ω d ) . Denote by b Ψ the Fourier transformation of Ψ. Then by direct computations (seeLemma 4.1 for details), the vector Z = n Z k = e − ik · ~θ b Ψ(Θ ′ + K k ~ω ) o k ∈ Z bd will satisfy h (Θ) Z = EK Z , where( h (Θ) Z ) k = X k ′ ∈ Z bd ε b V k − k ′ Z k ′ + d X i =1 (Θ i + k i · ~ω i ) Z k , ε = λK , Θ = Θ ′ K .
In addition, we can even show Z k grows at most polynomially in k for a.e. Θ.Thus the asymptotic property of Z k can be controlled by fast off-diagonal decayingproperties of the Green’s functions (if exists) G Λ ( E ; Θ) = ( R Λ ( h (Θ) − E ) R Λ ) − , Λ ⊂ Z bd . At this stage, it suffices to get certain off-diagonal decaying estimates on G Λ ( E ; Θ)as | Λ | → ∞ . For this purpose, it needs to make quantitative restrictions on ~ω , Θand even on E due to the small-divisors difficulty. Usually, the KAM and MSAmethods are powerful to overcome the small-divisors. In continuous quasi-periodicoperators case, Surace [Sur90] performed the first MSA scheme in momentum spaceΘ ∈ R but restricted to diagonal part of the form(Θ + k + k ω ) , ( k , k ) ∈ Z . Because of this special structure, Surace can even obtain uniform in E ∈ R estimate.However, the proof of [Sur90] depends heavily on eigenvalue variations and is hardto work for Θ in higher dimensions. For recent Θ ∈ R type MSA, we also refer to[DG14] in dealing with the inverse spectral theory for quasi-periodic operators on R . For general Θ ∈ R d in momentum space the only known MSA is developedin recent papers [KS13, KS19] for d = 2. However, for arbitrary d ≥ ∈ T d [BGS02, Bou07,BK19, JLS20, Shi19b] via techniques of semi-algebraic geometry arguments andsubharmonic estimates. As compared with compact momentum case (i.e. Θ ∈ T d ),there comes new difficulty for general Θ ∈ R d : For any finite Λ ⊂ Z bd ,sup k ∈ Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 (Θ i + k i · ~ω i ) − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ may frequently occur while | Θ | ≫ | E | ≫
1. To avoid this difficulty, one maymake the assumption that for some
C > , − C ≤ E ≤ C. (1.5)Once (1.5) is satisfied, Θ stays essentially in a compact set of size ∼ | Λ | , and thesmall divisors presented in (1.4) could be handled via methods of [Bou07, JLS20].The elimination frequencies argument of Bourgain [Bou07] based on semi-algebraicgeometry theory is likely applicable. Consequently, we may expect that H ( ~θ ) hasno eigenvalues in interval [ − CK , CK ] if 0 < ε ≤ ε ( C ).It is reasonable that the absence of eigenvalues for H ( ~θ ) should hold in a muchlonger energy interval, which of course needs a proof with new ideas. In this paperwe will provide such a proof based on the following observations: • The first step of MSA is in general based on perturbative argument, inparticular the Neumann series expansions. It is important that the largedeviation estimate (LDE) for Green’s functions is valid for every E ∈ R inthis step. • Once the first step of MSA is finished, we will deal with the second iterationstep. To propagate the LDE, some restrictions on ~ω are necessary, which isessentially necessary even in Θ ∈ T d case. What’s new here is that we canmake quantitative restrictions on E in this step. More precisely, the firststep MSA implies that LDE holds in scales interval N ≤ N ≤ | log ε | C .The second MSA step will start with N = | log ε | C . To avoid (1.4), we canactually set | E | ≤ | log ε | C . In this energy interval, we could perform theiteration similar to that of [Bou07, JLS20] together with some technicalimprovements. • Since | E | ≤ | log ε | C is negligible as compared with the later general MSAscales N ≫ | log ε | C , the iterations become similar to that done in thesecond step.In conclusion, our new aspect here is that we focus on the MSA (in the momentumspace Θ ∈ R d ) in the first two steps and make effective restrictions on E in thesecond step.Of course, the final aim is to show H ( ~θ ) has no eigenvalues on R , which seemsdifficult to handle via the present methods.The structure of the paper is as follows. Some preliminaries are introduced in §
2. The MSA in momentum space is established in §
3. In §
4, we finish the proofof Theorem 1.1. Some useful estimates are included in the Appendix.2.
Preliminaries
Some notation.
For any x ∈ R d , let | x | = max ≤ i ≤ d | x i | . For Λ ⊂ R d , weintroduce diam(Λ) = sup n,n ′ ∈ Λ | n − n ′ | , dist( m, Λ) = inf n ∈ Λ | m − n | . For Θ ∈ R d and 1 ≤ j ≤ d , let Θ ¬ j = (Θ , · · · , Θ j − , Θ j +1 · · · , Θ d ) ∈ R d − .For x ∈ R d and ∅ 6 = X ⊂ R d + d , define the x -section of X to be X ( x ) = { y ∈ R d : ( x, y ) ∈ X } . This assumption is essentially satisfied in [BGS02, Bou07, BK19, JLS20, Shi19b].
YUNFENG SHI
For example, X (Θ ¬ j ) = { Θ j ∈ R : (Θ j , Θ ¬ j ) ∈ X } if ∅ 6 = X ⊂ R d .Throughout this paper, we assume ρ ∈ (0 ,
1) for simplicity.2.2.
Aubry duality.
