On Two-Sided Estimates for the Nonlinear Fourier Transform of KdV

The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order. We further obtain quantitative estimates of the nonlinear Fourier transformed in arbitrary weighted Sobolev spaces.


Introduction
We consider the KdV equation u t = −u xxx + 6uu x on the circle T = R/Z with u real-valued. As is well known, the KdV equation can be written as an infinite-dimensional Hamiltonian system As an infinite-dimensional Hamiltonian system, the KdV equation is well known to be completely integrable. According to [12], this is true in the strongest possible sense meaning that KdV admits global Birkhoff coordinates (x n , y n ) n 1 .
To give a precise statement of our results, we define for u = n∈Z u n e i2nπx in The Birkhoff map u → (x n , y n ) n 1 is a bi-analytic, canonical diffeomorphism Ω : H 0 0 → h 0 , whose restriction Ω : H m 0 → h m is again bi-analytic for any integer m 1. On h 1 the transformed KdV Hamiltonian H • Ω −1 is a real-analytic function of the actions I n = (x 2 n + y 2 n )/2 alone and the equations of motion take the particular simple forṁ x n = −ω n y n ,ẏ n = ω n x n , ω n := ∂ In H.
The mapping Ω may thus be viewed as a nonlinear Fourier transform for the KdV equation. Furthermore, the derivative d 0 Ω of Ω at the origin is a weighted Fourier transform, and on H 0 which we refer to as Parseval's identity -cf. e.g. [12]. Our main result says that for higher order Sobolev norms the following version of Parseval's identity holds for the nonlinear map Ω. The estimates (i) and (ii) are reminiscent of the 1-smoothing property of the Birkhoff map Ω established in [14] as they are linear in the highest Sobolev norm u m and the highest weighted h m -norm Ω(u) m , respectively. The proof of Theorem 1 relies on estimates for the KdV action variables I(u) = (I n ) n 1 where I n = (x 2 n + y 2 n )/2, n 1. The decay properties of the actions are closely related to the regularity properties of u -cf. [9,10,5,21]. We quantify this relationship by providing two-sided estimates of the Sobolev norms of u in terms of weighted 1 -norms of I(u). For that purpose introduce for s real the weighted sequence space 1 s whose norm is defined by Note that in the estimates (i) and (ii) of Theorem 2 the corresponding exponents are of the same order since the action variables have to be viewed as quadratic quantities.
For the direct problem, that is estimates (i) of Theorems 1 & 2, we obtain estimates which hold for a larger family of spaces referred to as weighted Sobolev spaces -see [9,10] for an introduction. A normalized, symmetric, submultiplicative, and monotone weight is a function w : Z → R with w n 1, w n = w −n , w n+m w n w m , w |n| w |n|+1 , February 17, 2015 for all n, m ∈ Z. The class of all such weights is denoted by M and H w 0 is the space of H 0 0 functions u with finite w-norm Further, h w ⊂ h 0 denotes the elements (x, y) with x 2 w + y which give rise to the Gevrey spaces H s,a,σ 0 , as well as weights of the form that are lighter than Abel weights but heavier than Gevrey weights. We assume all weights w ∈ M to be piecewise linearly extended to functions on the real line w : Theorem 3 For any weight w = n r v with 0 r 1/2 and v ∈ M there exists an absolute constant c w,r > 0 such that the restriction of the Birkhoff map Ω to H w 0 takes values in h w and satisfies Moreover, we find for the action variables on H w The bounds given in (i) and (ii) are valid for all submultiplicative weights including those growing exponentially fast. They reflect the nature of the weight: If, for example, w grows polynomially, then the bounds (i) and (ii) are polynomial in u w , whereas if w grows exponentially, then the bounds are exponential in u w . Note that the bounds improve as soon as the weight w incorporates the factor n r with r > 0.
