aa r X i v : . [ m a t h . SP ] N ov ON PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGEROPERATORS
HELGE KR ¨UGER
Abstract.
Using relative oscillation theory and the reducibility result of Elias-son, we study perturbations of quasiperiodic Schr¨odinger operators. In par-ticular, we derive relative oscillation criteria and eigenvalue asymptotics forcritical potentials. Introduction
We will be interested in generalizing classical perturbation result of eigenvaluesto quasiperiodic operators. We first overview the classical results of interest. Most(if not all) of our results will be parallel to these. For this introduction let H be aself-adjoint realization of(1.1) H = − d dx + q ( x )on L (1 , ∞ ) with q ( x ) → x → ∞ and q bounded. A classical result of Weylnow tells us, that the essential spectrum of H , is equal to the one of − d dx , hence σ ess ( H ) = [0 , ∞ ). We give the generalization of this to quasiperiodic operators inTheorem 3.1.Kneser answered in [9], the question when 0 is an accumulation point of eigen-values below 0. One has if(1.2) lim sup x →∞ q ( x ) x < − x →∞ q ( x ) x > − N ( λ ) be the number of eigenvalues of − d dx + µx below λ , then(1.4) N ( λ ) = 14 π r µµ crit − | ln | λ || (1 + o (1)) , λ ↑ , µ crit = − . Mathematics Subject Classification.
Primary 34C10, 34B24; Secondary 34L20, 34L05.
Key words and phrases.
Sturm–Liouville operators, oscillation theory, quasi-periodic.
For − d dx + µx γ , 0 < γ <
2, we have N ( λ ) = 1 π Z { x, q ( x ) <λ } ( λ − q ( x )) / dx (1 + o (1)) , λ ↑ p µ/µ crit π − γ (cid:12)(cid:12)(cid:12) µλ (cid:12)(cid:12)(cid:12) (2 − γ ) / γ (1 + o (1)) , λ ↑ This result goes back to results in the sixties, seethe notes in [14]. The periodic case was answered by Schmidt [15] for γ = 2. Wewill answer this question in Theorem 3.7.Periodic operators have a spectrum made out of the union of finitely or infinitelymany bands. That is(1.6) σ ess ( − d dx + q ( x )) = [ E , E ] ∪ [ E , E ] ∪ . . . , E j < E j +1 , for q ( x + p ) = q ( x ), p >
0. Since, we now have several boundary points ofthe spectrum, one can also ask what happens at all, finitely many, . . . boundarypoints of σ ess ( H ). Rofe-Beketov gave the following answer to this question: Onlyfinitely many gaps can contain infinitely many eigenvalues for critical perturbations( q ( x ) = µ/x ) (see (6.145) in [8]). We will treat this question in Theorem 3.6.The organization of this paper is as follows. In Section 2, we will state theneeded results about quasiperiodic Schr¨odinger operators. In Section 3, we willstate our main results. Most proofs are easy enough to be directly stated. Only theeigenvalue asymptotics requires more work and is stated in the following section. InSection 5, we give an outline of Eliasson’s proof and derive some further estimates.In Appendix A, we will review relative oscillation theory, followed by another shortappendix on needed methods from the theory of differential equations.2. Quasiperiodic Operators
We will now recall the basic notations about quasiperiodic Schr¨odinger operators.Let T d be the d -dimensional torus, where T = R / (2 π Z ). Let Q : T d → R be a realanalytic function. We will consider the Schr¨odinger operator on L (1 , ∞ ) given by(2.1) H = − d dx + q ( x ) , q ( x ) = Q ( ωx )where ω ∈ T d is fixed. We will assume that ω is a Diophantine number, that isthere is some τ > d − κ >
0, such that(2.2) DC ( κ, τ ) : |h ω, n i| ≥ κ | n | τ , n ∈ Z d \{ } , holds.Recall the rotation number ρ ( E ) from [4]. Denote by ϑ ( x, E ) the Pr¨ufer angleof a solution u of H u = Eu . That is a continuous function of x such that(2.3) u ( x ) = r ( x ) sin ϑ ( x, E ) , u ′ ( x ) = r ( x ) cos ϑ ( x, E ) , ≤ ϑ (1 , E ) < π, for some continuous function r . The rotation number ρ ( E ) is now introduced by(2.4) ρ ( E ) = lim x →∞ ϑ ( x, E ) x . We obtain a factor different from [7] in the case γ = 2, since we are considering half lineoperators. This factor does not arise for 0 < γ <
2, since the domain of integration also shrinks.
