Estimates on Periodic and Dirichlet Eigenvalues for the Zakharov-Shabat System

Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system

B. Gr´ebert1, T. Kappeler2 December 19, 2002

1. UMR 6629 CNRS, Universite de Nantes, 2 rue de la Houssi`ere, BP 92208, 44322 Nantes cedex 3, France.

2. Institut fur¨ Mathematik, Universit¨at Zuric¨ h, Winterthurerstrasse 190, CH-8057 Zuric¨ h, Switzerland.

Abstract Consider the 2 2 ﬁrst order system due to Zakharov-Shabat, × 1 0 0 ψ LY := i Y 0 + 1 Y = λY 0 1 ψ 0 − 2

with ψ1, ψ2 being complex valued functions of period one in the weighted w w Sobolev space H HC . Denote by spec(ψ , ψ ) the set of peri- ≡ 1 2 odic eigenvalues of L(ψ1, ψ2) with respect to the interval [0, 2] and by specDir(ψ1, ψ2) the set of Dirichlet eigenvalues of L(ψ1, ψ2) when considered on the interval [0, 1]. It is well known that spec(ψ1, ψ2) and specDir(ψ1, ψ2) are discrete. Theorem Assume that w is a weight such that, for some δ > 0, w (k) = (1 + k )−δw(k) is a weight as well. Then for any bounded −δ | | subset B of 1-periodic elements in H w Hw there exist N 1 and M × ≥ ≥ 1 so that for any k N, and (ψ , ψ ) B, the set spec(ψ , ψ ) λ | | ≥ 1 2 ∈ 1 2 ∩{ ∈ 1 C λ kπ < π/2 contains exactly one isolated pair of eigenvalues | | − | } λ+, λ− and spec (ψ , ψ ) λ C λ kπ < π contains a { k k } Dir 1 2 ∩ { ∈ | | − | 2 } single Dirichlet eigenvalue µk. These eigenvalues satisfy the following estimates (i) w(2k)2 λ+ λ− 2 M; |k|≥N | k − k | ≤ (λ++λ−) (ii) P w(2k)2 k k µ 2 M. |k|≥N | 2 − k| ≤ P Furthermore spec(ψ , ψ ) λ, k N and spec (ψ , ψ ) µ 1 2 \{ k | | ≥ } Dir 1 2 \{ k | k N are contained in λ C λ < Nπ π/2 and its cardinality | | ≥ } { ∈ | | | − } is 4N 2, respectively 2N 1. − − When ψ = ψ (respectively ψ = ψ ), L(ψ , ψ ) is one of the op- 2 1 2 − 1 1 2 erators in the Lax pair for the defocusing (resp. focusing) nonlinear Schr¨odinger equation.

Contents

1 Introduction 3 1.1 Results ...... 3 1.2 Comments ...... 5 1.3 Method of proof ...... 7 1.4 Related work ...... 8

2 Periodic eigenvalues 8 2.1 Lyapunov-Schmidt decomposition ...... 8 2.2 P -equation ...... 11 2.3 Q-equation ...... 17 2.4 Estimates for α(n, z) ...... 19 2.5 Estimates for β(n, z) ...... 20 2.6 z-equation ...... 24 2.7 ζ-equation ...... 25 2.8 Proof of Theorem 1.1 ...... 26 2.9 Improvement of Theorem 1.1 for L selfadjoint ...... 30

2 3 Riesz spaces and normal form of L 32 3.1 Riesz spaces ...... 32 3.2 Normal form of L ...... 34

4 Dirichlet eigenvalues 36 4.1 Dirichlet boundary value problem ...... 36 4.2 Decomposition ...... 38 4.3 Proof of Theorem 1.2 ...... 40

A Appendix A: Spectral properties of L(ψ1, ψ2) 41

B Appendix B: Proof of Lemma 2.8 44

1 Introduction

1.1 Results Consider the Zakharov-Shabat operator (see [ZS])

1 0 d 0 ψ1 L(ψ1, ψ2) := i + 0 1 dx ψ2 0 − w w where ψ , ψ are 1-periodic elements in the weighted Sobolev space H HC 1 2 ≡ of 2-periodic functions