It is easy to check that ifΨ( x ) = e i Θ · x X k ∈ Z bd Ψ k e ik · ( ~θ + x~ω ) is a solution of ( − ∆ + εV ( ~θ + x~ω ))Ψ( x ) = E Ψ( x ) (i.e. the Floquet-Bloch solution),then { Ψ k } k ∈ Z bd satisfies the following equation X k ′ ∈ Z bd ε b V k − k ′ Ψ k ′ + d X i =1 (Θ i + k i · ~ω i ) Ψ k = E Ψ k . This motivates us to study the following unbounded quasi-periodic operators on Z d ( h (Θ) Z ) k = X k ′ ∈ Z bd ε b V k − k ′ Z k ′ + d X i =1 (Θ i + k i · ~ω i ) Z k . (2.1)We call h (Θ) the Aubry duality of − ∆ + εV ( ~θ + x~ω ) . For Aubry duality results indiscrete case, we refer to [BJ02, JK16, Shi19b].2.3.
Green’s functions and elementary regions.
If Λ ⊂ Z bd , denote h Λ (Θ) = R Λ h (Θ) R Λ , where R Λ is the restriction operator and h (Θ) is given by (2.1). Definethe Green’s function as G Λ ( E ; Θ) = ( h Λ (Θ) − E + i − . Denote by Q N an elementary region of size N centered at 0 (as in [JLS20]): Q N = [ − N, N ] bd or Q N = [ − N, N ] bd \ { k ∈ Z bd : k ij ∈ ς ij , ≤ i ≤ d, ≤ j ≤ b i } , where ς ij ∈ {{ n < } , { n > } , ∅} and at least two ς ij are not ∅ . Denote by E N the set of all elementary regions of size N centered at 0. Let E N be the set of alltranslates of elementary regions: E N = S k ∈ Z bd ,Q N ∈E N { k + Q N } . Semi-algebraic sets.Definition 2.1 (Chapter 9, [Bou05]) . A set
S ⊂ R n is called a semi-algebraic setif it is a finite union of sets defined by a finite number of polynomial equalities andinequalities. More precisely, let { P , · · · , P s } ⊂ R [ x , · · · , x n ] be a family of realpolynomials whose degrees are bounded by d . A (closed) semi-algebraic set S isgiven by an expression S = [ j \ ℓ ∈L j { x ∈ R n : P ℓ ( x ) ς jℓ } , (2.2)where L j ⊂ { , · · · , s } and ς jℓ ∈ {≥ , ≤ , = } . Then we say that S has degree at most sd . In fact, the degree of S which is denoted by deg( S ), means the smallest sd overall representations as in (2 . Lemma 2.2 (Tarski-Seidenberg Principle, [Bou05]) . Denote by ( x, y ) ∈ R d + d theproduct variable. If S ⊂ R d + d is semi-algebraic of degree B , then its projections Proj x S ⊂ R d and Proj y S ⊂ R d are semi-algebraic of degree at most B C , where C = C ( d , d ) > . Lemma 2.3 ([Bou07]) . Let
S ⊂ [0 , d = d + d be a semi-algebraic set of degree deg( S ) = B and mes d ( S ) ≤ η , where log B ≪ log η . Denote by ( x , x ) ∈ [0 , d × [0 , d the product variable. Suppose η d ≤ ǫ. Thenthere is a decomposition of S as S = S ∪ S with the following properties. The projection of S on [0 , d has small measure mes d (Proj x S ) ≤ B C ( d ) ǫ, and S has the transversality property mes d ( L ∩ S ) ≤ B C ( d ) ǫ − η d , where L is any d -dimensional hyperplane in R d s.t., max ≤ j ≤ d | Proj L ( e j ) | < ǫ, wherewe denote by e , · · · , e d the x -coordinate vectors. LDT for Green’s functions
The main result of this section is the following LDT for Green’s functions.
Theorem 3.1 (LDT) . There exists c ⋆ ( b, d ) > such that the following holds: Forany < c ≤ c ⋆ , there exists N = N ( c , ρ, b, d ) > such that if log log 1 ε ≥ N , then for all N ≥ N we have (i) There is some semi-algebraic set Ω N = Ω N ( c , ε, ρ, b, d ) ⊂ [0 , π ] bd with deg(Ω N ) ≤ N d , and as ε → , mes [0 , π ] bd \ \ N ≥ N Ω N → . (ii) If ~ω ∈ Ω N and E ∈ R satisfying | E | ≤ | log ε | c , then there exists some set X N = X N ( ~ω, E ) ⊂ R d such that sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N (Θ ¬ j )) ≤ e − N c , and for Θ / ∈ X N , Q ∈ E N , k G Q ( E ; Θ) k ≤ e √ N , | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ρ | n − n ′ | for | n − n ′ | ≥ N/ . Proof of Theorem 3.1.
The proof is based on the multi-scale analysis scheme asin [Bou07, JLS20]. However, we formulate it into three steps.
STEP 1: The First Step
In this initial step we use a perturbative argument. It is important that in thisstep we obtain uniform measure estimates both on E ∈ R and ~ω. Lemma 3.2.
Given δ > , let X N = [ | k |≤ N ( Θ ∈ R d : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X i =1 (Θ i + k i · ~ω i ) − E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ ) . YUNFENG SHI
Then sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N (Θ ¬ j )) ≤ C (2 N + 1) bd √ δ, where C > is an absolute constant.Moreover, if ε − ≥ δ − (2 N + 1) bd , then for any Θ / ∈ X N and Λ ⊂ [ − N, N ] bd , k G Λ ( E ; Θ) k ≤ δ − , | G Λ ( E ; Θ)( n , n ′ ) | ≤ δ − e − ρ | n − n ′ | . Proof.
The proof is perturbative and based on the Neumann series argument: Theproof of measure bound is based on Lemma 4.7 in [Shi19a], and the estimate onGreen’s functions can be found in the proof of Theorem 4.3 in [JLS20]. (cid:3)
If we take C (2 N + 1) bd √ δ = e − N c in the above lemma, we obtain Proposition 3.3.