Method of proof. The main ingredient into the proof of Theorem 1 is the estimate of the action variables I n , n 1, of Theorem 2. To establish the latter, we use the KdV action variables J n,m , n 1, on level m 0 introduced by McKean & Vaninsky [19] in context of the nonlinear Schrödinger (NLS) equation. They are defined in terms of spectral data of the corresponding Hill operator arising in the Lax-pair formulation of KdV -see Section 2. For m = 0, the actions J n,0 coincide with I n , whereas for m 1 they have the asymptotic behavior Furthermore, they satisfy the trace formula where H k denotes the kth Hamiltonian in the KdV hierarchy. The first two Hamiltonians of this hierarchy are given by More generally, these Hamiltonians have the form with p m being a canonically-determined polynomial.
Taking the asymptotic behavior of the actions J n,m , the corresponding trace formulae, and the representation of the Hamiltonians together we get which yields the bound stated in Theorem 2 (i). To obtain item (ii) of Theorem 2 we rewrite the mth Hamiltonian as Using the trace formula we derive a bound of H m in terms of I(u) 1 2m+1 . Since the p m -term depends only on derivatives of u up to order m − 1, an inductive argument then gives part (ii) of Theorem 2. We remark that the February 17, 2015 asymptotics of J n,m are derived from sufficiently accurate asymptotics of the periodic eigenvalues of the corresponding Hill operator, and that our method of proof does not involve any auxiliary spectral quantities such as the spectral heights.
To prove Theorem 3, we take a slightly different approach by estimating the action variables in terms of the spacing of the periodic eigenvalues of the associated Hill operator. For the latter, estimates in any weighted norm are available -see e.g. [9,10,5,21] -allowing us to obtain Theorem 3.
Related results. First results on global bounds of the KdV actions in terms of Sobolev norms were obtained by Korotyaev [15,16] .
By augmenting c m but without increasing the degree of the right hand side, this bound may be simplified to In comparison, our estimate Theorem 2 (i) is linear in u Note that N m grows factorially with m. Our estimate Theorem 2 (ii) considerably improves the latter one, since the bound is linear in I(u) 1 2m+1 , the exponent of the remainder (1 + I(u) 1 First results on bounds of the Birkhoff map for KdV on the weighted Sobolev spaces H w 0 can be found in Kappeler & Pöschel [13]. The authors proved where the ⇐ part holds true only for weights with subexponential growth. The estimates of Ω(u) h w presented in Theorem 3 quantify this relationship. February 17, 2015 The methods developed in this paper were introduced by the author [20] in the context of the nonlinear Fourier transform for the periodic defocusing NLS equation to show results corresponding to Theorems 1-3. We note that the result of Theorem 1 improves on the corresponding result obtained for the defocusing NLS equation since the exponents in the remainders of estimate (i) and estimate (ii) are equal. From a broader perspective we may view the weighted actions (2nπ)I n as a perturbation of the squared modulus |u n | 2 of the nth Fourier coefficient of u.
Our method of comparing the norms I(u) 1 2m+1 with the Hamiltonians of the KdV hierarchy consists in a separate analysis of Fourier modes of low and high frequencies. This idea has a long history in the analysis of nonlinear PDEs. Most recently, it lead Colliander, Keel, Staffilani, Takaoka & Tao [2,3,4] to invent the I-Method, which allows to obtain global well-posedness of subcritical equations in low-regularity regimes where the Hamiltonian (or other integrals) of the equation cease to be well defined. The idea is to damp all sufficiently high Fourier modes of a local solution such that the Hamiltonian can be controlled by weaker norms while still being an »almost conserved« quantity. The difficulty here is to choose the damping carefully enough such that the nonlinearity of the equation does not create a significant interaction of low and and high frequencies.
Our aim is so to say opposite to that of the I-Method: As we look for quantitative global estimates, the most delicate part of our analysis is to get control of the modes of low frequencies. It is achieved by an appropriate localization of the periodic eigenvalues of the Hill operator -see Proposition A.1 and the references mentioned in Appendix A.