∞ Hw := f(x) = fˆ(k)eiπkx f < { | k kw ∞} −∞ X with 2 2 1/2 f w := (2 w(k) fˆ(k) ) k k Z | | Xk∈ and w = (w(k))k∈Z a weight, i.e. a sequence of positive numbers with w(k) 1, w( k) = w(k) ( k Z) and the following submultiplicative property ≥ − ∀ ∈ w(k) w(k j)w(j) k, j Z. ≤ − ∀ ∈ 3 As an example of such a weight we mention the Sobolev weights sN N ≡ (s (k)) Z, s (k) := k , where, for convenience, n k∈ N h i k := 1 + k , h i | | or more generally, the Abel-Sobolev weight w (w (k)) Z a,b ≡ a,b k∈ w (k) := k aeb|k| (a 0; b 0). a,b h i ≥ ≥ wa,b ˆ iπkx An element ψ H is a complex valued function f(x) = k∈Z f(k)e , ∈ b which admits an analytic extension f(x + iy) to the strip y < π such that b b a | P| a 1 f(x + i ) and f(x i ) are both in the Sobolev space HC H ( ; C). De- π − π ≡ S note by spec(ψ1, ψ2) the periodic spectrum of L(ψ1, ψ2) when considered on the interval [0, 2] and by specDir(ψ1, ψ2) the Dirichlet spectrum of L(ψ1, ψ2) when considered on [0, 1]. It is well known that both, spec(ψ1, ψ2) and specDir(ψ1, ψ2) are discrete. The main purpose of this paper is to study the asymptotics of the large (in absolute value) eigenvalues in spec(ψ1, ψ2) and specDir(ψ1, ψ2) for 1-periodic w functions ψ1, ψ2 in H . To formulate our first result we need to introduce some more notation: we say that w is a δ-weight for δ > 0 if w (k) := k −δw(k) ∗ h i is a weight as well. Notice that the Abel-Sobolev weight wa,b is a δ-weight iff 0 < δ a. Let ≤ 1 1 δ := δ = inf(δ, ) . ∗ ∧ 2 2 Further let 1/2 ρ := (ψˆ (2n) + β+(n))(ψˆ ( 2n) + β−(n)) n 2 0 1 − 0 with an arbitrary, but fixed choice of the square root and ψˆ (k + n) ψˆ ( k j) β+(n) := 2 1 − − ψˆ (j + n) 0 (k n)π (j n)π 2 kX,j6=n − − ψˆ ( k n) ψˆ (k + j) β−(n) := 1 − − 2 ψˆ ( j n) 0 (k n)π (j n)π 1 − − kX,j6=n − −

The ﬁrst result concerns the periodic eigenvalues (cf. section 2).

4 Theorem 1.1 Let M 1, δ > 0 and w a δ-weight. Then there exist constants 1 C < and ≥1 N < so that the following statements hold: ≤ ∞ ≤ ∞ w For any n N and any 1-periodic functions ψ1, ψ2 H with ψj w M, | | ≥ C π ∈ k k ≤ the set spec(ψ1, ψ2) λ λ nπ < 2 contains exactly one isolated pair of eigenvalues ∩λ+{, λ∈− . These| | − eigenvalues| } satisfy { k k } (i) w(2n)2 λ+ λ− 2 C; |n|≥N | n − n | ≤ 3δ∗ 2 + − 2 (ii)P n w(2n) min (λ λ ) 2ρn C; |n|≥N h i | n − n | ≤ C π (iii)Pspec(ψ1, ψ2) λn n N is contained in λ λ < Nπ 2 and its cardinality\{is 4N| | | 2≥. } { ∈ | | | − } −

Theorem 1.2 Let M 1, δ > 0 and w be a δ-weight. Then there exist ≥ constants 1 C < and N N 0 < (with N given by Theorem 1.1) so that the following≤ statements∞ hold:≤ ∞ 0 w For any n N and any 1-periodic functions ψ1, ψ2 H with ψj w M, the set| |spec≥ (ψ , ψ ) λ C λ nπ < π ∈contains exactlyk k one≤ Dir 1 2 ∩ { ∈ | | − | 2 } eigenvalue denoted by µn. These eigenvalues satisfy: 2 + 2 (i) 0 w(2n) µ λ C; |n|≥N | n − n | ≤ (ii) spec (ψ , ψ ) µ n N 0 is contained in λ C λ < N 0π π P Dir 1 2 \{ n | | | ≥ } { ∈ | | | − 2 } and its cardinality is 2N 0 1. − Statement (iii) in Theorem 1.1 and (ii) in Theorem 1.2 are obtained in a standard way. For the convenience of the reader we prove it in Appendix A. In section 3, we consider the Riesz spaces En, i.e. the images of the Riesz projectors associated to L(ψ1, ψ2) for a small circle around nπ with n suf- + | | ﬁciently large. We analyze the restriction of L λn to En and study the asymptotic properties of eigenfunctions in E for−n . n | | → ∞

1.2 Comments

Operator L(ψ1, ψ2): The Zakharov-Shabat operator occurs in the Lax pair dM representation dt = [M, A] of the focusing (NLS−) and defocusing (NLS+) nonlinear Schr¨odinger equation i∂ ϕ = ∂2 2 ϕ 2ϕ, t − x | | 5 M := L(ϕ, ϕ) ; M := L(ϕ, ϕ) + − − (whereas the operators A are rather complicated third order operators, given in [FT]). One can show that spec L(ϕ, ϕ) respectively spec L(ϕ, ϕ) − is a complete set of conserved quantities for NLS+ respectively NLS−. We mention that L(ψ1, ψ2) is unitarily equivalent to the AKNS operator (see [AKNS], [MA])

0 1 d q p LAKNS := − + − 1 0 dx p q where ψ := q + ip ; ψ = q ip. 1 − 2 − − Hence the selfadjoint operator M+ corresponds to an operator LAKNS with the functions q, p being real valued. Selfadjoint case: We emphasize that Theorem 1.1 and Theorem 1.2 do not require L(ψ1, ψ2) be selfadjoint. However, in the selfadjoint case, the decay rate of the asymptotics in Theorem 1.1 (ii) can be improved from 3δ∗ to 4δ∗,