Let < c < / . Then there exists N = N ( c , ρ, b, d ) > such that the following: If N ≤ N ≤ | log ε | c , then for all E ∈ R and ~ω ∈ [0 , π ] bd there is some X N = X N ( E, ~ω ) ⊂ R d such that sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N (Θ ¬ j )) ≤ e − N c , (3.1) and, if Θ / ∈ X N and Q ∈ E N , k G Q ( E ; Θ) k ≤ e √ N , (3.2) | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ρ | n − n ′ | for | n − n ′ | ≥ N/ . (3.3) Proof.
Let X N be given by Lemma 3.2. Then the measure bound (3.1) can bederived from choosing C (2 N + 1) bd √ δ = e − N c , i.e., δ = C − (2 N + 1) − bd e − N c . (3.4)We then turn to the Green’s function estimates. Let Θ X N . Since (3.4), thecondition ε − ≥ δ − (2 N + 1) bd is equivalent to ε − ≥ C (2 N + 1) bd e N c It suffices to let N ≥ N ( c , b, d ) > ε − ≥ e N c ≥ C (2 N + 1) bd e N c , (3.5)that is N ( c , b, d ) ≤ N ≤ | log ε | c . From (3.5) and 0 < c < /
4, we have2 δ − ≤ e N c ≤ e √ N , which implies (3.2). Finally, the exponential decay bound(3.3) follows from: If | n − n ′ | ≥ N/
10 and N ≥ N ( c , ρ, b, d ) > , then | G Q ( E ; Θ)( n , n ′ ) | ≤ e √ N − ρ N e − ρ | n − n ′ | ≤ e − ρ | n − n ′ | . (cid:3) STEP 2: The Second Step
In the second step we will propagate LDT from (scales) h N , | log ε | c i to h | log ε | c , e | log ε | / i . For this purpose, further removal of ω is necessary and the semi-algebraic geometryarguments established by Bourgain [Bou07] play a key role. In what follows wealways assume E ∈ I ε = h − log ε | c , log ε | c i . (3.6)Denote B ( N ) = { Θ ∈ R d : | Θ | ≤ N } . Then we have Proposition 3.4.
Let | log ε | c ≤ N ≤ e | log ε | / , N ∼ (log N ) /c . Then thereexist constants < c ( b, d ) < c ( b, d ) < c ( b, d ) ≪ , and semi-algebraic set Ω N ⊂ [0 , π ] bd satisfying deg(Ω N ) ≤ N d , mes([0 , π ] bd \ Ω N ) ≤ N − c such that the following holds: If log log ε ≥ N ( c , b, d ) > and ~ω ∈ Ω N , then forall ( E, Θ) ∈ I ε × B (100 bN ) there is some N c < M < N c such that for all k ∈ [ − M, M ] bd \ [ − M bd , M bd ] bd , one has Θ + k ~ω / ∈ X N , here X N is given byProposition 3.3. Remark 3.1.
The conditions log log ε ≥ N ( c , b, d ) > N ≤ e | log ε | / canensure that N ( c , ρ, b, d ) ≤ N ≤ | log ε | c . Proof.
We will eliminate the variables (Θ , E ) ∈ B (100 bN ) × I ε . This needs make quantitative restrictions on ~ω .Let N c < L < N c , where 0 < c < c < S L be the set of all ( ~ω, Θ , E ) ∈ [0 , π ] bd × B (100 bN ) × I ε such that, for any n ∈ [ − L, L ] bd and ~ω ∈ [0 , π ] bd , Θ + n ~ω ∈ X N ( E, ~ω ) . Applying Proposition 3.3 givesdeg( S L ) ≤ L C ( b,d ) , (3.7)sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( R \ S L (Θ ¬ j )) ≤ e − N c / . (3.8)Fix I ⊂ { , · · · , d } and define A to be all ( ~ω, Θ , y, E ) ∈ [0 , π ] bd × B (100 bN ) × R I × I ε satisfying ( ~ω, (Θ j + y j ) j ∈ I , (Θ j ) j / ∈ I , E ) / ∈ S L . (3.9)Obviously, by (3.7) and (3.9), deg( A ) ≤ L C ( b,d ) . (3.10)Fix ~ω ∈ [0 , π ] bd and consider A := A ( ~ω ) ⊂ B (100 bN ) × R I × I ε . Assume for y I = ( y i ) i ∈ I , y I / ∈ B I (200 bN ) = { y ∈ R I : | y | ≤ bN } . Then forall n ∈ [ − L, L ] bd + E N , one has since | Θ | ≤ bN and | E | ≤ | log ε | c ≤ N , d X i =1 (Θ i + y i + n i · ~ω i ) − E ≥ X i ∈ I (Θ i + y i + n i · ~ω i ) − E ≥ N , (3.11)which shows that there is no resonances in this case. In other words, we must have A ⊂ B (100 bN ) × B I (200 bN ) × I ε . (3.12)By (3.10) and Tarski-Seidenberg principle (i.e., Lemma 2.2), we obtaindeg( A ) ≤ L C ( b,d ) . (3.13)From (3.8), for all (Θ , E ) ∈ B (100 bN ) × I ε , we havemes I ( A (Θ , E )) ≤ η := e − N c / . (3.14)At this stage, we need a lemma for eliminating multi-variable: Lemma 3.5.
Let
S ⊂ [0 , bN ] s + r be a semi-algebraic set of degree B and suchthat mes s ( S ( y )) < η for ∀ y ∈ [0 , bN ] r and log( BN ) ≪ log 1 η . Then the set ( x , · · · , x r ) ∈ [0 , bN ] s r : \ ≤ i ≤ r S ( x i ) = ∅ is semi-algebraic of degree at most B C and measure at most N C B C η s − r − r ( r − / , where C = C ( s, r, b ) > . Proof of Lemma 3.5.