N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 3
We remark that the integrated density of states k ( E ) satisfies(2.5) k ( E ) = 1 π ρ ( E ) . Johnson and Moser showed
Theorem 2.1. [ [4] ] The spectrum σ ( H ) is given by (2.6) σ ( H ) = { E, ρ ( E ) = 12 h ω, n i , n ∈ Z d } . Furthermore ρ is a continuous function, and constant outside the spectrum. Now we come to Eliasson’s result. Recall that we can rewrite the Schr¨odingerequation − u ′′ ( x ) + Q ( ωx ) u ( x ) = Eu ( x ) , as the first order system(2.7) U ′ ( x ) = (cid:18) Q ( ωx ) − E (cid:19) U ( x )where U ( x ) = (cid:18) u ( x ) u ′ ( x ) (cid:19) . Theorem 2.2. [ [3] ] There is an E , such that for E = h ω, m i > E a boundarypoint of the spectrum of H , there is a function Y : T d → SL (2 , R ) and A ∈ sl (2 , R ) with A = 0 such that (2.8) X ( x ) = Y Y ( ω x ) e Ax , Y = 12 √ E (cid:18) −√ E √ E (cid:19) is the fundamental solution of (2.7). Furthermore we have that for | m | ≥ | A | ≤ c | m | τ (2.9) | Y | ≤ C log | m | , | det( Y ) − | ≤ , (2.10) for constants c , C independent of m , and the spectrum of H is purely absolutelycontinuous above E . We will give an outline of Eliasson’s proof in Section 5, and derive the additionalestimates there. In fact Eliasson proved that (2.8) holds, when ρ ( E ) satisfies thenext Diophantine condition(2.11) | ρ − h n, ω i | ≥ ˜ κ | n | σ , n ∈ Z d \{ } , ˜ κ > , σ > . Eliasson also showed that the spectrum of H will be a Cantor set for genericfunctions Q : T d → R in the | . | s topology given by the norm(2.12) | Q | s = sup | Im( z ) |
We are interested in decaying perturbations of the quasiperiodic operator H .That is for some function ∆ q consider the operator(3.1) H = − d dx + q ( x ) , q ( x ) = q ( x ) + ∆ q ( x ) , for q ( x ) = Q ( ωx ) as described in Section 2. We then have the next basic stabilityresult of the essential spectrum. Theorem 3.1. If ∆ q → , then (3.2) σ ess ( H ) = σ ess ( H ) = R \ [ n G n , for open sets G n . If ∆ q is integrable, we have that the spectrum of H is purelyabsolutely continuous above E .Proof. The first part follows by Weyl’s Theorem and Theorem 2.1. For the secondpart, note that by Theorem 2.2, H has purely absolutely continuous spectrumabove E , and by Theorem 1.6. of [6] it is invariant under L perturbations. (cid:3) It is conjectured in [6], that there is still absolutely continuous spectrum for∆ q ∈ L , but it may not be pure. This was shown for the free case in [1] and forthe periodic one in [5]. See also the recent review in [2]. If we write G n = ( E − n , E + n )for the intervals of the last theorem, and call them gaps. We call E − n (resp. E + n ) alower (resp. upper) boundary point of the spectrum. The next relative oscillationcriterion follows. Theorem 3.2.
Assume that ∆ q → , and let E be a boundary point above E ofthe essential spectrum of H . Then there exists a constant K = K ( E ) such that E is an accumulation point of eigenvalues of H if (3.3) lim sup x →∞ K ∆ q ( x ) x < − and E is not an accumulation point of eigenvalues if (3.4) lim inf x →∞ K ∆ q ( x ) x > − . Furthermore
K > (resp. K < ) if E is a upper (resp. lower) boundary point.Proof. Everything follows from Theorem A.6, except for the existence of K . Wehave from (2.8) that u ( t ) = U ( ω t ) for a function U : T d → R . We will show K = lim inf l →∞ lim sup x →∞ ℓ Z x + ℓx u ( t ) dt = lim sup l →∞ lim inf x →∞ ℓ Z x + ℓx u ( t ) dt = Z T d U ( z ) dz. Now note, that (2.2) implies that the system ( T d , T t , µ ), where T t = ω t and µ is thenormalized Lebesgue measure is uniquely ergodic. By Birkhoff’s ergodic theorem,we have that lim l →∞ ℓ Z x + ℓx U ( ω t ) dt = Z T d U ( z ) dz. By unique ergodicity, we know that the limit is uniform in x . Hence, the resultfollows. (cid:3) N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 5
We even have a whole scale of relative oscillation criteria. To state this, we recallthe iterated logarithm log n ( x ) which is defined recursively vialog ( x ) = x, log n ( x ) = log(log n − ( x )) . Here we use the convention log( x ) = log | x | for negative values of x . Then log n ( x )will be continuous for x > e n − and positive for x > e n , where e − = −∞ ande n = e e n − . Abbreviate further L n ( x ) = 1log ′ n +1 ( x ) = n Y j =0 log j ( x ) , ˜ Q n ( x ) = − K n − X j =0 L j ( x ) . From Theorem 2.10. of [12].