4δ∗ 2 + − 2 n w(2n) min (λn λn ) 2ρn M. h i | − | ≤ |nX|≥N (This is proved in section 2.9). L2-case: Theorem 1.1 (i) and Theorem 1.2 (i) no longer hold for H w = L2 (i.e. w(k) = 1 k Z) as the number N in Theorem 1.1 cannot be chosen uniformly for 1-p∀ erio∈ dc functions ψ , ψ L2 in a L2-bounded set. This 1 2 ∈ can be easily deduced from the examples considered by Li - McLaughlin [LM] : Assume that Theorem 1.1 (i) holds for L2. Given M > 0, choose 2 N as in Theorem 1.1 and ψ1, ψ2 L with ψj ψj L2 = M. Deﬁne 2πikx −2πikx ∈ k k ≡ k k (ψ1,k, ψ2,k) = (e ψ1, e ψ2) (k Z). Then ψj,k L2 = ψj L2 ( k) and, for n N, k 0 ∈ k k k k ∀ ≥ ≥ λn+k(ψ1,k, ψ2,k) = λn (ψ1, ψ2) + kπ which leads for appropriate choices of ψ1, ψ2 to a contradiction. For L selfadjoint, a local version of Theorem 1.1 and Theorem 1.2 have been established, using diﬀerent methods, in [GG]. Most likely, the analysis presented in this paper can be used to obtain a local version of Theorem 1.1 (i) and Theo- rem 1.2 (i) for L arbitrary.

6 Submultiplicative property of weights: Notice that the requirement of a weight to be submultiplicative excludes weights of super-exponential growth exp(a k α) with α > 1. Most likely, the conclusions of Theorem 1.1 and Theorem| 1.2| do not hold for such weights (cf. [KM] for the case of Schrödinger operators). Boundary conditions: Similarly as in [KM] the method for proving Theo- rem 1.2 can be applied to a whole class of boundary conditions (cf. section 4 in [KM] where this class has been described for the Schrödinger operator d2 2 + V ). − dx Smoothness vs. decay of gap length: For selfadjoint Zakharov-Shabat operators L(ψ, ψ), Theorem 1.1 has a partial inverse. In this case, the eigenvalues (λn )n∈Z = spec L(ψ, ψ) are real and can be ordered such that . . . λ+ < λ− λ+ < λ− . . . ; λ = nπ + o(1). ≤ n−1 n ≤ n n+1 ≤ n Given a weight w and K 0, denote by w the weight w (n) := n Kw(n). ≥ K K h i Proposition 1.3 Let w be a δ-weight for some δ > 0, K 0 and ϕ H w. ≥ ∈ Then ϕ HwK iff ∈ w (2n)2 λ+ λ− 2 < K | n − n | ∞ n∈Z X where λ λ(ϕ, ϕ). n ≡ n In the non selfadjoint case, the smoothness is not characterized by properties of the periodic spectrum alone (cf. [ST] for an analysis in the case of Schrödinger operators).

1.3 Method of proof + − Typically, asymptotic estimates on the gap’s lengths (λ λ ) Z of k − k k∈ spec(L(ψ1, ψ2)) are obtained from asymptotic expansions of the eigenvalues c−1 λk = kπ + k + . . . (cf. e.g. [Ma]). This approach, however does not allow to obtain the results of Theorem 1.1 and Theorem 1.2 for weights with exponential decay such as the Abel-Sobolev weight. The new feature in the proof of our results is to use as in [KM] a Lyapunov-Schmidt type decomposition described in detail in section 2.1.

7 1.4 Related work Similar results as the ones presented here for the Zakharov-Shabat operator L(ψ1, ψ2) have been obtained previously for the Schrödinger operator d2 2 + V in [KM]. In this paper we document that the same methods, with − dx adjustments, can be applied to L. At first sight this is astonishing, as, un- like in the case of the Schrödinger operator, the distance between adjacent + − + − pairs of eigenvalues (λn , λn ) and (λn+1, λn+1) does not get unbounded for n , a fact which was used in an essential way in [KM]. We explain in |section| → ∞2.1 how this problem for L can be overcome. A weaker version of Theorem 1.1 has been reported in [GKM] (cf. also [GK]). For Sobolev weights, the asymptotics of the eigenvalues λn and hence of the gap length γ := λ+ λ− have been obtained in the selfadjoint case by n n − n Marchenko [Ma] (cf. also [GG], [Gre], [Mis], [LS]). In the non selfadjoint case only a few results have been known so far (see [LM], [Ta1], [Ta2]).

2 Periodic eigenvalues

2.1 Lyapunov-Schmidt decomposition Consider the Zakharov-Shabat operator

1 0 d 0 ψ1 L(ψ1, ψ2) := i + 0 1 dx ψ2 0 − w where ψ1 and ψ2 are in H . For ψ1 = ψ2 = 0, the periodic eigenvalues are + − Z + − given by λk , λk k with λk = λk = kπ and an orthonormal basis of corresponding{ eigenfunctions| ∈ } in L2[0, 2] L2[0, 2] are given by × 1 0 1 1 e+(x) = eikπx, e−(x) = e−ikπx. (2.1) k √ 1 k √ 0 2 2 0 ψ Considering the multiplication operator 1 as a perturbation of the ψ 0 2 1 0 Dirac operator i d we will see that for k suﬃciently large L has a 0 1 dx − 8 pair of eigenvalues near kπ, isolated from the remaining part of the spectrum of L. Our aim is to obtain an estimate for the distance between the two eigenvalues and to compare the eigenvalues and corresponding eigenfunctions (or root vectors) with the corresponding ones for ψ1 = ψ2 = 0. We express the eigenvalue equation