The degree bound follows from Tarski-Seidenberg prin-ciple Lemma 2.2. The measure bound is derived as follows. We first divide[0 , bN ] s + r into about N C ( s,r,b ) many unit cubes. Then applying Lemma 1.18of [Bou07] gives measure bound B C η s − r − r ( r − / on each such unit cube. Finally,it suffices to take account of all those cubes. (cid:3) From (3.13) and (3.14), we have by using Lemma 3.5 (with s = | I | , r = d +1 , B = L C and η = e − N c / ) that A = n ( y ( i ) ) ≤ i ≤ d +1 : T ≤ i ≤ d +1 A ( y ( i ) ) = ∅ o is a semi-algebraic set withdeg( A ) ≤ L C ( b,d, | I | ) , mes( A ) ≤ η := N C ( b,d, | I | ) η | I | − d − − d ( d +1) / . (3.15)Define B = (cid:16) ~ω, ( y ( i ) ) ≤ i ≤ d +1 (cid:17) : ~ω ∈ [0 , π ] bd , \ ≤ i ≤ d +1 A ( ~ω, y ( i ) ) = ∅ ⊂ [0 , π ] bd × [0 , bN ] | I | d +1 . By Lemma 2.2, deg( B ) ≤ L C ( d, | I | ) . Write ω = ( ~ω i ) i/ ∈ I and e ω := ( ~ω i ) i ∈ I . Then byFubini Theorem and (3.15),mes( B ( ω )) ≤ η , deg( B ( ω )) ≤ L C . (3.16)Notice that for any C ⋆ >
1, one has η ≫ N C if N ≥ N ( C ⋆ , c , b, d, | I | ) >
0. Oneconsiders the set B containing e ω , which is defined by the following: there is somesequence n (1) , · · · , n (2 d +1 ) ∈ Z | I | b I ( b I = P i ∈ I b i ) satisfying L C α ≤ min i ∈ I, ≤ j ≤ b i | n ( α ) ij | ≤ | n ( α ) | ≤ L C α (1 ≤ α ≤ d +1 ) , (3.17) such that ( e ω, n (1) e ω, · · · , n (2 d +1 ) e ω ) ∈ B ( ω ) , where ≪ C ≪ C ≪ · · · ≪ C d +1 ≪ C d +1 . At this stage, we will introduce a key lemma for eliminating frequencies:
Lemma 3.6.
Let
S ⊂ R sr be a semi-algebraic set of degree B and mes( S ) < η with η > . Let b = P ri =1 b i , b i ∈ N . For ~ω i ∈ [0 , π ] b i , ~ω = ( ~ω , · · · , ~ω r ) ∈ [0 , π ] br and n = ( n , · · · , n r ) ∈ Z br ( n i ∈ Z b i ) , define n ~ω = ( n · ~ω , · · · , n r · ~ω r ) . For any
C > , define N , · · · , N s − ⊂ Z br to be finite sets with the followingproperty: min ≤ i ≤ r, ≤ j ≤ b i | n ij | > ( B max ≤ i ≤ r | m i | ) C , where n ∈ N l , m ∈ N l − (2 ≤ l ≤ s − .Then there is some C = C ( r, s, b ) > such that for max n ∈N s − | n | C < η , one has mes( { ~ω ∈ [0 , π ] br : ∃ n ( i ) ∈ N i s.t., ( ~ω, n (1) ~ω, · · · , n ( s − ~ω ) ∈ S} ) ≤ B C ( min n ∈N min i,j | n ij | ) − . Proof of Lemma 3.6.
The proof is similar to that of Lemma 1.20 in [Bou07] byiterating Lemma 2.3. The main difference is that we allow S to vary in R sr ratherthan in the unit cube. Bourgain’s proof remains applicable since Lemma 2.3 permitsre-scaling. Moreover, at i -th ( i ≤ s −
1) iteration step, the valid sets are essentiallycontained in [0 , max n ∈N s − i | n | ] r ( s − i +1) . We omit the details here. (cid:3) Then from (3.16) and Lemma 3.6 (set r = | I | , s = 2 d +1 + 1), we havemes( B ) ≤ L − C L C ≤ L − . (3.18)Define G L := \ ∅6 = I ⊂{ , ··· ,d } (cid:8) ~ω ∈ [0 , π ] bd : ~ω = ( ω, e ω ) , e ω / ∈ B (cid:9) . Thus ~ω / ∈ G L if and only if, there are ∅ 6 = I ⊂ { , · · · , d } and some sequence n (1) , · · · , n (2 d +1 ) ∈ Z | I | b I satisfying (3.17) such that, ( ω, n (1) e ω, · · · , n (2 d +1 ) e ω ) ∈ B . This implies that G L is a semi-algebraic set. Furthermore, by (3.17) and (3.18),mes([0 , π ] bd \ G L ) ≤ L − , deg( G L ) ≤ L CC d +1 . We should remark that G L also depends on C , · · · , C d +1 .Finally, for 0 < c ( b, d ) ≪ c ( b, d ) ≪
1, we can choose appropriate L l , C , · · · , C d +1 and then iterate along every axis direction (i.e., induction on I ) as done by Bour-gain [Bou07] (see the proof of the Claim ) if ~ω ∈ G L l . The number of all possible L l , C , · · · , C d +1 is finite and depends only on b, d . In particular, we havedeg(Ω N ) ≤ CN c CC d +1 ≤ N d , mes([0 , π ] bd \ Ω N ) ≤ CN − c ≤ N − c , where Ω N = T L l ,C , ··· ,C d +1 G L l and 0 < c ( b, d ) ≪ c ( b, d ). In addition, if ~ω ∈ Ω N ,then for all ( E, Θ) ∈ I ε × B (100 bN ) there is some N c < M < N c such thatfor all k ∈ [ − M, M ] bd \ [ − M bd , M bd ] bd , one has Θ + k ~ω / ∈ X N . (cid:3) Combining the above proposition and Cartan’s estimate on subharmonic func-tions, we can finish the proof of LDT in [
N, N ].Since (3.6) and (3.11), it suffices to consider in caseΘ ∈ B (1000 bN ) . (3.19) Proposition 3.7.