Theorem 3.3.
Assume the assumptions of the last theorem, and that for some n ∈ N (3.5) lim x →∞ L n − ( x ) − (∆ q ( x ) − ˜ Q n − ( x )) = − K .
Then E is an accumulation point of eigenvalues of H if (3.6) lim sup x →∞ KL n ( x ) (∆ q ( x ) − ˜ Q n ( x )) < − and E is not an accumulation point of eigenvalues if (3.7) lim inf x →∞ KL n ( x ) (∆ q ( x ) − ˜ Q n ( x )) > − , with the same K as in the last theorem. The next lemma gives us an estimate on K . Lemma 3.4.
The constant K of Theorem 3.2, satisfies (3.8) | K ( E ) | ≤ C | m | ˜ τ √ E , < ˜ τ < τ where m ∈ Z d is such that E ∈ ρ − ( h ω, m i ) .Proof. From Theorem 2.2, we know the existence. We note that det( Y ) = 1. By(2.9), we have that | A | ≤ c | m | − τ , where the m is the one such that ρ ( E ) = h ω, m i .Hence, we obtain that | β | ≤ c | m | − τ . Now | K | ≤ c R T U ( z ) dz | m | τ √ E .
The claim now follows by (2.10). (cid:3)
Remark 3.5.
One can hope that the estimate (3.10) on K ( E ) can be improved. Itwas shown in [3] that the matrix A and then β would satisfy the bound | β | ≤ C | E + − E − | for some constant C . Then it was shown in [13] , that | E + − E − | ≤ ce − γ | m | forsome constants c and γ . Hence one should expect K ( E ) to decrease exponentiallyin m . Unfortunately, the estimate of [13] depends on further arithmetic propertiesof m . Hence, it is not clear if it holds at all band edges. HELGE KR¨UGER
For simplicity, we will now restrict our attention to perturbations of the form(3.9) ∆ q ( x ) = µx γ , µ = 0 , γ > . We will denote the operator H + µx γ by H γµ . Now, we come to the question howmany gaps above E can contain infinitely many eigenvalues. This question is a bitodder than the one for periodic operators, since there are bounded intervals thatcontain infinitely many gaps.Introduce µ crit by(3.10) µ crit ( E ) = − K ( E ) . Then
E > E is an accumulation point of eigenvalues of H µ if and only if µ/µ crit >
1. For H γµ , 0 < γ <
2, this requirement is µ/µ crit >
0. Now, we come to
Theorem 3.6. If γ > , then no boundary point of σ ( H ) above E is an accumu-lation point of eigenvalues of H γµ = H + γx α . If γ < , then if µ < (resp. µ > ),then all upper (resp. lower) boundary points above E are accumulation points ofeigenvalues of H γµ .If γ = 2 , we can add infinitely many eigenvalues to each gap by choosing µ largeenough. However, for every value of µ only finitely many gaps contain infinitelymany eigenvalues of H γµ .Proof. The first claim follows from Theorem 3.2. The second claim follows fromthe last lemma and Theorem 3.2. (cid:3)
Now, we come to the eigenvalue asymptotics. Let E be again a boundary pointof the spectrum of H . Introduce, if the set ( ˜ E, E ) ∩ σ ( H ) = ∅ , by N ( λ ) thenumber(3.11) N ( λ ) = tr( P ( ˜ E,λ ) ( H γµ )) , ˜ E < λ < E with the obvious modification for ( E, ˜ E ) ∩ σ ( H ) = ∅ . Theorem 3.7.