LF = λF (2.2) in the basis e+, e−(k Z) deﬁned in (2.1): Given F in the Sobolev space H 1, k k ∈ write ˆ + ˆ − F (x) = F2(k)ek (x) + F1( k)ek (x) (2.3) Z − Xk∈ and ˆ ikπx ˆ ikπx ψ1(x) = ψ1(k)e ; ψ2(x) = ψ2(k)e . (2.4) Z Z Xk∈ Xk∈ Substituting (2.3) - (2.4) into (2.2) leads to

ˆ + ˆ − LF (x) = kπ F2(k)ek (x) + F1( k)ek (x) Z − Xk∈ (2.5) ˆ ˆ − ˆ ˆ + + ψ1( k j)F2(j)ek (x) + ψ2(k + j)F1( j)ek (x). Z − − − kX,j∈

Hence λ is a periodic eigenvalue of L(ψ1, ψ2), when considered on the interval 2 2 [0, 2], iﬀ there exists (Fˆ1, Fˆ2) ` ` with (Fˆ1, Fˆ2) = (0, 0) such that, for all k Z, ∈ × 6 ∈

(kπ λ)Fˆ2(k) + ψˆ2(k + j)Fˆ1( j) = 0 (2.6) − Z − Xj∈ ˆ (kπ λ)Fˆ1( k) + ψ1( k j)Fˆ2(j) = 0. (2.7) − − Z − − Xj∈ Here `2 `2(Z; C) denotes the Hilbert space of complex valued `2-sequences ≡ (a(k))k∈Z. In order to solve equations (2.6) - (2.7) we consider a Lyapunov- Schmidt type decomposition. For n Z ﬁxed, we look for eigenvalues near π ∈ nπ, λ = nπ + z, with z 2 . The linear system (2.6) - (2.7) is then decom- posed into a two dimensional| | ≤ system consisting of (2.6) - (2.7) with k = n, referred to as the -equation, and an inﬁnite dimensional system consisting of (2.6) - (2.7) withQ k Z n , referred to as the -equation. ∈ \{ } P 9 First we introduce some more notation. For K Z and a weight w denote by `2 (K) the complex Hilbert space `2 (K) `2∈(K, C), w w ≡ w `2 (K) := (a(k)) a < w { k∈K | k kw ∞} 1/2 2 where a = (a, a)w and, for a, b ` , k kw ∈ w 2 (a, b)w := w(k) a(k)b(k). kX∈K Most frequently, we will use for K the set Z or Z n Z n . If necessary \ 2 ≡ \{ } for clarity, we write aK for a sequence (a(k))k∈K `w(K). 2 2 ∈ For a linear operator A : `w1 (K1) `w2 (K2) we denote by A(k, j) its matrix elements, → (Aa)(k) := A(k, j)a(j) (k K ). ∈ 2 jX∈K1 Further we introduce the shift operator S and an involution operator J S : `2(Z `2(Z), (Sa)(k) := a(k + 1) k Z. → ∀ ∈ J : `2(Z `2(Z), ( a)(k) := a( k) k Z. → J − ∀ ∈ 2 2 The restriction of S to `w(K) with values in `Snw(K) is again denoted by S and Sn := S . . . S denotes the n0th iterate of S. Notice that ◦ ◦ n 2 2 2 2 S a `2 (K) = w(k + n) a(k + n) a `2 (Z). k k Snw | | ≤ k k w kX∈K For (Fˆ , Fˆ ) `2 `2, write 2 1 ∈ × F F Fˆ2 = (x , F˘2), x := Fˆ2(n); F˘2 := (Fˆ2(k))k∈Z\n F F Fˆ = (y , JF˘ ), y := Fˆ ( n); F˘ := (Fˆ (k)) Z . 1 1 1 − 1 1 k∈ \n Using the above introduced notation, the equations (2.6) - (2.7) read as follows:

zxF + ψˆ (2n)yF + Snψˆ , JF˘ = 0 (2.8) − 2 h 2 1i ψˆ ( 2n)xF zyF + SnJψˆ , F˘ = 0 (2.9) 1 − − h 1 2i and F n ˆ y (S ψ2)Z\n F˘2 F n + (An z) = 0. (2.10) x (S Jψˆ )Z − JF˘ 1 \n 1 10 The equations (2.8) - (2.9) together form the -equation and (2.10) is the -equation. The operator A is given by Q P n ˆ ((k n)πδkj)k,j∈Z\n ψ2(k + j) A = − k,j∈Z\n n  ˆ  (Jψ1(k + j) ((k n)πδkj)k,j∈Z\n k,j∈Z\n −     and , , Z is defined by (no complex conjugation) h· ·i ≡ h· ·i \n a c n , n := (a (k)c (k) + b (k)d (k)) . b d n n n n h n n i Z kX∈ \n For ψ = ψ = 0 and z π , the operator (z A ) is invertible as (kπ 1 2 | | ≤ 2 − n − (nπ z)) = 0 for k = n. By a perturbation argument we will show that (z −A ) can6 be inverted6 for z π and n sufficiently large which then − n | | ≤ 2 | | allows to solve the -equation (2.10) for (F˘ , JF˘ ) for any xF , yF C. This P 2 1 ∈ solution is substituted into (2.8) - (2.9) which leads to a homogeneous linear system of two equations for xF and yF with coefficients which depend on the parameter z. Hence λ = nπ + z is a periodic eigenvalue of L(ψ1, ψ2) iff the corresponding determinant is equal to 0. The nature of the latter equation + − allows to obtain asymptotics for the difference λn λn without having to + − − compute the asymptotics of λn and λn (cf. section 2.6 - 2.7).