Assume the assumptions of Proposition 3.4 are satisfied. Let Θ satisfy (3.19) . Then for ~ω ∈ Ω N , we have (1) Fix ≤ j ≤ d and Θ ¬ j ∈ B d − (1000 bN ) . Write Θ = (Θ j , Θ ¬ j ) ∈ R d .Assume there exist e N ∈ [ N c / , N c ] and ¯Λ ⊂ Λ ∈ E e N with diam(¯Λ) ≤ e N bd and Λ ⊂ B (1000 bN ) such that, for any k ∈ Λ \ ¯Λ , there exists some E N ∋ W ⊂ Λ \ ¯Λ such that dist( k , Λ \ ¯Λ \ W ) ≥ N / , and Θ + k ~ω / ∈ X N .Let Y Θ = (cid:26) y ∈ R : | y − Θ j | ≤ e − ρN , | Θ j | ≤ bN , k G Λ ( E ; ( y, Θ ¬ j )) k ≥ e √ e N (cid:27) . Then mes( Y Θ ) ≤ e − e N . (2) Fix N ⋆ ∈ [ N, N ] . If E ∈ I ε and < c < c / , then there exists someset X N ⋆ = X N ⋆ ( E, ~ω ) ⊂ R d such that sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N ⋆ (Θ ¬ j )) ≤ e − N c ⋆ , and for Θ / ∈ X N ⋆ , Q ∈ E N ⋆ , | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ( ρ − C ( ρ,b,d ) √ N ) | n − n ′ | for | n − n ′ | ≥ N ⋆ / . Proof. (1) Let D be the e − ρN neighbourhood of Θ j in the complex plane, i.e., D = { y ∈ C : |ℑ y | ≤ e − ρN , |ℜ y − Θ j | ≤ e − ρN } . From assumptions, we have for all k ∈ Λ \ ¯Λ and Q ∈ E N , k G Q ( E ; Θ + k ~ω ) k ≤ e √ N , (3.20) | G Q ( E ; Θ + k ~ω )( n , n ′ ) | ≤ e − ρ | n − n ′ | for | n − n ′ | ≥ N / . (3.21)Note that for all n , n ′ ∈ [ − N , N ] bd , e − ρN < e − ρN − ρ | n − n ′ | . Then by Lemma B.1, (3.20) and (3.21), we have for any y ∈ D , Q ∈ E N and k ∈ Λ \ ¯Λ, k G Q ( E ; (Θ j + y, Θ ¬ j ) + k ~ω ) k ≤ e √ N , (3.22) | G Q ( E ; (Θ j + y, Θ ¬ j ) + k ~ω )( n , n ′ ) | ≤ e − ρ | n − n ′ | for | n − n ′ | ≥ N / . (3.23)Applying Lemma B.2 with M = M = N implies for any y ∈ D , k G Λ \ ¯Λ ( E ; (Θ j + y, Θ ¬ j )) k ≤ N + 1) bd e √ N ≤ e √ N . (3.24)We then can use the following matrix-valued Cartan’s estimate to propagate the“smallness of measure”: Lemma 3.8 (Cartan’s estimate, [Bou05]) . Let T ( θ ) be a self-adjoint N × N matrixfunction of a parameter θ ∈ [ − δ, δ ] satisfying the following conditions: (i) T ( θ ) is real analytic in θ ∈ [ − δ, δ ] and has a holomorphic extension to D δ,δ = { θ ∈ C : |ℜ θ | ≤ δ, |ℑ θ | ≤ δ } satisfying sup θ ∈D δ,δ k T ( θ ) k ≤ K , K ≥ . (ii) For all θ ∈ [ − δ, δ ] , there is subset V ⊂ [1 , N ] with | V | ≤ M such that k ( R [1 ,N ] \ V T ( θ ) R [1 ,N ] \ V ) − k ≤ K , K ≥ . (iii) Assume mes { θ ∈ [ − δ, δ ] : k T − ( θ ) k ≥ K } ≤ − δ (1 + K ) − (1 + K ) − . Let < ǫ ≤ (1 + K + K ) − M . Then mes (cid:8) θ ∈ [ − δ/ , δ/
2] : k T − ( θ ) k ≥ ǫ − (cid:9) ≤ Cδe − c log ǫ − M log( K K K , (3.25) where C, c > are some absolute constants. We will apply Lemma 3.8 with T ( y ) = h Λ ((Θ j + y, Θ ¬ j )) − E, δ = δ = 2 e − ρN . (3.26)It suffices to verify the assumptions of Lemma 3.8. By assumptions (3.19), Λ ⊂ B (1000 bN ) and (3.24), one has K = O ( N ) , M = | ¯Λ | ≤ C ( b, d ) e N / , K = e √ N . (3.27)Since LDT holds at scale N for y being outside a set of measure at most e − N c .Applying Lemma B.2 yields k T − ( y ) k ≤ N + 1) bd e √ N ≤ e √ N = K for y being outside a set of measure at most(2 e N + 1) bd e − N c ≤ e − N c / . It follows from 100 N < N / < N c that10 − δ (1 + K ) − (1 + K ) − ≥ e − N c / . This verifies (iii) of Lemma 3.8. For ǫ = e − √ e N , one has by (3.26) and (3.27), ǫ < (1 + K + K ) − M . By (3.25) of Lemma 3.8,mes( Y Θ ) ≤ e − c √ f NN f N /