Let E be a boundary point of the spectrum of H , which is anaccumulation point of eigenvalues of H γµ . Then if γ = 2(3.12) N ( λ ) = 14 π r µµ crit − · | log | E − λ || · (1 + o (1)) , and if < γ < N ( λ ) = 1 π − γ r µµ crit (cid:18) | µ || E − λ | (cid:19) (2 − γ ) / γ · (1 + o (1)) . where N ( λ ) is the number of eigenvalues near E . We will give a proof in Section 4. In difference to the proof of [15], our proof onlyuses the decay of the potential and the behavior of the solution at the boundarypoint of the spectrum. In fact everything carries over to general elliptic situations.That is, where one has two solutions u , u such that u ( x ) and u ( x ) − xu ( x ) arebounded functions. Remark 3.8.
It was already shown in Corollary 6.6 in [8] that µ crit ( E ) has todiverge as E → ∞ . We also remark that [8] develops a different approach torelative oscillation criteria than was used in [12] . N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 7 Proof of Theorem 3.7
We will now give explicit bounds on the spectral projections.
Lemma 4.1.
Let ψ be a solution of (4.1) ψ ′ ( x ) = − ∆ q ( x )( u ( x ) cos ψ ( x ) − v ( x ) cos ψ ( x )) . Then we have that (4.2) ψ ( x ) = (cid:18) r µµ crit − o (1) (cid:19) log | x | if ∆ q ( x ) = µ/x . If ∆ q ( x ) = µ/x γ , < γ < , (4.3) ψ ( x ) = (cid:18)r µµ crit − γ + o (1) (cid:19) x − γ/ . Proof.
Use in (A.5) α = x , to obtain if γ = 2 the next equation ϕ ′ = 1 x (sin ϕ + cos ϕ sin ϕ + µu cos ϕ ) + O ( 1 x ) , whose asymptotics can be evaluated with Lemma B.2 and Lemma B.1.In the case 0 < γ <
2, we choose α = 1 / p | ∆ q | K , then also the sin ϕ cos ϕ termbecomes of lower order, hence we obtain by averaging ϕ ′ ( x ) = p K ∆ q ( x ) + O (∆ q + ∆ q ′ ∆ q )which implies the claim for ∆ q of the particular form. (cid:3) Then, we have that
Lemma 4.2.
Let the Wronskian W ( u ( E ) , u ( E )) have n zeros on { x, ∀ y > x, | ∆ q | ≤ | E − λ |} . Then we have that (4.4) N ( λ ) ≤ n + 3 . Proof.
Observe that by the comparison theorem for Wronskians, we have that W ( u ( λ ) , u ( λ )) can have at most one zero left of x n .Hence, we obtaindim Ran P ( −∞ ,λ ) ( H ) ≤ (1 ,x n ) ( u ( λ ) , u ( λ )) + 1Now, by the triangle inequality for Wronskians, we obtain (1 ,x n ) ( u ( λ ) , u ( λ )) ≤ (1 ,x n ) ( u ( λ ) , u ( E )) + 1. It, now suffices to note that (1 ,x n ) ( u ( λ ) , u ( E )) isbounded by (1 ,x n ) ( u ( E ) , u ( E )) + 1 by using the comparison theorem for Wron-skians. (cid:3) Note, that the last two lemmas imply the next bound on the eigenvalues if γ = 2(4.5) N ( λ ) ≤ π r µµ crit − · | log | E − λ || · (1 + o (1)) , and if 0 < γ < N ( λ ) ≤ π − γ r µµ crit (cid:18) | µ || E − λ | (cid:19) (2 − γ ) / γ · (1 + o (1)) . The next lemma shows that we have equality. Hence with it Theorem 3.7 is proven.
HELGE KR¨UGER
Lemma 4.3.
Let < δ < and < γ ≤ , then if γ = 2 , (4.7) N ( λ ) ≥ π r µµ crit (1 − δ ) − · | log | E − λ || · (1 + o (1)) , and if < γ < , (4.8) N ( λ ) ≥ π − γ r µµ crit (1 − δ ) (cid:18) | µ || E − λ | (cid:19) (2 − γ ) / γ · (1 + o (1)) . Proof.