2.2 P -equation

Let us ﬁrst introduce some more notation. Denote by ∆n the diagonal part of An Dn 0 ∆n := ; Dn := ((k n)πδkj)k,j∈Z\n 0 Dn − and set B := A ∆ . n n − n Notice that for z π , (z ∆ )−1 is invertible. Hence we may introduce | | ≤ 2 − n (2) −1 0 Rn Tn Tn,z := Bn(z ∆n) = (1) (2.11) ≡ − Rn 0 !

11 (j) (j) 2 2 where Rn Rn,z : ` (Z n) ` (Z n) are deﬁned by ≡ \ → \ R(1)(a) := J(ψˆ (z D )−1)a; R(2)(a) := ψˆ J(z D )−1a. (2.12) n 1 ∗ − n n 2 ∗ − n (1) (2) Rn and Rn have the following matrix representations

ψˆ ( k j) ψˆ (k + j) R(1)(k, j) := 1 − − ; R(2)(k, j) := 2 (k, j Z n). (2.13) n z (j n)π n z (j n)π ∈ \ − − − − Formally, for any xF , yF C, the -equation (2.10) can be solved ∈ P F˘ yF Snψˆ 2 = (z A )−1 2 JF˘ − n xF SnJψˆ 1 1 with (z A )−1 = (z ∆ )−1(Id T )−1. (2.14) − n − n − n To justify the formal considerations above it is to show that (Id Tn) is 2 − 2 invertible. Unfortunately, the norm Tn of Tn in (`Snw) (with `Snw 2 Z C2 k k L ≡ `Snw( n; )) does not become small as n . However, it turns out \ | | → ∞ 2 that, assuming an additional condition on the weight, the norm of Tn is small for n . The invertibility of (Id T ) then follows from the identity | | → ∞ − n Id = (Id T ) (Id + T )(Id T 2)−1. (2.15) − n ◦ n − n w 2 2 2 Given ϕ H , denote by Φn the operator in (` ) (with ` ` (Z; C)) deﬁned by∈(n Z; a `2(Z; C)) L ≡ ∈ ∈ ϕˆ(k + j) (Φna)k) := a(j) ( k Z), Z n j ∀ ∈ Xj∈ h − i where k = 1 + k . h i | | Recall that a weight w is called a δ-weight (δ 0) if w (k) := k −δw(k) ≥ −δ h i is a weight. For convenience we denote the weight w−δ by w∗. The two key lemmas for proving that lim T 2 = 0 are the following ones: n→∞ k n k

1 Z Lemma 2.1 Let w be a δ-weight with 0 δ < 2 and n . Then there exists C = C(δ) such that ≤ ∈

2 2 Φn L(` ;` n ) C ϕ w. k k S−nw∗ S w ≤ k k

12 2 2 Proof For a ` −n and b ` n , ∈ S w∗ ∈ S w

(b, Φ a) n | n S w| ≤ w(k + n) b(k) w (j n) a(j) w(k + j) ϕˆ(k + j) ≤ | | ∗ − | | | |· Xj,k w(k + n) 4 . w (j n)w(k + j) n j ∗ − h − i Using that w is submultiplicative, one gets w(k + n) w(n j) 1 1 − = j n δ (4 n j + )δ 2 n j + δ. w (j n)w(k + j) ≤ w (j n) h − i ≤ | − 2| ≤ | − 2| ∗ − ∗ − and hence, by the Cauchy-Schwartz inequality

(b, Φ a) n | n S w| ≤ 1/2 4 ϕˆ(k + j) 2w(k + j)2 n −n b S w a S w∗ | | 2(1−δ) ≤ k k k k n j ! Xk,j h − i C b n a −n ϕ ≤ k kS wk kS w∗ k kw 1/2 4 1 with C C(δ) := 2 2 < as δ < . ≡ k hki − δ ∞ 2 P Lemma 2.2 Let δ 0, w be a δ-weight and n Z. Then there exists C > 0, ≥ ∈ independent of δ, such that

ϕ w∗ Φ 2 2 C k k n L(` n ;` −n ) δ∧1 k k S w S w∗ ≤ n h i where as usual δ 1 = min(1, δ). ∧

2 2 Proof For a ` n and b ` −n , ∈ S w ∈ S w∗

13 As w∗ submultiplicative and symmetric, w (k n) w (k + j)w (j + n) ∗ − ≤ ∗ ∗ which leads to (use deﬁnition of w∗)

1/2 4 4 (b, Φ a) −n b −n a n ϕˆ . | n S w∗ | ≤ k kS w∗ k kS wk kw∗ j + n 2δ j n 2 j ! X h i h − i The claimed estimate then follows from the following elementary estimate

1/2 1 1 1 C j + n 2δ j n 2 ≤ n δ∧1 j ! X h i h − i h i for some C, independent of δ.

