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On the spectrum of skew-shift Schr\"odinger operators

I prove that the spectrum of a skew-shift Schr\"odinger operator contains larges interval in the semi-classical regime. In the semi-classical limit, these intervals approach the range of the potential.


Introduction
In this paper, I consider the discrete Schrödinger operator H h = h∆ + V on 2 (Z) with potential where α is a Diophantine number and f a one-periodic real-analytic function. In the semi-classical regime, that is for h > 0 sufficiently small, I will show that the spectrum of the operator H h contains large intervals (see Theorem 2.2), which approach the range of f as h → 0. I will refer to the Schrödinger operator with potential given by (1.1) as the skew-shift Schrödinger operator, since its potential can be generated by evaluating a sampling function along the orbit of the skew-shift: T α (x, y) = (x + 2α, x + y) (mod 1), (1.2) where T = R/Z is the unit circle. It is expected that for all h > 0 the spectrum of the operator H h is an interval, see the end of Chapter 15 in Bourgain's book [8] and the end of this introduction. Let me mention at this point that understanding the model above is of physical relevance due to its relation to the kicked-rotor problem, see Chapter 16 in [8].
My result extends the result of Bourgain from [5], [6] that the spectrum has positive measure. As this result, the proof proceeds by analyzing a parametrization of the eigenvalues of finite restrictions. One might wonder, why it is hard to prove that the spectrum consists of intervals, since it is true for simple examples like the free Laplacian ∆ or periodic Schrödinger operators. Maybe, the most obvious obstructions are the results of Avila, Bochi, and Damanik [1], [2], which show that generic potentials have Cantor spectrum. Before discussing results on Cantor spectrum further, I will comment on previous results showing interval spectrum.
Since the spectrum of random Schrödinger operators is the union of periodic spectra, these consists of intervals. See [17], [20], [21] for implementations of this. Also my results concerning the potential V (n) = f (n ρ ) for ρ > 0 not an integer from [18] are an implementation of this fact, since they boil down to showing that there are arbitrarily long stretches of n, where V (n) ≈ f (x) for any x ∈ T. It is also possible to construct limit-periodic examples with spectrum containing intervals see [14] and [22]. Most relevant to us is the work of Chulaevsky and Sinai [11], where they show that the spectrum of a two-periodic quasi-periodic Schrödinger operators is a single interval for a set of frequencies A, which approaches all possible frequencies as h → 0. However, Bourgain has shown in [6] that these models also exhibit gaps for arbitrary h > 0 but extreme frequencies.
My result that the spectrum of the skew-shift Schrödinger operator contains large intervals distinguishes these from one-frequency quasi-periodic Schrödinger operators, where the potential is given by (1.3) V QP (n) = f (αn).
In fact for these Goldstein and Schlag [15] have shown that the spectrum is a Cantor set for almost every frequency α in the regime of h > 0 small. At this point, let me also point out that Avila and Jitomirskaya have solved the so called Ten Martini Problem [3], [4], which asked to show that for f (x) = 2 cos(2πx) the operator with potential given by (1.3) has Cantor spectrum for any irrational α and any h > 0. As already mentioned, Avila, Bochi, and Damanik [1], [2] have shown that for generic continuous sampling function and a large class of base transformations, one has Cantor spectrum. This result applies in particular to the skew-shift potential (1.1) with any irrational frequency α. I will further comment on the results from [1] when discussing optimality of my results.
If the sampling functions only takes finitely many values, it is known some generality, that the spectrum has zero Lebesgue measure, and thus is a Cantor set. This follows from the work of Damanik and Lenz [12], [13].
Before coming to the technical discussion of the results, let me come to an aspect of the proof, I find surprising. If one iterates the the skew-shift defined in (1.2), one finds (1.4) (T α ) n (x, y) = (x + 2nα, y + nx + n(n − 1)α) (mod 1).
One might expect that the relevant part of the dynamics is encoded in the αn 2 term, which is expected to behave like random variables [23]. However, we will not make use of this, but exploit the nx term, to obtain independence of events, which are far enough apart. This is possible, since x enters the problem as a fast variable, see Section 7 for the implementation of this fact. However, the equidistribution properties of the sequence αn 2 enters the proof of the large deviation estimates, see [7], [8], and [10].
Last, let me mention that it is easy to produce overwhelming numerical evidence for that the spectrum of H h = h∆ + V is an interval for all h > 0. I will discuss this in Appendix A.