Let x max be given by x max ( λ ) = δ (cid:18) | µ || E − λ | (cid:19) /γ . Let ϕ λ ( x ) be a Pr¨ufer angle of W ( u ( E ) , u ( λ )). By the triangle inequality forWronskians, it is clear that ϕ λ is close to the Pr¨ufer angle of W ( u ( λ ) , u ( λ )).Now, for x < x max ( λ ), we have that ϕ ′ λ ( x ) ≥ µ (1 − δ ) x γ ( u cos ϕ λ ( x ) − v sin ϕ λ ( x )) . This is the same equation for all λ . As x → ∞ , the solution has the claimedasymptotics by using Lemma 4.1. Hence, the claim follows. (cid:3) Outline of Eliasson’s proof
We now give an outline of Eliasson’s proof of reducibility in [3]. The next lemmais an easy computation.
Lemma 5.1.
The equation (5.1) X ′ ( x ) = (cid:18) Q ( ωx ) − E (cid:19) X ( x ) can be transformed by (5.2) X ( x ) = Y − X ( x ) , Y = (cid:18) −√ E √ E (cid:19) to (5.3) X ′ ( x ) = ( A + F ( ωx, √ E )) X ( x ) , where (5.4) A = √ EJ, J = (cid:18) − (cid:19) , F ( z, √ E ) = Q ( z )2 √ E (cid:18) − −
11 1 (cid:19) . Furthermore A , F satisfy Hypothesis 5.2. Hypothesis H. 5.2.
Let A ∈ sl (2 , C ) and F : T d → M at (2 , C ) satisfy tr( ˆ F (0)) = 0 , (5.5) | A − √ EJ | < | F | r < ε , (5.7) for some ε > , small. N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 9
We have now seen that we can reduce our system to one of the form(5.8) X ′ ( x ) = ( A + F ( x )) X ( x ) , where F is small. This system although close to a constant coefficient one cannotbe solved explicitly. However, we can reduce it to a system(5.9) X ′ ( x ) = ( A + F ( x )) X ( x )where F is smaller than F , as follows. We will construct A , F , and a solution Y of the system(5.10) Y ′ ( x ) = ( A + F ) Y − Y ( A + F ) . Then for X a solution to (5.9), we have that Y X will solve (5.8). Of course, wecannot hope that (5.9) will be explicitly solvable, however we will be able to iteratethe above procedure to obtain better and better approximate solutions.Since, we will require that F k →
0, and then our final X ∞ ( x ) = e xA . Here A = lim k →∞ A k . So the final solution will be(5.11) ∞ Y k =1 Y k ( ω x ) e Ax We will not attempt to solve (5.10) in this paper, and refer to [3] for the details.However, we will draw further conclusions from Eliasson’s method to control ourquantities.Fix 0 < ε < < σ <
1, and let(5.12) ε j +1 = ε σj = ε (1+ σ ) j . Furthermore, assume that r > r > r > . . . is a decreasing sequence of positivenumbers, satisfying(5.13) r j +1 ≤ r j − r j +1 .r j will play the role of the neighborhood, where we suppose to have analyticity.Introduce N j by(5.14) N j = 2 σr j − r j +1 log( ε − j ) = 2 σ (1 + σ ) j r j − r j +1 log( ε − ) ≤ C (2 + 2 σ ) j , C > . Furthermore, one also sees that N j ≥ ˜ C (1 + σ ) j for some other constant ˜ C . Hence N j → ∞ as j → ∞ . Furthermore, we have(5.15) ε σj ≤ (cid:18) σr (1 + σ ) log( ε − )(2 + 2 σ ) j (cid:19) − τ ≤ N − τj if ε is small enough. Proposition 5.3.
Assume Hypothesis 5.2 with ε small enough, then there arefunctions Y j : 2 T d → GL (2 , R ) , A j ∈ sl (2 , R ) , and F j : T d → M at (2 , R ) , for j ≥ .Furthermore, there are numbers m j that satisfy (5.16) ε σj ≤ | α j − h ω, m j i| ≤ ε σj , < | m j | ≤ N j , or m j = 0 if (5.16) cannot be satisfied. Here α j is the rotation number of A k .Furthermore A j , F j , and Y j satisfy h Y ′ j +1 ( x ) , ω i = ( A j + F j ( x )) Y j +1 ( x ) − Y j +1 ( x )( A j +1 + F j +1 ( x )) , (5.17) | (cid:18) Y j +1 ( . ) − exp (cid:18) h m k , . i α j A j (cid:19)(cid:19) | ≤ ε / j , (5.18) | (cid:18) A j +1 − (cid:18) − h ω, m j i α j (cid:19) A j (cid:19) | ≤ ε / j , (5.19) tr( ˆ F j +1 (0)) = 0 , | F j +1 | r j +1 < ε j +1 , (5.20) | A j +1 | ≤ | α j +1 | N τj +1 , if | α j +1 | ≥ N − τj +1 . (5.21) Proof.