As an application of Lemma 2.1 and 2.2 we obtain estimates for the norms (j) 2 of Rn , Tn and Tn . By deﬁnition

2 (2) (2) (1) 2 0 Rn Rn Rn 0 Tn = (1) = (1) (2) (2.16) Rn 0 ! 0 Rn Rn ! and it is useful to introduce the operators

(2) (1) (1) (2) Pn := Rn Rn ; Qn := Rn Rn . (2.17) 2 2 Z C 2 To make notation easier we write `Snw for both, `Snw( n; ) and `Snw (Z n; C2). \ \

Corollary 2.3 Let δ 0, M 1 and w be a δ-weight. Then, for any 1- ≥ w≥ periodic functions ψ1, ψ2 H with ψj w M (j = 1, 2), the following statements hold: ∈ k k ≤

1 Z (i) If 0 δ < 2 , there exists C C(δ) > 0 so that for 1 j 2, n , and≤z π , ≡ ≤ ≤ ∈ | | ≤ 2 (j) 2 2 Rn L(` ;` n ) CM; k k S−nw∗ S w ≤ 2 2 Tn L(` ;` n ) CM. k k S−nw∗ S w ≤

14 (ii) If δ 0, there exists C > 0 such that for 1 j 2, n Z, and z π , ≥ ≤ ≤ ∈ | | ≤ 2

(j) CM R 2 2 n L(` n ,` −n ) δ∧1 k k S w S w∗ ≤ n h i CM T 2 2 . n L(` n ,` −n ) δ∧1 k k S w S w∗ ≤ n h i (iii) If 0 δ < 1 , then there exists C C(δ) so that for n Z and z π , ≤ 2 ≡ ∈ | | ≤ 2 CM 2 CM 2 Pn L(`2 ) ; Qn L(`2 ) ; k k Snw ≤ n δ k k Snw ≤ n δ h i h i CM 2 CM 2 P 2 ; Q 2 . n L(` −n ) δ n L(` −n ) δ k k S w∗ ≤ n k k S w∗ ≤ n h i h i

(j) Proof The claimed estimates for Rn (j = 1, 2) follow from Lemma 2.1 and (2) 0 Rn Lemma 2.2. As Tn = (1) , these estimates then imply the ones Rn 0 ! for Tn. The estimates in (iii) are obtained by combining the estimates in (i) (j) and (ii) for Rn .

Under the assumptions of Corollary 2.3 deﬁne, for 0 < δ < 1 and M 1, 2 ≥ N N (δ, M, w) := max 1, (2CM 2)1/δ (2.18) 0 ≡ 0 with C given as in Corollary 2.3 (iii).

Proposition 2.4 Let 0 < δ < 1 , M 1 and w be a δ-weight. Then, for any 2 ≥ 1-periodic functions ψ , ψ Hw with ψ M, n N and z π/2, 1 2 ∈ k jkw ≤ | | ≥ 0 | | ≤

(i) 1 1 Pn L(`2 ) ; Qn L(`2 ) ; k k Snw ≤ 2 k k Snw ≤ 2 (ii) (Id P ) and (Id Q ) are invertible and − n − n −1 −1 (Id Pn) L(`2 ) 2; (Id Qn) L(`2 ) 2. k − k Snw ≤ k − k Snw ≤

15 (iii) Id T 2 is invertible and − n 2 1 2 −1 Tn L(`2 ) ; (Id Tn ) L(`2 ) 2. k k Snw ≤ 2 k − k Snw ≤ (iv) Statements (i) - (iii) remain true if one replaces the weight Snw by −n S w∗.

1 Proof (i) By Corollary 2.3 (iii) Pn satisﬁes the estimate (as 0 < δ < 2 ) CM 2 Pn L(`2 ) . k k Snw ≤ n δ h i Hence for n N0 | | ≥ CM 2 1 P 2 . n L(` n ) δ k k S w ≤ N0 ≤ 2 1 h i Similarly, one obtains Qn 2 . L(`Snw) 2 (ii) follows immediatelykfromk (i) and≤(iii) follows from (i) - (ii) and the identity

2 Pn 0 −n Tn = . Finally, statements (i) - (iii) for the weight S w∗ are 0 Qn provedin a similar way as for Snw. Summarizing the results obtained in this section, we obtain, with k · k ≡ L(`2 (Z\n;C2)): k · k Snw 1 Corollary 2.5 Let 0 < δ < 2 , M 1 and w be a δ-weight. Then there exists ≥ w C > 0 such that, for any 1-periodic functions ψ1, ψ2 H with ψj w M (j = 1, 2), n N and z π/2 ∈ k k ≤ | | ≥ 0 | | ≤ (i) T C; k nk ≤ 2 2 −1 (ii) (Id T ) is invertible in (` n (Z n; C )) and (Id T ) C; − n L S w \ k − n k ≤ 2 2 −1 (iii) (z A ) is invertible in (` n (Z n; C )) and (z A ) C. − n L S w \ k − n k ≤ (2) 0 Rn Proof (i) Recall that T = (1) . By standard convolution es- Rn 0 ! timates, there exists an absolute constant C > 0 so that for n Z and ∈ z π/2, ψj w M | | ≤ k k ≤ T CM. k nk ≤ Therefore (ii) and (iii) follow immediately from (2.14) - (2.15) and Proposi- tion 2.4.