Statement of the results
I will now make the statement of my result precise. First, let me specify that I will assume the frequency α satisfies for some c > 0 the Diophantine condition for all integers q ≥ 1, where x = dist(x, Z). Next, we will need the following result, which can be proven by the methods of Bourgain, Goldstein, and Schlag [10] or the ones of Bourgain [7]. [10]). There exists h 0 = h 0 (c, f ) > 0 such that for 0 < h < h 0 , we have that large deviation estimates for the Green's function hold.
I will give a precise meaning to the large deviation estimates in Theorem 5.3. The requirement that the large deviation estimates hold is the first smallness condition on h, I impose.
The second smallness condition on h > 0 is required to verify the initial condition of the inductive scheme. Given δ > 0 define a set of energies by This set is clearly an union of intervals. I am now ready to state the main result of the paper if the large deviation estimates for the Green's function hold.
Here, σ(H h ) denotes the spectrum of the operator H h = h∆ + V , where V was given by (1.1). The smallness condition in this theorem does not depend on the Diophantine condition (2.1), but the condition that the large deviation estimates hold, imposes such a condition through Theorem 2.1. In Appendix B, I will demonstrate that (2.4) which shows that Theorem 2.2 covers most of the spectrum.
I will prove a more precise result than Theorem 2.2. Recall that f : T → R is a non-constant real-analytic function. For x, y ∈ T and α Diophantine, we introduce the family of potentials (2.5) V x,y (n) = f ((T n α (x, y)) 2 ) = f (y + nx + n(n − 1)α). Let now h > 0, then we introduce the family of skew-shift Schrödinger operators by In short notation, we have H h,x,y = h∆ + V x,y . We also note that H h = H h,α,0 . The more precise result is Theorem 2.3. There exists h 1 = h 1 (δ, f ) > 0. Let 0 < h < h 1 such that the large deviation estimates for the Green's function hold. Then for E ∈ E δ , there exist (x, y) ∈ T 2 such that E is an eigenvalue of H h,x,y .
By minimality of the skew-shift, we have that σ(H h,x,y ) = σ(H h,x,ỹ ) for any x, y,x,ỹ ∈ T 2 . Using this, it is easy to show that Theorem 2.3 implies Theorem 2.2. Minimality of the skew-shift even implies that that the set of (x, y) ∈ T 2 such that E is an eigenvalue of H h,x,y is dense. However, it follows from the general theory of ergodic Schrödinger operators, that this set has zero measure.
It should be mentioned here that it is an open question whether the operator H h,x,y has pure point spectrum for almost every (x, y) or not. The results of Bourgain, Goldstein, and Schlag only imply pure point spectrum for almost every frequency α. So in some sense, Theorem 2.3 exhibiting at least one eigenvalue for some x, y is a step towards proving that H h,x,y has pure point spectrum for almost every (x, y) ∈ T 2 .
2.1. Optimality of the results. It might seem that the choice of potential in (2.5) is somewhat arbitrary, since we assume that the function only depends on the second coordinate. However, when one considers potentials of the more general form for a real analytic function g : T 2 → R, one faces obstruction to the spectrum being an interval. Consider g of the form for some small κ > 0. For κ = 0, the operator H h will just be the Almost-Mathieu operator, which is known to have Cantor spectrum, in particular it has at least one gap of size at least η for η > 0 sufficiently small. Hence, if κ < η 2 also the operator with skew-shift potential depending non-trivially on the second coordinate has at least one gap in its spectrum.
For fixed h > 0 and any continuous function f : T 2 → R, Avila, Bochi, and Damanik have shown in [1] that there exists a continuous function f 1 : is arbitrarily small and the Schrödinger operator H h,1 = h∆+V 1 with has Cantor spectrum. In particular, the spectrum contains a gap of size 2η, that is there is some E 0 such that It is classical that there exists now an analytic function f 2 such that . Then standard perturbation theory shows that where V 2 (n) = f 2 (T n (0, 0)). Hence, this operator has a gap in the spectrum. However, an inspection of the argument of my proof, shows that my result is stable under perturbing the sampling function f in the C 1 (T 2 ) topology as long as the large deviation estimates continue to hold. For this it is necessary, that the domain of analyticity of f stays the same.