The results on the analyticity and asymptotic expansion of the integral F (λ) introduced in Section 2 have also been applied in [8] concerning convexity properties of the KdV Hamiltonian. They were put together in a joint effort.
Organization of the paper. In Section 2 the KdV action variables on integer levels m 0 are defined and the trace formulae relating them to the hierarchy of KdV Hamiltonians are proven. In Sections 3 and 4 Theorem 2 (i) and (ii), respectively, are obtained by use of the localization of the Hill spectrum, which for the convenience of the reader is proved in Appendix A. Finally, in Section 5 we prove Theorem 3 by obtaining a uniform estimate of the actions in terms of the spacing of the periodic eigenvalues.

Setup
In this section we briefly recall the definition of the action variables as well as the main properties of the spectral quantities used to define them. We follow the exposition [12] -see also [22,21,5,6]. Consider Hill's operator on the interval [0, 2], endowed with periodic boundary conditions and q being a complex potential in H 0 0,C := H 0 0 (T, C). The spectrum of L(q), called the periodic spectrum of q, is pure point and complex in general as the operator is not self-adjoint. We may order the eigenvalues lexicographically -first by their real part and second by their imaginary part -such that Their asymptotic behavior is λ ± n (q) = n 2 π 2 + 2 n , and we define for any n 1 the gap length n stands for an 2 -sequence. For convenience we set γ 0 = ∞. To obtain a suitable characterization of the periodic spectrum of q, we denote by y 1 (x, λ, q) and y 2 (x, λ, q) the standard fundamental solutions of L(q)y = λy, and by ∆(λ, q) the discriminant ∆(λ, q) := y 1 (1, λ, q) + y 2 (1, λ, q).
To simplify matters, we may drop some or all of its arguments from the notation whenever there is no danger of confusion. The periodic spectrum of q is precisely the zero set of the entire function ∆ 2 (λ) − 4, and we have the product representation Hence, the discriminant is uniquely determined by the periodic spectrum. We also need the λ-derivative ∆ • := ∂ λ ∆ whose zeros are denoted by λ • n and satisfy λ • n = n 2 π 2 + 2 n . This derivative has the product representation For each real potential q there exists an open neighborhood W q within H 0 0,C such that for every p ∈ W q the closed intervals , n 1, are disjoint from each other. Even more, there exist mutually disjoint neighborhoods U n ⊂ C, n 0, called isolating neighborhoods, which satisfy: for n sufficiently large.
Throughout this text W q denotes a neighborhood of q such that a common set of isolating neighborhoods for all p ∈ W q exists. The union of all W q for q real defines an open and connected neighborhood of H 0 0 within H 0 0,C and is denoted by W.
Following the approach of Flaschka & McLaughlin [6], one can define action variables for the KdV equation by Arnold's formula Here a n denotes a cycle around (λ − n , λ + n ) on the spectral curve on which the square root ∆ 2 (λ) − 4 is defined. This curve is another spectral invariant associated with q, and an open Riemann surface of infinite genus if and only if the periodic spectrum of q is simple. To avoid the technicalities involved with this curve, we fix proper branches of the square root which allows us to reduce the definition of the actions to standard contour integrals in the complex plane -see also [12,11]. Denote by + √ the principal branch of the square root on the complex plane minus the ray (−∞, 0]. Furthermore, for q ∈ W the standard root is defined by the condition The standard root is analytic in λ on C \ G n and in (λ, Finally, we define the canonical root This root is analytic in The nth KdV action variable of q ∈ W is then given by where Γ n denotes any sufficiently close circuit around G n . More generally, we define the nth KdV action variable on level m 0 by Note that the action on level zero J n,0 equals the action I n .
We proceed with the analysis of the analytical properties of the action integrand. To this end, we define for any A path in the complex plane is said to be admissible for q if, except possibly at its endpoints, it does not intersect any non collapsed gap G n (q).