This is Lemma 1 and 2 in [3]. (cid:3)
Remark 5.4.
The requirement of ε being small enough, will in fact determine ourlower bound on allowed energies E . Since for E > E | F | r = C √ E < C √ E = ε for some constant C . Hence by making E large, we can make ε arbitrarily small. Lemma 5.5.
Assume that Y j , A j and F j satisfy the conditions given in Proposi-tion 5.3. If for all j ≤ k , m j = 0 , then (5.22) | A k − λJ | < . We furthermore obtain, if K is the largest integer less than k such that m K = 0 ,that (5.23) | A k | ≤ C N τK < , k ≥ K where C doesn’t depend on K .Proof. By m k = 0, we have that from (5.16) | α k − h ω, n i| ≥ ε σk , < | n | ≤ N j . For m j = 0, j = 1 , . . . , k , we have by (5.19) that | A k | ≤ ε / + · · · + ε / k − < . This shows the first part.For the second claim, let l ≤ k be maximal such that the m l = 0. Then, weobtain a bound on | A j ( λ ) | by | A k | ≤ ε / l + · · · + ε / k − + | (cid:18) − h ω, m l i α l (cid:19) A l |≤ ε / l + 2 ε σl | A l α l | ≤ N τl ε σl , where we used (5.16) in the middle and (5.21) in the last step. (5.23) follows from(5.15). The last claim is evident. (cid:3) Let us now consider ˜ ρ = P ∞ k =1 h m k , ω i + α , where α = lim j →∞ α j . Further-more, ρ j +1 = P jk =1 h m k , ω i + α j +1 , Furthermore, we know that inside the gap α = 0 from [3]. We now obtain N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 11
Lemma 5.6. ρ j +1 → ˜ ρ uniformly. If ρ is rational, m j = 0 for j large. Furthemore, (5.24) X k,m k =0 m k = m, holds.Proof. The first two parts are Lemma 3 in [3]. The last part follows, since α → m = P k,m k =0 m k , one has0 = ˜ ρ − h ω, m i = 12 h ω, ˜ m − m i . Hence ˜ m = m by the Diophantine condition. (cid:3) Proof of (2.9).
We will now show how (5.24) can be used to make the bound from(5.23) only depend on m . By definition | m k | ≤ N k , we have by (5.14) | m | ≤ X k,m k =0 | m k | ≤ K X k =1 N k ≤ C (2 σ + 2) K +1 . Hence K ≥ log | m | log(2+2 σ ) − C and by (5.14) N K ≥ C p | m | and then (5.23) implies theclaim, since it holds for all large k . (cid:3) We have that
Lemma 5.7. If m j = 0 for j large, we have that Q Y j converges to some Y uniformly on compact subsets. Furthermore A j → A and F j → . Furthermore(2.10) holds.Proof. Since r j is decreasing and positive, it has a limit r ≥
0. By (5.20), we havethat | F j | r j →
0. Since m j = 0 for large j , we have that | A j +1 − A j | ≤ ε / j from(5.19). Hence, A j → A , since P ∞ j = N ε / j < ∞ .By (5.18), we have that | Y j − I | ≤ ε / j , if m j = 0, which implies Q Y j → Y bya similar argument. If m j = 0, we have that | Y j +1 ( ω t ) − I | ≤ ε / j + | cos( h m j , t i I + sin( h m j , t i A j α j − I | , where the last term ≤
3. Since, we can bound the number of these terms bylog m , we obtain the claim. By (5.18), we have that Y j +1 − I , if m j = 0, resp.exp( −h m j , t i A j /α j ) Y j +1 − I are bounded by ε / j . Hence, we can bound | det( Y j +1 ) − | ≤ ε j , from which the estimate on the determinant follows. (cid:3) Appendix A. Relative Oscillation Theory
As introduced in [10], relative oscillation theory is a tool to compute the differ-ence of spectra of two different Schr¨odinger operators. Let q , q ∈ L loc and(A.1) H j = − d dx + q j , j = 0 , L (1 , ∞ ). Introduce ∆ q = q − q , whichwe will assume to be sign-definite. Denote by u , u ) the number of zeros ofthe Wronskian W ( u , u ) = u u ′ − u ′ u on (1 , ∞ ), for solutions τ j u j = λ j u j .Let ψ j, − ( λ ) be the solution of τ j ψ j, − ( λ ) = λψ j, − ( λ ), which obeys the boundary condition at 1 (e.g. ψ j, − ( λ )(1) = 0). Similarly let ψ j, + ( λ ) be the solution satisfying ψ j, + ( λ ) ∈ L (1 , ∞ ). Then [10] tells us: Theorem A.1.