2.2.
Discussion of the proof. I will now try to explain the main ideas in this paper. Let me begin by pointing out that checking the initial condition of the multi-scale scheme in Section 4, is done by a computation, I used in [19], to show that all gaps [E − , E + ] of σ(H h ) must satisfy E + − E − = O(h 2 ) as h → 0.
As mentioned above the proof proceeds by a multi-scale scheme. This scheme bears some similarities to the one used by Bourgain in [5], [6] to prove that the measure of the spectrum of quasi-periodic Schrödinger operators in the localization regime is positive. A key difference is that the arguments of this paper use analytic perturbation theory to show that a fixed number E 0 belongs to the spectrum, see Section 9. This is necessary, since extending an eigenvalue from scale to scale slightly perturbs it.
Maybe the key insight was that since the nth iterate of the skew-shift is (2.12) T n α (x, y) = (x + 2nα, y + nx + n(n − 1)α) (mod 1), one has that x enters the problem as a fast variable, since it gets multiplied by n in the second coordinate. This realization will allow us to prove an elimination of x argument in Section 7, which is used to eliminate double resonances. At this point let me also mention that the argument of Section 7 is an adaptation of the frequency elimination argument of Bourgain and Goldstein from [9].
As a further point of interest, let me point out that of the arguments to prove the presence of gaps in the spectrum, my argument is most related to the one of Goldstein and Schlag [15], see also [16] for a non-technical discussion. My argument shows that there are always simple resonances, so that I can eliminate double resonances, whereas Goldstein and Schlag show that there are certain simple resonances, that must also be double resonances, and not triple. The formation of double resonances then implies that gaps must open.
I furthermore wish to point out, what the methods of this paper would yield for the one-frequency quasi-periodic model, so consider the potential (2.13) V QP x,α (n) = f (nα + x) and the associated Schrödinger operator H QP h,x,α = h∆ + V QP x,α . Now, as in [9], α will play the role of a fast-variable. Translating the statement of Theorem 2.3, one obtains that for every E ∈ E δ , there exists a set A = A(E, h) such that (i) |A| → 1 as h → 0.
(ii) For α ∈ A, there exists x ∈ T such that E is an eigenvalue of H QP h,x,α . It is clear that this is compatible with quasi-periodic Schrödinger operators having gaps in their spectrum. . Given E 0 ∈ E δ , we will construct inductively a sequence N j and curves ξ j : T → T such that for a positive measure set X j , we have whenever x ∈ X j . The main problem with this approach is to pass from scale to scale. In order to discuss some aspects of this problem, I have included Figure 1, which shows the set of (x, y) such that This is illustrated in Figure 2, where these lines are shown dashed. Making these assertions precise is the content of the first step in the proof of the initial condition given in Section 4. I have also included Figure 3, which shows the same situation as Figure 1 except for N = 2 instead of N = 1.
The explanations so far explain, why there are eigenvalues close to the line y = 0.25 for many x. Let me now mention that using analytic perturbation theory, one can construct a function ξ 1 such that for many x, we have for some N 1 ≥ 1. This function ξ 1 will satisfy that |ξ 1 (x)| and |ξ 1 (x) − 0.25| are both small. In order to pass to the scale N 2 , we will exploit that x is a fast variable and the large deviation estimates for the Green's functions. These will allow us to show that for we have )) is not too small. This is what is called elimination of double resonances. Using this, we are able to show that for many x, we have h,x,ξ1(x) . Then, we construct ξ 2 similarly to ξ 1 and repeat the process.
In order to show that E 0 is indeed an eigenvalue of H h,x,y for some y, we will show that there are many x that are good for all j. Furthermore, for these we have that and also the eigenfunctions ψ j corresponding to the eigenvalue E 0 are convergent to some ψ ∞ . This implies that and thus that E 0 is an eigenvalue of H h,x,y for some y.