Lemma 1 For each q ∈ W, the 1-form ω has the following properties: In particular, for any closed circuit Γ n in U n around G n , Proof. (i) ∆ • is analytic on C × H 0 0,C and the canonical root is analytic on (C \ k 0 U k ) × W q and does not vanish there. Therefore, ω is analytic on (iii) We first consider the case of q being real-valued. Clearly, the functional W q → C, p → ∂Un ω is analytic. Further, for any p ∈ W q real-valued, one has (−1) n ∆(λ) 2 on G n so, after deforming the contour of integration to G n , according to the definition of the canonical root (see [12] for a sign table) 1 With the convention γ 0 = ∞. February 17, 2015 Thus ∂Un ω vanishes on W q ∩ H 0 0 and hence on all of W q by Lemma D.1. In view of W = q∈H 0 0 W q we conclude that Γn ω = 0 for any closed circuit Γ n in U n around G n and any q ∈ W. Now fix q ∈ W arbitrary. The identity then we may define the left hand side G + n and the right hand side G − n of G n by The canonical root admits opposite signs on G ± n that is Defining the contour Γ n by going from λ − n to λ + n along G − n and then going back By contour deformation it then follows that Writing the action variables as makes the claimed analyticity on W evident. To proceed, we define for any q ∈ W and any n 0 on Clearly, this improper integral exists, as ω has the integrable singularity 1/ λ − λ + n near λ + n . By Lemma 1 the integral is also independent of the chosen admissible path.

Lemma 2
For every q ∈ W and every n 0, we have that In particular, F 0 extends continuously to all points λ + 0 and λ ± n , n 1. One has F 0 (λ + 0 ) = 0 and G k for every n 0. February 17, 2015 (iv) If q is real, then for any n 1 and any real λ − n λ λ + n , Clearly, f n is continuous on G n , strictly positive on (λ − n , λ + n ), and vanishes at the boundary points.
(v) At the zero potential one has F n (λ, 0 Proof. (i) Let us first consider the case where n = 0. Since ω is analytic on Then by the product representation (4) Since λ + 0 is analytic on W and χ 0 is analytic on U 0 × W q -cf. [11, Appendix A] and [7, Corollary 12.8] -the claimed analyticity of F 0 follows. In Lemma C.1 we obtain the analyticity of F n , n 1. This proof is a bit more work due to the fact that in this case λ + n may be a double eigenvalue.
To proceed, first consider the case where q is real-valued. In this case i and hence (iii) In view of item (i) it remains to show that F 2 n admits also for γ n = 0 an analytic extension from U n \ G n to all of U n . For n = 0 write (4) in the form February 17, 2015 and similarly, for n 1 write (4) in the form The mappings χ n , n 0, are analytic on U n -cf. [11,Appendix A]. Moreover, the roots + λ − λ + 0 and ς n (λ), n 1, respectively, admit opposite signs on opposite sides of G 0 and G n , n 1, respectively. Therefore, in view of F n (λ) = λ λ + n ω, for any n 0 and λ ∈ G n , Consequently, F 2 n is continuous and hence analytic on all of U n . (iv) If q is real-valued, then for any (v) At the zero potential, ω(λ, 0) = 1/ 2i + √ λ dλ which follows directly from the product representation (4) of ω.
we drop the subscript of F 0 in the sequel to simplify notation and denote Remark. By exactly the same arguments one can show that ω and F n , n 0 satisfy the properties stated in Lemma 1 & 2 when W is extended to the open neighborhood of H −1 0 within H −1 0,C constructed in [11], and W q is chosen as an open neighborhood of q within H −1 0,C such that a common set of isolating neighborhoods exists. However, we will not make use of this fact.
If q is real-valued, then we can integrate by parts and subsequently shrink the contour of integration to the interval G n to obtain Thus for real-valued potentials all action variables J n,m are real, and those on even levels are nonnegative. Moreover, by the mean value theorem, for some ζ n,m ∈ G n . By (1) the actions on level zero satisfy In the sequel we derive similar formulae expressing the actions on any level m 0 in terms of Hamiltonians of the KdV hierarchy. To this end, we need the following expansion of F for real-valued finite-gap potentials, which follows from the expansion of the discriminant ∆ obtained in Appendix B.
where an empty sum denotes zero, and H k denotes the kth Hamiltonian in the KdV hierarchy.