Assume that [ λ , λ ] ∩ σ ess ( H ) = ∅ . Then, we have that tr P [ λ ,λ ) ( H ) − tr P ( λ ,λ ] ( H )= ( ( ψ , ± ( λ ) , ψ , ∓ ( λ ) − ψ , ± ( λ ) , ψ , ∓ ( λ )) , ∆ q < − ( ψ , ± ( λ ) , ψ , ∓ ( λ ) − ψ , ± ( λ ) , ψ , ∓ ( λ )) , ∆ q > Here tr P [ λ ,λ ) ( H ) denotes the number of eigenvalues of H in [ λ , λ ) . Since one has the next triangle inequality for Wronskians(A.3) u , u ) + u , u ) − ≤ u , u ) ≤ u , u ) + u , u ) + 1 , one can replace ψ j, ± ( λ ) by any other solution of τ j u = λu , up to a finite error. Wefurthermore remark that the next two comparison theorems hold. The first one isfound in [11]. Theorem A.2 (Sturm’s Comparison theorem) . Let q − q > , and H j u j = 0 , j = 0 , . Then between any two zeros of u or W ( u , u ) , there is a zero of u .Similarly, between two zeros of u , which are not at the same time zeros of u ,there is at least one zero of u or W ( u , u ) . The next result is found in [10].
Theorem A.3 (Comparison theorem for Wronskians) . Suppose u j satisfies τ j u j = λ j u j , j = 0 , , , where λ r − q ≤ λ r − q ≤ λ r − q .If c < d are two zeros of W x ( u , u ) such that W x ( u , u ) does not vanish identi-cally, then there is at least one sign flip of W x ( u , u ) in ( c, d ) . Similarly, if c < d are two zeros of W x ( u , u ) such that W x ( u , u ) does not vanish identically, thenthere is at least one sign flip of W x ( u , u ) in ( c, d ) . We call H relatively oscillatory with respect to H at E if for any solutions of H j u j ( E ) = Eu j ( E ), j = 0 ,
1, we have that u ( E ) , u ( E )) is infinite. Otherwisewe call it relatively nonoscillatory. Now, we come to relative oscillation criteria. Lemma A.4.
Let lim x →∞ ∆ q ( x ) = 0 . Then σ ess ( H ) = σ ess ( H ) and H isrelatively nonoscillatory with respect to H at E ∈ R \ σ ess ( H ) . By Theorem A.1, this is equivalent if E is a boundary point of the essentialspectrum of H , to E being an accumulation point of eigenvalues of H . In orderto state a relative oscillation criterion at a boundary point of the spectrum, somepreparations are needed. Definition A.5.
A boundary point E of the essential spectrum of H will be calledadmissible if there is a minimal solution u of ( τ − E ) u = 0 and a second linearlyindependent solution v with W ( u , v ) = 1 such that (cid:18) u p u ′ (cid:19) = O ( α ) , (cid:18) v p v ′ (cid:19) − β (cid:18) u p u ′ (cid:19) = o ( αβ ) for some weight functions α > , β ≶ , where β is absolutely continuous such that ρ = β ′ β > satisfies ρ ( x ) = o (1) and ℓ R ℓ | ρ ( x + t ) − ρ ( x ) | dt = o ( ρ ( x )) . N PERTURBATIONS OF QUASIPERIODIC SCHR ¨ODINGER OPERATORS 13
It is shown in Lemmas 4.2 and 4.3 of [12], that there exists a Pr¨ufer angle ψ for W ( u , u ) such that it obeys(A.4) ψ ′ ( x ) = − ∆ q ( x )( u ( x ) cos( ψ ( x )) − v ( x ) sin( ψ ( x ))) . Through the transformation cot ψ = α cot ϕ + β , this can then be transformed to(see Lemma 4.6 of [12]) ϕ ′ = α ′ α sin ϕ cos ϕ + β ′ α sin ϕ − ∆ q · αu cos ϕ (A.5) + O (∆ q ) + O (∆ q/α ) . Through an application of the methods of Appendix B, one comes to the mainresult of [12].