Outline of the proof
In this section, we explain the inductive construction, which we use in the proof. The following two sections contain the explanations of how to do check the initial condition and how to obtain the induction step. The proof is concerned with parameterizing isolated eigenvalues and E is simple.
In the setting of one-dimensional Schrödinger operators this condition is always satisfied, see Remark 1.10 in [24].
Isolated eigenvalues are important, since they behave well under perturbations. This can for example be seen in Lemma 9.2. The conclusions of this lemma can be summarized that a ε-isolated eigenvalue is stable under perturbations of A of size ε 2 . Next, we define what we mean by a parametrization: In order to examine this definition, let us look at the simplest example of M = 0. Then, we have that H h,x,y is just the multiplication operator by f (y). Hence, if we define The essential part of the induction scheme will be to show that given a parametrization at scale M , we can extend it to a parametrization at scale R ≈ e M c for some positive c > 0. However, this alone will not carry a sufficient amount of informations, we will also want that the eigenfunctions will have something to do with each other. For this, we introduce corresponding to the eigenvalue E 0 . We have for some |a| = 1 Here, we take W (n) = 1 + n 2 and we define the norm which is always well defined, since for us u and v are non-zero for only finitely many n. The reason for adding the weight W is that, we will want to control ψ, where the (x, y) are always the ones so that the ψ in ψ, ∂ We define in the following In order to understand one way, in which we will use Definition 3.3, we prove Lemma 3.4. Let (ξ, X) be a δ-extension of (ξ 0 , X 0 ). Let x ∈ X and ψ be a normalized eigenfunction of H . Then we have that Proof. Let x ∈ X. By condition (iv) of Definition 3.3, there exists a choice of a normalized eigenfunction ψ of H This implies that The claim now follows by the choice of δ.
This lemma shows, how we will use the condition the parametrization extends (ξ 0 , X 0 ). We will use these conditions to obtain some control on the ξ in the parametrization, in particular that ξ L ∞ (T) ≤ 1 3 , which we need to eliminate double resonances.
We will pass from scale to scale using the next theorem.
Since M needs to be large, (ξ 0 , X 0 ) does not satisfy the assumptions of this theorem. So, we will need We now begin the proof of Theorem 2.3. Choose M so large that Theorem 3.5 holds, finitely many additional largeness conditions might be imposed below. For For M large enough, ∞ =j+1 ε ≤ ε j and ε 1 ≤ 1 6 We will now inductively construct (ξ j , X j ) such that We construct (ξ 1 , X 1 ) using Theorem 3.6. We can require here that h is small enough such that We now see that the assumptions of Theorem 3.5 hold for (ξ 1 , X 1 ), so we can construct (ξ 2 , X 2 ). Using the following lemma, one can now construct (ξ j+1 , X j+1 ) from (ξ j , X j ) using Theorem 3.5.
Proof. This follows by being an extension is transitive.
We will obtain this way a sequence of compact subsets Since the X j are compact, we have for some x ∞ that Define y j = ξ j (x ∞ ). We clearly also have y j → y ∞ . By condition (iv) of Definition 3.3, we can choose eigenfunctions ψ j of H corresponding to the eigenvalue E 0 , which form a Cauchy sequence. We have where ψ ∞ = lim j→∞ ψ j . This finishes the proof of Theorem 2.3.