Proof. Note that n 1 n 2m |γ n | 2 < ∞ uniformly on bounded subsets of H m 0,Csee [17,10] and also Appendix A. Moreover, uniformly on bounded subsets of W ∩ H m 0,C -see [12]. Thus the sum of the actions on level m converges locally uniformly to a real-analytic function on H m 0 . Since also the right hand side of (12) is real-analytic on H m 0 , it suffices to prove the claim on the dense subset of real-valued finite-gap potentials.
Suppose q is a real-valued finite-gap potential, then there exists N 1 such that γ n = 0 for n > N . Let C r denote a circle around the origin of sufficiently large radius r so that all gaps G n , 1 n N are enclosed. Then F 2 = F 2 0 is analytic outside C r by Lemma 2 and by contour deformation Since F 2 n is analytic on U n , the even powers of F n in the expansion of F 2 = (F n − inπ) 2 do not contribute to the contour integral, thus On the other hand, according to Proposition 3 for λ n = (n + 1/2) 2 π 2 February 17, 2015 as n → ∞. One infers directly from (4) that for a finite-gap potential, ω = O(1/ √ λ) and hence F 2 (λ) = O(λ) as |λ| → ∞. Therefore, F 2 (λ) is meromorphic at infinity and we obtain from (13) using Cauchy's Theorem This proves the trace formula. v As an immediate consequence we obtain for q ∈ H m where · · · comprises only lower order derivatives of q. In Sections 3 and 4 this identity is used to compare the sum n 1 (2nπ)J n,m of the actions on level m with the corresponding Sobolev norm of the potential. To proceed, we need an estimate of the action J n,m on level m 0 in terms of the weighted action (nπ) 2m I n . (1 + q 0 ) q 0 ζ n,m 256 q 2 0 . v

Estimating the actions
In this section we obtain an estimate of all weighted 1 -norms I(q) 1 2m+1 , m 1, in terms of the Sobolev norms q m of q. This will be done in two steps. First, we estimate I(q) 1 2m+1 in terms of the sum n 1 (2nπ)J n,m of the actions on level m and a remainder depending solely on the L 2 -norm q 0 . Second, we express n 1 (2nπ)J n,m through Hamiltonians of the KdV hierarchy using the trace formula. The polynomial structure of these Hamiltonians then allows us to prove the claimed estimate. The remaining actions may then be estimated by Both estimates together with 2 I(q) 1 1 = q 2 0 give the claim. v Proof of Theorem 2 (i) for m = 1. Lemma 6 and the trace formula (12) yield Using the L ∞ -estimate q L ∞ q x 0 on H 1 0 , we obtain for the Hamiltonian Thus we arrive at To proceed with the general case m 2, we need an estimate of the Hamiltonian H m (q) for m 2. To this end, we recall some well known facts about the Hamiltonians in the KdV hierarchy -see for example [18, Theorem 1].

Lemma 7 The mth Hamiltonian in the KdV hierarchy has the form
with p m being a homogeneous polynomial of degree m+2, without constant term, where q counts as 1 degree and differentiation as 1/2 degree.
Consequently, each monomial p of p m may be estimated by with some positive constant c p and integers µ 0 , . . . , µ m−1 , where we denote q (m) := ∂ m x q to simplify notation. It turns out to be convenient to use exponents which are multiples of two, that is Thus, we obtain the estimate with positive reals c σ and I m ⊂ (Z 0 /2) m being the set of all multi-indices satisfying the constraint (14). The majorant P m allows us to obtain detailed estimates of p m and therefore H m .

Proof of Theorem 2 (i) for m 2.