16 2.3 Q-equation

Using the notations introduced in section 2.2, we have for n N0 and z π/2 | | ≥ | | ≤ −1 −1 −1 (2) −1 −1 (z Dn) (Id Pn) (z Dn) Rn (Id Qn) (z An) = − −1 (1) − −1 − −1 − −1 . − (z D ) Rn (Id P ) (z D ) (Id Q ) − n − n − n − n ! Hence the P -equation (2.10) leads to the following formulas

F˘ = yF (z D )−1(Id P )−1Snψˆ (2.19) 2 − n − n 2 + xF (z D )−1R(2)(Id Q )−1SnJψˆ − n n − n 1 JF˘ = yF (z D )−1R(1)(Id P )−1Snψˆ (2.20) 1 − n n − n 2 + xF (z D )−1(Id Q )−1SnJψˆ . − n − n 1 These solutions are substituted into the Q-equation (2.8) - (2.9) to obtain for z π , n n the following homogeneous system | | ≤ 2 | | ≥ 0 ( z + α+(n, z))xF + (ψˆ (2n) + β+(n, z))yF = 0 (2.21) − 2 (ψˆ ( 2n) + β−(n, z))xF + ( z + α−(n, z))yF = 0, (2.22) 1 − − where

α+(n, z) := Snψˆ , (z D )−1(Id Q )−1SnJψˆ (2.23) h 2 − n − n 1i β+(n, z) := Snψˆ , (z D )−1R(1)(Id P )−1Snψˆ (2.24) h 2 − n n − n 2i α−(n, z) := SnJψˆ , (z D )−1(Id P )−1Snψˆ (2.25) h 1 − n − n 2i β−(n, z) := SnJψˆ , (z D )−1R(2)(Id Q )−1SnJψˆ . (2.26) h 1 − n n − n 1i π (j) Notice that α (n, z) and β (n, z) are analytic for z < 2 as Rn , Pn and Qn are analytic for z < π . An important simpliﬁcation| | of the equations (2.21) | | 2 - (2.22) results from the following observation

Lemma 2.6 For z π and n N , | | ≤ 2 | | ≥ 0 α+(n, z) = α−(n, z).

17 Proof In view of (2.23) and (2.25) it is to show that

(z D )−1(Id Q )−1 = (Id P )−1 t (z D )−1 (2.27) − n − n − n − n where At denotes the transpose of A,

(At)(k, j) := A(j, k) (no complex conjugation).

The equation (2.27) can be reformulated,

((Id Q )(z D ))−1 = (z D )(Id P t) −1 . − n − n − n − n which holds iﬀ Q (z D ) = (P (z D ))t. (2.28) n − n n − n The identity (2.28) follows easily from

(Q (z D ))(j, k) = (R(1)R(2))(j, k)(z (k n)π) n − n n n − − ψˆ ( j `) = 1 − − ψˆ (` + k) z (` n)π · 2 X` − − and

(P (z D ))t(j, k) = (P (z D ))(k, j) n − n n − n = (R(2)R(1))(k, j)(z (j n)π) n n − − ψˆ (k + `) = 2 ψˆ ( ` j). z (` n)π 1 − − X` − −

In view of Lemma 2.6 we write

α(n, z) := α+(n, z) (= α−(n, z)). (2.29)

In subsequent sections we estimate the coeﬃcients α(n, z), β+(n, z) and β−(n, z).

18 2.4 Estimates for α(n, z) 1 Lemma 2.7 Let 0 < δ 2 , M 1 and w be a δ-weight. Then, for any ≤ w ≥ 1-periodic functions ψ1, ψ2 H with ψj w M, n N0 (N0 N0(δ, M) given by (2.18)) and z ∈π k k ≤ | | ≥ ≡ | | ≤ 2 4M 2 α(n, z) . | | ≤ n 2δ h i

n −1 −1 n Proof Write α(n, z) = S ψˆ2, (z Dn) a with a := (Id Qn) S Jψˆ1 2 h − i − ∈ `Snw. By Proposition 2.4,

a n 2 ψˆ 2M. k kS w ≤ k 1kw ≤ Hence n 2δ n 2δ α(n, z) h i ψˆ (k + n) a(k) h i | | ≤ n k | 2 || | Xk6=n h − i n 2δ h i ψˆ (k + n) a(k) ≤ n | 2 || | |k+Xn|<|n| h i n 2δ + h i w(k + n) ψˆ (k + n) w(k + n) a(k) w(k + n)2 | 2 | | | |k+Xn|≥|n| 2 ψˆ a n ≤ k 2kwk kS w δ δ where we used that 2δ 1 and w(k + n) = w∗(k + n) k + n n for k + n n . ≤ h i ≥ h i | | ≥ | |