Theorem A.6.
Suppose E is an admissible boundary point of the essential spec-trum of τ , with u , v and α , β as in Definition A.5. Furthermore, suppose thatwe have (A.6) ∆ q = O (cid:0) β ′ α β (cid:1) . Then τ − E is relatively oscillatory with respect to τ − E if (A.7) inf ℓ> lim sup x → b ℓ Z x + ℓx β ( t ) β ′ ( t ) u ( t ) ∆ q ( t ) dt < − and relatively nonoscillatory with respect to τ − E if (A.8) sup ℓ> lim inf x → b ℓ Z x + ℓx β ( t ) β ′ ( t ) u ( t ) ∆ q ( t ) dt > − . We finish this section with a closing remark.
Remark A.7.
The requirement made that ∆ q is of definite sign is not necessary.However, a general theory requires a more difficult definition of u , u ) . We referthe interested reader to [10] for details. Appendix B. Averaging ordinary differential equations
In this section we collect the required results for these ordinary differential equa-tions. Proofs and further references can be found in [12].
Lemma B.1.
Suppose ρ ( x ) > (or ρ ( x ) < ) is not integrable near b . Then theequation (B.1) ϕ ′ ( x ) = ρ ( x ) (cid:18) A sin ϕ ( x ) + cos ϕ ( x ) sin ϕ ( x ) + B cos ϕ ( x ) (cid:19) + o ( ρ ( x )) has only unbounded solution if AB > and only bounded solution if AB < . Inthe unbounded case we have (B.2) ϕ ( x ) = (cid:18) sgn( A )2 √ AB − o (1) (cid:19) Z x ρ ( t ) dt. In addition, we also need to look at averages: Let ℓ >
0, and denote by(B.3) g ( x ) = 1 ℓ Z x + ℓx g ( t ) dt. the average of g over an interval of length ℓ . Lemma B.2.
Let ϕ obey the equation (B.4) ϕ ′ ( x ) = ρ ( x ) f ( x ) + o ( ρ ( x )) , where f ( x ) is bounded. If (B.5) 1 ℓ Z ℓ | ρ ( x + t ) − ρ ( x ) | dt = o ( ρ ( x )) then (B.6) ϕ ′ ( x ) = ρ ( x ) f ( x ) + o ( ρ ( x )) Moreover, suppose ρ ( x ) = o (1) . If f ( x ) = A ( x ) g ( ϕ ( x )) , where A ( x ) is boundedand g ( x ) is bounded and Lipschitz continuous, then (B.7) f ( x ) = A ( x ) g ( ϕ ) + o (1) . Condition (B.5) is a strong version of saying that ρ ( x ) = ρ ( x )(1 + o (1)) (it isequivalent to the latter if ρ is monotone). It will be typically fulfilled if ρ decreases(or increases) polynomially (but not exponentially). For example, the conditionholds if sup t ∈ [0 , ρ ′ ( x + t ) ρ ( x ) → A ( x ) has a limit, A ( x ) = A + o (1), then A ( x ) can bereplaced by the limit A and we have the next result Corollary B.3.
Let ϕ obey the equation (B.8) ϕ ′ = ρ (cid:18) A sin ( ϕ ) + sin( ϕ ) cos( ϕ ) + B cos ( ϕ ) (cid:19) + o ( ρ ) with A, B bounded functions and assume that ρ = o (1) satisfies (B.5). Then theaveraged function ϕ obeys the equation (B.9) ϕ ′ = ρ (cid:18) A sin ( ϕ ) + sin( ϕ ) cos( ϕ ) + B cos ( ϕ ) (cid:19) + o ( ρ ) . Note that in this case ϕ is bounded (above/below) if and only if ϕ is bounded(above/below). Acknowledgments.
I thank D.Damanik for bringing [3], [2] to my attention, andmany helpful discussions.
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Department of Mathematics, Rice University, Houston, TX 77005, USA
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