For σ ∈ I m we have by (14) m + 2 (m + 1)σ m−1 , hence P m is at most quadratic in q (m−1) . Together with the L 1 -estimate q (m−1) L 1 q (m−1) 0 we obtain for the derivative of highest order Estimating the remaining factors using the L ∞ -estimate Since 2σ 0 + · · · + 2σ m−1 m + 2, we obtain for the Hamiltonian

Estimating the Sobolev norms
We now turn to the problem of controlling the Sobolev norm q m of the potential in terms of weighted norms of its actions. As a starting point we use the identity inferred from Lemma 7, and proceed in two steps. First, we estimate the sum  (9), and further by (12) for any m 1 Lemma 8 gives an estimate of the first summand, while we inductively obtain for the second This proves the claim. v February 17, 2015 Proof of Theorem 2 (ii). We begin with the case m = 1. Clearly, and, together with q 2 0 = 2 I(q) 1 1 , we therefore obtain Finally, by the preceding Theorem, To proceed with the general case m 2, we recall from (16) that Theorem 4 provides an estimate of H m , while for p m we have by (15) where the majorant P m only contains derivatives of q up to order m − 1 and the multi-indices σ satisfy the constraint (14). We use this to prove by induction for any m 2 Clearly, this holds true for m = 0, 1. Consider the inductive step m − 1 → m. Recall from the proof of Theorem 2 (i) that P m is at most quadratic in |q (m−1) | for m 2. Thus, we have the following expansion P m = P m;2 (q, . . . , q (m−2) )|q (m−1) | 2 + P m;1 (q, . . . , q (m−2) )|q (m−1) | + P m;0 (q, . . . , q (m−2) ), where for 0 k 2 the entity P m;k |q (m−1) | k incorporates those multi-indices σ ∈ I m with 2σ m−1 = k. In particular, P m;2 = c|q| with some c 0. These three terms will be estimated separately.

Estimating the Actions in Weighted Sobolev Spaces
The case of estimating the actions in arbitrary weighted Sobolev spaces H w 0 differs significantly from the case of integer Sobolev spaces H m 0 since for an arbitrary weight w there is no identity known to exist relating q w to Hamiltonians of the KdV hierarchy. Albeit, even in the case of weighted Sobolev spaces, the regularity properties of q are well known to be closely related to the decay properties of the gap lengths γ n (q) -see e.g. [21,5,9] and Appendix A. Moreover, the asymptotic relation is known to hold locally uniformly on H 0 0 -see [12]. In this section we obtain a quantitative version of (17) which is uniform in q 0 on all of H 0 0 . This together with the estimates of the gap lengths given in the appendix allows us to prove Theorem 3.
For q ∈ H 0 0 we recall from (5) that February 17, 2015 Here the latter identity follows from the closedness of ω around the gap. In the case I n = 0, or equivalently γ n = 0, we shrink the contour Γ n to the straight line [λ − n , λ + n ] and insert the product representation (4) of ω, to obtain Parametrizing the gap G n = [λ − n , λ + n ] by λ t = τ n + tγ n /2 gives 8nπI n γ 2 where we set t n = 2(λ • n − τ n )/γ n . Since |τ n − λ • n | γ n /2 we conclude |t n | 1, and hence The following uniform estimate of (nπ)|χ n | Gn allows us to proof the desired quantitative version of (17) in the sequel.

Lemma 9
On H s 0 with 0 s 1/2 for any n 8 q Proof. For n = m with n, m 4 q 0 and λ ∈ G n we have by Lemma A.5 Further, |γ m | 6 q 0 by Proposition A.7, hence Suppose N 4 q s , then by Proposition A.7 To estimate the remaining part of the product we note that by the ordering of the eigenvalues By Proposition A.1 and Lemma A.5 we have for n N and λ ∈ G n , while on the other hand by Proposition A.1 |λ • 1 − λ| |λ + 0 − λ| n 2 π 2 + 4 q 0 + (1 + q 0 ) q 0 12n 2 .