Proof of the initial condition
In this section, we prove the initial condition, that is Theorem 3.6. In order to make the statements look nice, we introduce A convenient choice for us will be η = ε 10 for most of this work. However, leaving η as an independent parameter has a big advantage. If we consider η <η <ε < ε, then we have that η-approximate ε-isolated impliesη-approximateε-isolated. Similarly to Definition 3.3, we will define what it means for an approximately isolated eigenvalue to extend an eigenvalue.
We note that this definition implies that However, as noted in the last section, condition (iii) is crucial to control various quantities needed in our multi-scale scheme. At the end of this section, we will prove  h,x,ξ0(x) .
We recall from (3.3) that ξ 0 (x) = y 0 . In the following, we will refer to the fact described in the previous theorem as (ξ 0 , X 0 ) extends to an η 1 -approximate ε 1 -parametrization on scale M 1 on X 1 . It should be clear how to generalize this definition to the more general situation, we are interested in. Furthermore, one can check that Lemma 3.4 remains valid in this setting. We now come to the last result, we need to prove the initial condition. I already formulate it in the way, we will need it for the inductive step.

Assume
(4.7) (ξ, X) extends to an ε 5 · η-approximate ε-parametrization on scale R on X andL = L + ε ≤ 1 3 . Then there exist (ξ, X) such that The proof of this theorem will be given at a later point. Before giving the proof of Theorem 4.3, we will give the proof of Theorem 3.6.
Proof of Theorem 3.6. This follows from the previous two theorems.
We now begin the proof of Define ψ h,x,y = h,x,y , the normalized version of the above vector. It should be noted that in order for (4.10) to make sense, we need that V x,y (±1) = 0. We will be able to ensure this with the following lemma In order to prove this lemma, we need to recall some things about analytic functions. Since f is analytic there exist F > 0 and α > 0 such that for every E ∈ R and ε > 0 Since V x,y (n) = f (y + nx + n(n − 1)α), this implies that for all E ∈ R, ε > 0, y ∈ T, n ∈ Z\{0}. For n = 0, (4.13) fails, since V x,y (0) = f (y).
Proof of Lemma 4.5. By (4.13), we can find a set X such that for x ∈ X , we have
We summarize the findings so far as  .

Proof. A computation shows that for
By the previous lemma, we have for j = that |E j −E 0 | 2 ≥ 1 2 h 1 2 . The claim follows by some computations.

We now come to
Proof of Theorem 4.3. The previous lemma implies that ϕ − aψ h,x,y ≤ h 1 4 for some |a| = 1. This finishes the proof.