10 log f N ≤ e − e N / . (2) Fix N ⋆ ∈ [ N, N ] and E ∈ I ε . As done in (3.11), to prove LDT at scale N ⋆ it suffices to restrict Θ ∈ B (10 bN ⋆ ) ⊂ B (10 bN ).Fix 1 ≤ j ≤ d, Θ ¬ j ∈ R d − and Θ = (Θ j , Θ ¬ j ) ∈ B (10 bN ⋆ ). Then Θ + n ~ω ∈ B (100 bN ) for all | n | ≤ N ⋆ . By using Proposition 3.4 ( with Θ replaced by Θ+ n ~ω ),for such Θ and any n ∈ Q ∈ E N ⋆ , there exist N c ≤ e N ≤ N c , Λ ∈ E e N and ¯Λ,such that n ∈ ¯Λ ⊂ Λ ⊂ Q, dist( n , Q \ Λ) ≥ e N / , diam(¯Λ) ≤ e N bd . Moreover, for any k ∈ Λ \ ¯Λ, Θ + k ~ω / ∈ X N and there exists some E N ∋ W ⊂ Λ \ ¯Λsuch that k ∈ W, dist( k , Λ \ ¯Λ \ W ) ≥ N / . We should remark that in the aboveargument we had applied essentially methods of [JLS20] in dealing with elementaryregions (of size e N ) near the boundary of Q .We now fix above e N, ¯Λ , Λ throughout the set { ( y, Θ ¬ j ) ∈ R d : | y − Θ j | ≤ e − ρN } . Recalling Lemma B.1 and the above constructions, we have by (1) ofProposition 3.7 that there exists a Y ⊂ { y ∈ R : | y − Θ j | ≤ e − ρN } such thatmes( Y ) ≤ e − e N / , (3.28)and for Θ j / ∈ Y , k G Λ ( E ; Θ) k ≤ e √ e N . Applying Lemma B.3 yields | G Λ ( E ; Θ)( n , n ′ ) | ≤ e − ( ρ − C √ N ) | n − n ′ | for | n − n ′ | ≥ e N/ . Cover [0 , bN ⋆ ] by pairwise disjoint e − ρN -size intervals and let X N ⋆ (Θ ¬ j ) = [ Q ∈E N⋆ , n ∈ Q, Θ=(Θ j , Θ ¬ j ) Y. (3.29)We remark that while Θ = (Θ j , Θ ¬ j ) varies on a line for fixed Θ ¬ j , the total numberof Y is bounded by 10 N ⋆ e ρN . Thus by (3.28), (3.29) and c < c /
10, one hasmes( X N ⋆ (Θ ¬ j )) ≤ C (2 N ⋆ + 1) bd e ρN e − e N / ≤ e − N ⋆c / ≤ e − N ⋆c . Suppose now Θ / ∈ X N ⋆ . Applying Lemma B.2 yields k G Q ( E ; Θ) k ≤ N c + 1) bd e √ N c ≤ e √ N ⋆ . Recalling Lemma B.3, we obtain | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ( ρ − C ( ρ,b,d ) √ N ) | n − n ′ | for | n − n ′ | ≥ N ⋆ / . (cid:3) STEP 3: The General Step
The proof is based on similar arguments as in
STEP 2 . We define for N ≥ e | log ε | / the following scales N ∼ (log N ) /c , N ∼ N /c . Then we have
Proposition 3.9.
Let Ω N i ( i = 1 , be semi-algebraic set satisfying deg(Ω N i ) ≤ N di and let ¯ ρ i ∈ [ ρ/ , ρ ) . Assume further the following holds: If ~ω ∈ Ω N i and E ∈I ε , then there exists some semi-algebraic set X N i ⊂ R d satisfying deg( X N i ) ≤ N Ci such that sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N i (Θ ¬ j )) ≤ e − N c i , and for Θ / ∈ X N i , Q ∈ E N i , k G Q ( E ; Θ) k ≤ e √ N i , | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ¯ ρ i | n − n ′ | for | n − n ′ | ≥ N i / , ( i = 1 , . Then exists some semi-algebraic set Ω N ⊂ Ω N ∩ Ω N with deg(Ω N ) ≤ N d and mes((Ω N ∩ Ω N ) \ Ω N ) ≤ N − c such that, if ~ω ∈ Ω N , then for N ≥ N ( c , ρ, b, d ) > For all E ∈ I ε and Θ ∈ B (100 bN ) , there is N c < M < N c such thatfor all k ∈ [ − M, M ] bd \ [ − M bd , M bd ] bd , Θ + k ~ω / ∈ X N . (ii) Fix N ⋆ ∈ [ N, N ] . Then there exists some set X N ⋆ = X N ⋆ ( E, ω ) ⊂ R d suchthat sup ≤ j ≤ d, Θ ¬ j ∈ R d − mes( X N ⋆ (Θ ¬ j )) ≤ e − N c ⋆ , and for Θ / ∈ X N ⋆ , Q ∈ E N ⋆ , k G Q ( E ; Θ) k ≤ e √ N ⋆ , | G Q ( E ; Θ)( n , n ′ ) | ≤ e − ( ρ − C ( ρ,b,d ) √ N ) | n − n ′ | for | n − n ′ | ≥ N ⋆ / . Proof.
The proof is similar to that in
STEP 2 , and we omit the details here. (cid:3)
From the above arguments, we can propagate LDT from (scales) h N, N /c i to h e N c / , e N c / i . Thus a standard MSA induction (on scales) can establish LDT on the whole interval[ N , ∞ ] (see [BGS02] or [JLS20] for details). This finishes the proof of Theorem3.1. (cid:3) Proof of Theorem 1.1
In this section we will prove Theorem 1.1 by using LDT and Aubry duality.We begin with a useful lemma:
Lemma 4.1.
Let Ψ ∈ H ( R d ) satisfy ( − ∆ + εV ( ~θ + x~ω ))Ψ( x ) = E Ψ( x ) . (4.1) Let b Ψ be the Fourier transformation of Ψ . Then for a.e. Θ ∈ R d and any ~θ ∈ T bd the following holds: Let Z = { Z k } k ∈ Z bd satisfy Z k = e − ik · ~θ b Ψ(Θ + k ~ω ) . Then h (Θ) Z = EZ, (4.2) where h (Θ) is given by (2.1) . Moreover, There is C = C (Θ , Ψ , b, d ) ∈ (0 , ∞ ) suchthat | Z k | ≤ C (1 + | k | ) bd for ∀ k ∈ Z bd . (4.3) Proof.