A Appendix -Spectral Theory
In this appendix we review, for the convenience of the reader, the localization of the periodic spectrum of Hill's operator as well as an estimate of its gap lengths, which both are used in the various parts of this paper. We follow the exposition in [21] -see also [9,10,5]. Consider the operator on the interval [0, 2] endowed with periodic boundary conditions and q being a complex-valued, 1-periodic L 2 -potential with vanishing mean value, that is q ∈ H 0 0,C = H 0 0 (T, C). The spectrum of L(q) for q = 0 consists of λ + 0 = 0 and the double eigenvalues λ + n = λ − n = n 2 π 2 , n 1. For q ∈ H 0 0,C arbitrary and λ sufficiently large, the equation −f + qf = λf may be regarded as a perturbation of the free equation −f = λf , hence one can expect the eigenvalues to come asymptotically in pairs λ ± n satisfying λ ± n ∼ n 2 π 2 as n → ∞. The following localization of the eigenvalues is well known -see e.g. [21,5,9].
The remaining eigenvalues for 4 q 0 > n satisfy The lower bound of the remaining eigenvalues is obtained directly from the quadratic form associated to L.
For q = 0 the eigenspace of the double eigenvalue λ + n = λ − n = n 2 π 2 , n 1, is spanned by e n := e inπx and e −n := e −inπx . When n is sufficiently large, then for λ ∈ U n the dominant modes of a solution of Lf = λf are thus expected to be e ±n . Therefore, it makes sense to separate these modes from the others by a Lyapunov-Schmidt reduction. To this end, denote the space of 2-periodic complex-valued H w -functions by H w ,C and consider the splitting H w ,C = P n ⊕ Q n = sp{e k : |k| = n} ⊕ sp{e k : |k| = n}.
The projections onto P n and Q n are denoted by P n and Q n , respectively. We write the eigenvalue equation Lf = λf in the form where V denotes the operator of multiplication with q. Since A λ is a Fouriermultiplier, by writing February 17, 2015 called the P -and the Q-equation.
We first solve the Q-equation on each strip U n by writing V v as a function of u. With foresight to estimating the gap lengths, we consider operator norms induced by shifted weighted norms [21]. For u in H w ,C the i-shifted H w -norm is given by Lemma A. 3 If q ∈ H w 0,C , then for any n 1 and λ ∈ U n , Hence, the restriction of A λ to Q n is boundedly invertible, and the function is well defined. By Hölder's inequality we obtain for the weighted 1 -norm Finally, by Young's inequality for the convolution of sequences Consequently, T n is a 1 2 -contraction on H w ,C for all n 4 q w . If we multiply the Q-equation from the left by V A −1 λ , then which may be written as February 17, 2015 Hence, for n 4 q w one finds a unique solution of the Q-equation. Substituting this solution into the P -equation yields Writing the latter as we immediately conclude that there exists a one-to-one relationship between a nontrivial solution of S n u = 0 and a nontrivial 2-periodic solution of Lf = λf . Hence, a complex number λ ∈ U n is a periodic eigenvalue of L if and only if the determinant of S n vanishes. Recall that P n is the orthogonal projection onto the two-dimensional space P n . The matrix representation of an operator B on P n is given by ( Be ±n , e ±n ) ±,± .
Therefore, we find for S n the representation with σ n = n 2 π 2 , and the coefficients of the latter matrix given by a n := T n V e n , e n , c n := T n V e −n , e n .
Moreover, by inspecting the expansions of a n and a −n using the representations of T n and V in Fourier space, one concludes that a n = T n V e n , e n = T n V e −n , e −n = a −n .
Hence the diagonal of S n is homogenous, We introduce the following notion for the sup-norm of a complex-valued function Lemma A.4 If n 4 q w , then |a n | Un 2 T n w;n q w and w 2n |c ±n − q ±n | Un 2 T n w;±n q w . February 17, 2015 Proof. WithT n = I + T nTn we obtain c n = q n + T nTn V e −n , e n . Furthermore, since f, e n = f e n , e 2n for any function f , we conclude The claim follows with T n T n V e −n w;n 2 T n w;n V e −n w;n , and V e −n w;n = V e 0 w = q w . The proof for a n and c −n is the same. v The preceding lemma implies that the determinant of S n det S n = (λ − σ n − a n ) 2 − c n c −n is an analytic function in λ, which is close to (λ − σ n ) 2 for n sufficiently large. This is what we need to localize the eigenvalues.