5.
The proof of the multi-scale step; Theorem 3.5 I will begin by introducing the notion of suitability for a Schrödinger operator H acting on 2 (Z), and then introduce the large deviation estimates. After this, I will discuss the proof of Theorem 3.5. Given an interval [a, b] ⊆ Z, we introduce H [a,b] as the restriction of H to 2 ({a, . . . , b}). For E ∈ R and k, ∈ [a, b], we introduce the Green's function by where {e } ∈Z denotes the standard basis of 2 (Z). That is (5.2) e (n) = 1, = n; 0, otherwise.
We will quantify the properties of the Green's function with the following definition.  h,E,γ,Γ,p denotes the set of all (x, y) ∈ T 2 such that (5.5) [−N, N ] is not (γ, Γ, p)-suitable for H h,x,y − E.
Lemma 6.2 will derive a certain geometric structure for the set U, whereas the next theorem shows that the measure of this set is small. Proof. A proof of this theorem can be extracted from [8], [7], [10].
We will refer to the assumption that the conclusions of the previous theorem hold, as that the large deviation estimates hold. The specific form of the constants in the theorem above is of little importance to us, in particular the exponent 1 10 in e −N 1 10 could be any number > 0. I am using this concrete value to reduce the number of constants in the proofs. The next theorem is the basic conclusion, we will draw from it. such that for M 10 ≤ |n| ≤ R, we have It should be pointed out here, that this theorem does not tell us that double resonances happen with small probability. It tells us that resonances happen with small probability along curves satisfying certain estimates. We will use these curve to parametrize resonances of H . This way, we can eliminate the double resonances relevant to us. It would be interesting to obtain a true double resonance elimination theorem, since it would imply Anderson localization. I will still refer to the previous result as double resonance elimination.
I also wish to point out that the geometric content of Theorem 5.4 should be surprising. The information that the large deviation estimates hold, tells us that the measure of a subset of T 2 is small, then the output tells us that certain curves intersect this set with small probability. This is possible since, the set U has further geometric structure and the availability of a fast variable.
Having eliminated double resonances, we have the following result, which tells us that eigenvalues extend The proof of this theorem directly follows from the more abstract Theorem 8.1. We have no provided the abstract methods for proving Theorem 3.5. We need Lemma 5.6. We have that Proof. We have that ξ 0 (x) = 0 for all x. Since (ξ, X) is a 1 3 -extension of (ξ 0 , X 0 ), the claim follows.
Proof of Theorem 3.5. Apply Theorem 5.4 and introduce By Theorem 5.5, we have for x ∈ X 1 and η = e − 1 5 M that (ξ, X) extends to an η-approximate ε 1000 By Theorem 4.4, we obtain a parametrization for (ξ, X) such that The claims follows using that ϕ W ≤ (1 + R 2 ) ϕ .

Suitability
In this section, we discuss the notion of suitability defined in Definition 5.1 in more detail. We begin with the following stability result Then Proof. This is Lemma 5.3. in [20].
For the mechanism to eliminate double resonances, we will need to understand the horizontal slices of the set of unsuitability. For y ∈ T, we introduce for any set U ⊆ T 2 . We will need Lemma 6.2. For N large enough, there exists a set U such that h,E,γ,γN,5 and for y ∈ T (6.5) U (y) consists of less then N 10 intervals.
Since f is analytic, we have for some positive c > 0 that f − f R L ∞ (T) ≤ e −cR for R large enough. Define U to be the set of all x such that (i) We have Here . HS denotes the Hilbert-Schmidt norm, that is where A is a L × L matrix. We note that Choosing R = N 2 , the inclusions of the sets follow by Lemma 6.1.
Proof of Lemma 6.2. It remains to discuss the bound on the number of intervals of the sections U (y). By construction and choice of R, we have that for |n| ≤ N where deg(.) denotes the degree of a trigonometric polynomial in x. Since, using Cramer's rule, one can rewrite the conditions defining U as less than 10N conditions involving polynomials of degree ≤ 5N in the V x,y,R (n), the claim follows.