It suffices to consider Ψ ∈ C ∞ ( R d ). We note first (4.1) is equivalent to k ξ k b Ψ( ξ ) + [ εV Ψ( ξ ) = E b Ψ( ξ ) , (4.4)where k ξ k = d P i =1 | ξ i | . Since V is quasi-periodic (and analytic), we have V ( ~θ + x~ω ) = X k ∈ Z bd b V k e ik · ( ~θ + x~ω ) , and as a result, d V Ψ( ξ ) = Z R d V ( ~θ + x~ω )Ψ( x ) e − i x · ξ d x = X m ∈ Z bd Z R d b V m e im · ( ~θ + x~ω ) Ψ( x ) e − i x · ξ d x = X m ∈ Z bd b V m e im · ~θ Z R d Ψ( x ) e − i x · ( ξ − m ~ω ) d x = X m ∈ Z bd b V m e im · ~θ b Ψ( ξ − m ~ω ) . (4.5)Combining (4.4) and (4.5) yields k ξ k b Ψ( ξ ) + ε X m ∈ Z bd b V m e im · ~θ b Ψ( ξ − m ~ω ) = E b Ψ( ξ ) . (4.6)Given k ∈ Z bd , we set ξ = Θ + k ~ω in (4.6). Then e − i k · ~θ k Θ + k ~ω k b Ψ(Θ + k ~ω ) + ε X m ∈ Z bd b V k − m e − im · ~θ b Ψ(Θ + m ~ω )= Ee − i k · ~θ b Ψ(Θ + k ~ω ) , which implies (4.2). We then deal with the polynomial bound (4.3). Since b Ψ ∈ L ( R d ), we have Z R d X k ∈ Z bd | Z k | (Θ)(1 + | k | ) bd dΘ = X k ∈ Z bd | k | ) bd Z R d | b Ψ(Θ + k ~ω ) | dΘ= k b Ψ k L X k ∈ Z bd | k | ) bd < ∞ , which implies for a.e. Θ ∈ R d , X k ∈ Z bd | Z k | (Θ)(1 + | k | ) bd := C (Θ , Ψ , b, d ) < ∞ . This means that for a.e. Θ ∈ R d , | Z k | ≤ C (Θ , Ψ , b, d )(1 + | k | ) bd for ∀ k ∈ Z bd . (cid:3) We then prove our main result Theorem 1.1
Proof of Theorem 1.1.
Let e Θ = Θ
K , e E = EK , ε = λK . Suppose now H ( ~θ ) admits some eigenvalue E ∈ h − K (log K λ ) c , K (log K λ ) c i ,i.e., there is some Ψ ∈ H ( R d ) such that H ( ~θ )Ψ = E Ψ , Ψ = 0. Then we have e E ∈ I ε (see (3.6)) and ( − ∆ + λV ( ~θ + Kx~ω ))Ψ( x ) = E Ψ( x ) . Using Lemma 4.1, we know Z (Θ) = { Z k = e − k · ~θ b Ψ(Θ + K k ~ω ) } k ∈ Z bd satisfies X k ′ ∈ Z bd λ b V k − k ′ Z k ′ (Θ) + d X i =1 (Θ i + K k i · ~ω i ) Z k (Θ) = EZ k (Θ) . (4.7)Obviously, (4.7) is equivalent to X k ′ ∈ Z bd λK b V k − k ′ Z k ′ (Θ) + d X i =1 (Θ i /K + k i · ~ω i ) Z k (Θ) = EK Z k (Θ) , or in the new coordinate ( e Θ , e E, ε ), X k ′ ∈ Z bd ε b V k − k ′ e Z k ′ ( e Θ) + d X i =1 ( e Θ i + k i · ~ω i ) e Z k ( e Θ) = e E e Z k ( e Θ) , (4.8)where e Z k ( e Θ) := Z k (Θ) for ∀ k ∈ Z bd . (4.9)Note that (4.8) has the form of h ( e Θ) e Z = e E e Z with h ( e Θ) being given by (2.1).Furthermore, recalling (4.3) and (4.9), we have for a.e. e Θ, | e Z k ( e Θ) | ≤ C ( e Θ , Ψ , b, d )(1 + | k | ) bd . (4.10) Let δ > < ε = λK ≤ ε ( δ, c , ρ, b, d ) , mes [0 , π ] bd \ \ N ≥ N Ω N ≤ δ. We fix ~ω ∈ T N ≥ N Ω N . Then since e E ∈ I ε and N ≥ N , there exists some set X N = X N ( ~ω, e E ) ⊂ R d such thatsup ≤ j ≤ d, e Θ ¬ j ∈ R d − mes( X N ( e Θ ¬ j )) ≤ e − N c , (4.11)and for e Θ / ∈ X N , Q ∈ E N , k G Q ( e E ; e Θ) k ≤ e √ N , (4.12) | G Q ( e E ; e Θ)( n , n ′ ) | ≤ e − ρ | n − n ′ | for | n − n ′ | ≥ N/ . (4.13)We should remark that (4.12) and (4.13) obviously hold true if | e Θ | ≥ bN . With-out loss of generality, we may assume X N ⊂ B (10 bN ). As a result, we have byFubini Theorem and (4.11) thatmes( X N ) ≤ CN d − e − N c ≤ e − N c / and X N ≥ N mes( X N ) < ∞ . Then we obtain by using Borel-Cantelli Theorem thatmes( X ∞ ) = 0 , where X ∞ = \ N ≥ N [ M ≥ N X M . Assuming e Θ / ∈ X ∞ , then there exists N ( e Θ) ≥ N such that e Θ X N for all N ≥ N . Recall that the Poisson’s identity: For h ( e Θ) e Z = e E e Z and n ∈ Λ ⊂ Z bd , e Z n ( e Θ) = − ε X n ′ ∈ Λ , n ′′ / ∈ Λ G Λ ( e E ; e Θ)( n , n ′ ) b V n ′ − n ′′ e Z n ′′ ( e Θ) . Thus for a.e. e Θ and N ≥ N , | e Z ( e Θ) | = | X | n |≤ N, | n ′ | >N G [ − N,N ] bd ( e E ; e Θ)( , n ) b V n − n ′ e Z n ′ ( e Θ) |≤ C ( e Θ , Ψ , b, d ) X | n |≤ N, | n ′ | >N e − ρ | n | + ρN + √ N e − ρ | n − n ′ | | n ′ | bd ≤ C ( e Θ , Ψ , b, d ) N bd X | n ′ | >N e − ρ | n ′ | + ρN + √ N | n ′ | bd ≤ e − ρN . (4.14) Letting N → ∞ in (4.14), we have e Z ( e Θ) = 0 for a.e. e Θ. This implies0 = Z R d | e Z ( e Θ) | d e Θ = K − d Z R d | b Ψ(Θ) | dΘ , and Ψ = 0, a contradiction.We have shown H ( ~θ ) has no eigenvalues in h − K (log K λ ) c , K (log K λ ) c i . (cid:3) Acknowledgements
I am grateful to Yulia Karpeshina for pointing out the periodicity of the potentialin the previous version of the paper. I am grateful to Qi Zhou for his usefulsuggestions. This work was supported by NNSFC grant 11901010.