Lemma A.5 If n 4 q 0 , then the determinant of S n has exactly two complex roots ξ − , ξ + in U n , which are contained in Proof. Let h = λ − σ n − a n . The preceding lemma together with Lemma A.3 gives Thus it follows from Rouché's Theorem that h has a single root in D n , just as (λ − σ n ). In a similar fashion, we infer from that h 2 and det S n have the same number of roots in D n , namely two, while det S n clearly has no root in U n \ D n . v Proof of Proposition A.1. For each n 4 q 0 Lemma A.5 applies giving us two roots ξ + and ξ − of det S n which are contained in D n ⊂ U n . Since the strips U n cover the right complex halfplane, and λ ± n ∼ n 2 π 2 as n → ∞, it follows by a standard counting argument that these roots have to be the periodic eigenvalues λ ± n . Thus Moreover, these are the only eigenvalues contained in n 4 q 0 U n . Hence if we choose N 1 such that N 4 q 0 > N − 1 then for any 1 n < N , February 17, 2015 We now turn our attention to estimating the gap lengths γ n = λ + n − λ − n , which by the preceding considerations satisfy for n 4 q 0 with ξ ± being the complex roots of det S n on U n .
Lemma A. 6 If n 4 q 0 , then Proof. We write det S n = g + g − with where the branch of the root is immaterial. Each root ξ of det S n is either a root of g + or g − , respectively, and thus satisfies ξ = σ n + a n (ξ) ± ϕ n (ξ). To estimate the distance of the two roots ξ ± , we note that a n is analytic on U n , so Cauchy's estimate gives |∂ λ a n | Dn |a n | Un dist(D n , ∂U n ) The claim now follows with The coefficients c ±n have the same decay as the Fourier coefficients of q ±n , hence the regularity of the potential is reflected in the decay of its gap lengths.
and further w 2n |γ n | 6 q w for all n N .

C Appendix -Analyticity of F
In this appendix we prove analyticity properties of the functionals F n , n 1, introduced in Section 2. We use the notation established in Section 2. In particular, the open neighborhoods W and W q .
Lemma C.1 Let q ∈ W and n 1. Then for any ν ∈ ∂U n , within H −1 0,C constructed in [11], and W q is chosen as an open neighborhood of q within H −1 0,C such that a common set of isolating neighborhoods exists.
Proof. We want to apply [7,Theorem A.6]. First note that for any µ, ν ∈ ∂U n , µ ν ω is analytic on W q by Lemma 1. Hence at each step of the proof we might change ν at our convenience.
In a second step we show that the restriction of F n to Z n ∩W q is weakly analytic. Note that on Z n ∩W q , F n coincides with the function 1 2i ν τn χ n (λ) dλ with χ n given by (24), where the path of integration is chosen in U n but otherwise arbitrary. Thus F n | Zn∩Wq is weakly analytic.
In a third and final step we prove that F n is continuous on W q . By the considerations above, F n is continuous in each point of W q \ Z n and the restriction of F n to Z n ∩ W q is continuous. Hence it remains to show that for any p ∞ ∈ Z n ∩ W q and any sequence (p k ) k 1 ⊂ W q \ Z n with p k → p ∞ in W q it follows that F n (p k ) → F n (p ∞ ) as k → ∞. By [12,Proposition B.13] (see also [11,Proposition 2.9]) λ • n − τ n = O(γ 2 n ) locally uniformly around p ∞ . Since γ n (p k ) → 0 as k → ∞, there exists k 0 1 so that |λ • n (p k ) − τ n (p k )| |γ n (p k )|/2, k k 0 .