Elimination of the fast variable
In this section, I discuss a variant of the frequency elimination argument from the work [9] by Bourgain and Goldstein. There are two differences. First for us y will not be fixed, but depend on x. Second, we have some additional terms, since we vary x, the fast variable of the skew-shift, and not the frequency of a rotation. Proposition 7.1. Let U ⊆ T 2 and assume that for y ∈ T, we have that consists of at most M intervals. Furthermore let ξ : T → T be a continuously differentiable function satisfying for x ∈ T Then for R ≥ 1, we have Here we denote by ∼ R that R ≤ ≤ 2R. If one were to relax (7.2) to |ξ (x)| ≤ L, one would need to replace the meaning of this to = R + 3r L for 0 ≤ r ≤ R − 1. For simplicity, I have decided to work with (7.2 and that the sections U (y) consist of less then M 1 10 many intervals. Define a sequence R j by R j = 2 j−1 M 10 for j = 1, . . . , j max , where j max is defined to be minimal such that R jmax > R. We can apply Proposition 7.1 to U and these R j and see that the claim holds as long as M is large enough.
The main idea of the proof of Proposition 7.1 is that the second coordinate of T (x, ξ(x)) is to 1, whereas the first one is 1 to 1. This create sufficient independence of the two coordinates to imply the claim.
We begin now with fleshing out the details. Define for x ∈ T and ∼ R We will need Lemma 7.2. The map ψ is to 1 and satisfies for x ∈ T Proof. Since |ξ (x)| ≤ 1 3 , we have that for any x, y ∈ T The claim about ψ being to 1 follows. The claim about the derivative is a computation.
By the previous lemma, there exist maps θ ,p : T → T such that for every x ∈ T, there exists a unique 1 ≤ p ≤ such that (7.6) x = θ ,p (ψ (x)).
From this, we have for these that and that for any y ∈ T and 1 ≤ p ≤ (7.8) ψ (θ ,p (y)) = y.
Performing the change of variables y = ψ (x), we obtain that The claim follows.
The previous considerations imply that if we choose (7.15) γ = 1 10R 3 |U |, we obtain that The last piece is Proof. By assumption and y / ∈ B γ 1 , we may write for M ≤ M and I j intervals of length ≤ γ. Since y / ∈ B γ 2 , we have that for each j there is at most one pair ( , p) such that ϕ (θ p, (y)) ∈ I j .
The claim follows.
Proof of Proposition 7.1. This follows by the previous lemma and the equation preceding it.

Controlling a single eigenvalue
This section develops a mechanism to keep control of a single eigenvalue, when changing from scale to scale. The basic idea is the following: A single of the restrictions of the operator has an eigenvalue in a given energy interval, here H [−M,M ] in the energy interval [E 0 − ε, E 0 + ε]. If all other restrictions are suitable for this energy interval, then the same holds for the larger interval.
Since ϕ and ψ are normalized, the claim follows.
It should be pointed out at this point that the quantity ψ, Aψ is independent of the normalized eigenfunction chosen. This follows from that for two normalized eigenfunctions ψ,ψ of H [−R,R] to the eigenvalue E, one has ψ = cψ for some |c| = 1, since the eigenspaces of H [−R,R] to a single eigenvalue are one-dimensional.
Let us now begin with actually proving Theorem 8.1. The following lemma is a consequence of Lemma 6.1. Proof. This follows by Lemma 6.1, since ε ≤ e −3γN .
Having this lemma at hand, we will now first show that H [−R,R] has at least one eigenvalue in the interval [E 0 − ε 2 , E 0 + ε 2 ]. This follows by the next lemma, since 2e − 1 5 γM ≤ ε 20 . This implies the claim.
We now turn to show that H [−R,R] has only one eigenvalue in the energy interval We have that Lemma 8.6. For j = 0, we have Proof. We have that Now the claim follows, since for j = u 1 , we have |E j − E| ≥ 1 2 ε ≥ e − γ 10 N .
Hence, we have that Proof of Theorem 8.1. Since we may achieve by replacing ψ by cψ, where c = ϕ0,u1 | ϕ0,u1 | that the claim of the theorem follows.