Appendix A. Lemma A.1.
Let H = − ∆+ V ( x ) be a Schr¨odinger operator with M = sup x ∈ R d | V ( x ) | < ∞ . Then for any E ≥ , we have σ ( H ) ∩ [ E − M, E + M ] = ∅ , where σ ( H ) denotes the spectrum of H. Proof.
Let E ( · ) be the (projection-valued) spectral measure of H = − ∆ + V ( x ).Then the Spectral Theorem reads as H = Z σ ( H ) λ d E ( λ ) . Note that σ ( − ∆) = σ ess ( − ∆) = [0 , ∞ ). From Weyl criterion, for each E ≥ { F n ( x ) } n ∈ N ⊂ H ( R d ) such that( − ∆ − E ) F n → n → ∞ , k F n k L = 1 for ∀ n. For n ∈ N , we know µ n ( · ) = h E ( · ) F n , F n i is a positive Borel measure and µ n ( σ ( H )) = k F n k L = 1, where h F, G i = R R d F ( x ) G ( x )d x . Obviously, k V F n k L ≤ M . Henceinf λ ∈ σ ( H ) | λ − E | k F n k L ≤ Z σ ( H ) | λ − E | d µ n ( λ )= k ( − ∆ + V − E ) F n k L ≤ ( k ( − ∆ − E ) F n k L + M ) . (A.1)Letting n → ∞ in (A.1), we getdist( E, σ ( H )) ≤ M, (A.2)which shows [ E − M, E + M ] ∩ σ ( H ) = ∅ . For otherwise, there must be dist( E, σ ( H )) ≥ M , which contradicts with (A.2). (cid:3) Appendix
B.We write G ( · ) = G ( · ) ( E ; Θ) for simplicity.We have the following lemmas: Lemma B.1 (Lemma A.1, [Shi19a]) . Fix ¯ ρ > . Let Λ ⊂ Z bd satisfy Λ ∈ E N andlet A, B be two linear operators on C Λ . We assume further k A − k ≤ e √ N , | A − ( n , n ′ ) | ≤ e − ¯ ρ | n − n ′ | for | n − n ′ | ≥ N/ . Suppose that for all n , n ′ ∈ Λ , | ( B − A )( n , n ′ ) | ≤ e − ρN − ¯ ρ | n − n ′ | . Then k B − k ≤ k A − k , | B − ( n , n ′ ) | ≤ | A − ( n , n ′ ) | + e − ¯ ρ | n − n ′ | . Lemma B.2 (Lemma 3.2, [JLS20]) . Let ¯ ρ ∈ ( ǫ, ρ ] , M ≤ N and diam(Λ) ≤ N +1 .Suppose that for any n ∈ Λ , there exists some W = W ( n ) ∈ E M with M ≤ M ≤ M such that n ∈ W ⊂ Λ , dist( n , Λ \ W ) ≥ M/ and k G W k ≤ e √ M , | G W ( n , n ′ ) | ≤ e − ¯ ρ | n − n ′ | for | n − n ′ | ≥ M/ . We assume further that M ≥ M ( ǫ, ρ, b, d ) > . Then k G Λ k ≤ M + 1) bd e √ M . Lemma B.3 (Theorem 3.3, [JLS20]) . Let Λ ⊂ Λ ⊂ Z bd satisfy diam(Λ) ≤ N + 1 , diam(Λ ) ≤ N bd . Let M ≥ (log N ) and ¯ ρ ∈ (cid:2) ρ , ρ (cid:3) . Suppose that for any n ∈ Λ \ Λ , there exists some W = W ( n ) ∈ E M with M ≤ M ≤ N / such that n ∈ W ⊂ Λ \ Λ , dist( n , Λ \ Λ \ W ) ≥ M/ and k G W k ≤ e √ M , | G W ( n , n ′ ) | ≤ e − ¯ ρ | n − n ′ | for | n − n ′ | ≥ M/ . Suppose further that k G Λ k ≤ e √ N . Then | G Λ ( n , n ′ ) | ≤ e − (¯ ρ − C √ M ) | n − n ′ | for | n − n ′ | ≥ N/ , where C = C ( ρ, b, d ) > and N ≥ N ( ρ, b, d ) > . References [Agm70] S. Agmon. Lower bounds for solutions of Schr¨odinger equations.
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