Eigenvector perturbations
In this section, I discuss how to parametrize eigenvalues in a small neighborhood. I have decided to work in a somewhat general setting, in hope that this clarifies some of the arguments. Consider for t > 0 the square We will be considering a continuously differentiable family of self-adjoint operator H(x, y) defined for (x, y) ∈ D t . We will always assume that for (x, y) ∈ D t . We will be interested in small perturbations, that is in the language above that t is small. We begin with Theorem 9.1. Let ε > η > 0 with 4η < ε. Assume (9.2) and that Furthermore denote by ψ x,y the associated normalized eigenfunction of H(x, y). Then for any (x, y), (x,ỹ) ∈ D s , there exists |a| = 1 such that We recall that E 0 is an η-approximate ε-isolated eigenvalue of H(0, 0), if there exists an eigenvalue λ of H(0, 0) such that Instead of proving Theorem 9.1 directly, I will first prove the following more abstract lemma.
Proof. By (9.12) and the previous lemma, we have that Again by the previous lemma, we thus obtain that The claim follows since λ(x, y) is continuous.
We are now ready for Proof of Theorem 9.3. The intermediate value theorem implies that there exists some ξ(x) such that λ(x, ξ(x)) = E 0 . It follows from the implicit function theorem that The claim follows by some more computations.
Proof of Theorem 9.6. Define a = inf(X), b = sup(X) and introduce The claims now follow by some computations.

Proof of Theorem 4.4
In order to prove Theorem 4.4, we will need the following proposition, which allows us to construct continuously differentiable functions T → T. Let furthermore I 1 , . . . , I Q ⊆ T be disjoint intervals satisfying |I q | ≥ ε and ξ q : I q → T be continuously differentiable functions satisfying for x ∈ I q Then there exists a subset Q ⊆ {1, . . . , Q} such that and a continuously differentiable function ξ : T → T satisfying for q ∈ Q that for x ∈ I q we have ξ(x) = ξ q (x) and the bound We begin by constructing the set Q. For intervals I,Ĩ ⊆ T, we write (10.5) dist(I,Ĩ) = inf x∈I,x∈Ĩ x −x .
Lemma 10.2. There exists Q such that (i) For q,q ∈ Q with q =q, we have (10.6) dist(I q , Iq) ≥ ε. Proof. Write I q = [a q , b q ] and order the intervals such that 0 ≤ a 1 < b 1 < a 2 < b 2 < · · · < a Q < b Q .
We may have b Q > 0. For Q = 2P even define the three sets For Q = 2P + 1 odd, define Since |I q | ≥ ε and we ordered the intervals, property (i) holds. Furthermore, since the sets Q 1 , Q 2 , Q 3 are disjoint, there exists j ∈ {1, 2, 3} such that property (ii) holds. Chose Q = Q j , finishing the proof.
Having established this preliminary proposition, we will now proceed to prove Theorem 4.4. The first step is to apply Theorem 9.3. Define Q = Proof. It is easy to see that X ⊆ q good I q , which implies the claim.
For each good q, we choose x q ∈ I q ∩ X and define (10.11) h,xq+x,ξ(xq+x)+y . We note that, we have that (10.12) We have Lemma 10.5. Let d be defined as in (3.9). Then for ψ the eigenfunction of H(0, 0) corresponding to the eigenvalue in the interval [E 0 − ε, E 0 + ε], we have Proof. By Lemma 3.4, we have that We thus have that the conditions of Theorem 9.1 and Theorem 9.3 hold. Using these, we obtain Lemma 10.6. Let q be good. Then there exists a curve ξ q : I q → T such that for x ∈ I q ∩ X, we have h,x,ξq(x) .
Proof. By Theorem 9.6, we obtain for each good q a mapξ q defined on the interval [− 1 Q , 1 Q ] such that E 0 is a ε 2 -isolated eigenvalue of H(x,ξ q (x)).
Define ξ q (x) =ξ q (x − x q ) + ξ(x). It is easy to check that the previous equation implies for x ∈ I q that h,x,ξq(x) . Furthermore, we have that |ξ q (x) − ξ(x)| ≤ η., We will now check (10.14). First, one can describe the conclusions in the furthermore of Theorem 9.1 as E 0 extends to a ε 2 -isolated eigenvalue of H(x,ξ q (x)) from H(x, y) for any y. This combined with assumption of Theorem 4.4 implies (10.14).

(B.2)
It is easy to see that E + (h) ≤ max(f ) + 2h. Choose now y 0 such that f (y 0 ) = max(f ) and consider which is all we claimed.