2.5 Estimates for β(n, z) In this section we provide estimates for β(n, z). The β(n, z) - they turn out to be quite small - determine the asymptotics of the sequence of gap lengths given in Theorem 1.1. As β+(n, z) (cf (2.24)) and β−(n, z) (cf. (2.26)) are analyzed in a similar fashion we focus on the estimate for β+(n, z). Writing (Id P )−1 = ∞ P k we obtain for β+(n, z) the following convergent series − n k=0 n P ∞ + β (n, z) = βk(n, z). (2.30) Xk=0 19 where β (n, z) := Snψˆ , (z D )−1R(1)P kSnψˆ . (2.31) k h 2 − n n n 2i 1 2 The convergence of series (2.30) follows from Pn L(` ) 2 ( n N0, k k Snw ≤ | | ≥ Proposition 2.4). We begin by analyzing βk(n, z). (1) With Rn deﬁned by (cf (2.12))

(Jψˆ )(j + `)a(`) (R(1)a)(j) = J(ψˆ (z D )−1a)(j) = 1 n 1 ∗ − n z (` n)π X`6=n − − 1 and inf π z (` n)π ` n (for any ` = n) we get |z|≤ 2 | − − | ≥ 2 h − i 6 Jψˆ (j + `) (R(1)a)(j) 2 | 1 | a(`) | n | ≤ ` n | | X` h − i which leads to

ψˆ (n + j) Jψˆ (j + `) β (n, z) 4 | 2 | | 1 | (S−nP kSnψˆ )(` + n) . (2.32) | k | ≤ j n ` n | n 2 | j X h − i X` h − i Given three nonnegative sequences (i.e. sequences of nonnegative numbers), a, b, d in `2(Z) we deﬁne, for any n Z, the sequence Ψ Ψ (a, b, d) by ∈ n ≡ n a(k + j) b(j + `) Ψ (k + n) := d(` + n). n j n ` n j X h − i X` h − i 2 Then Ψn is a nonnegative sequence in ` (Z) and can be used to rewrite (2.32): Introduce, for n N and z π , | | ≥ 0 | | ≤ 2 η := 4Ψ ( ψˆ , Jψˆ , ψˆ ) (2.33) n,0 n | 2| | 1| | 2| and, for k 0, ≥ η := 4Ψ ( ψˆ , Jψˆ , η ). (2.34) n,k+1 n | 2| | 1| n,k 2 where, for any a = (a(j)) Z ` (Z), we denote by a the sequence ( a(j) ) Z. j∈ ∈ | | | | j∈

20 As for any z π , | | ≤ 2 (S−nP Sna)(k + n) = (P Sna)(k) | n | | n | = (R(2)R(1)Sna)(k) | n n | = (ψˆ (J(z D )−1(R(1)S−na)))(k) | 2 ∗ − n n | 1 Jψˆ (j + `) 4 ψˆ (k + j) | 1 | a(` + n) ≤ | 2 | j n ` n | | j X h − i X` h − i = 4Ψ ( ψˆ , Jψˆ , a ) n | 2| | 1| | | it follows, by an induction argument, from (2.32) and −n k n ˆ −n n −n k−1 n S Pn S ψ2 = (S PnS )(S Pn S ψ2) that, for any k 0, ≥ sup βk(n, z) ηn,k(2n). (2.35) π |z|≤ 2 | | ≤

To estimate ηn,k(2n), we need the following auxilary lemma concerning the operator Ψn. For δ > 0 and w be a δ-weight, deﬁne δ = δ 1/2. ∗ ∧

Lemma 2.8 Let w a δ-weight, and, for any n Z, dn a positive sequence in 2 ∈ `w so that n αd (j) d(j) n, j Z h i n ≤ ∀ ∈ 2 for some α 0 and some positive sequence d in `w. Then there exist C Cδ∗ , only depending≥ on δ , and e `2 so that for any positive sequences a, b≡ `2 , ∗ ∈ w ∈ w (i)

2(2δ∗+α) 2 2 n w(2n) (4Ψn(a, b, dn)(2n)) Zh i Xn∈ C a b d ; ≤ k kwk kwk kw (ii) for any n, j Z, ∈ n α+δ∗ 4Ψ (a, b, d )(j) e(j) ; h i n n ≤ e C a b d . k kw ≤ k kwk kwk kw

21 Proof Cf. Appendix B.

From Lemma 2.8 we obtain, in view of the deﬁnition (2.33) - (2.34) and the estimate (2.35) the following

Corollary 2.9 Let M 1 and w be a δ-weight. Then for any 1-periodic functions ψ , ψ Hw with≥ ψ M (j = 1, 2) 1 2 ∈ k jkw ≤ (i) for k 0 ≥

2(2+k)δ∗ 2 2 k+1 2k+3 n w(2n) sup βk(n, z) C M π h i |z|≤ 2 | | ≤ |nX|≥N0 where 1 C C < is given by Lemma 2.8 ≤ ≡ δ ∞ (ii) n 6δ∗ w(2n)2 sup β˜(n, z) 2 C0 π h i |z|≤ 2 | | ≤ |nX|≥N0 ˜ 0 where β(n, z) := k≥1 βk(n, z) and 1 C < is a constant depending only on M and δ. ≤ ∞ P

Proof We apply Lemma 2.8 to each of the βk’s in an inductive fashion to obtain (i). Statement (ii) then follows from (i) by the Cauchy-Schwartz inequality.

+ + To simplify further the asymptotics of β write β0(n, z) β0 (n, z) = β0 (n)+ + ≡ zβ#(n, z) where

1 β0 (n) := β0 (n, 0); β#(n, z) := ∂zβ0 (n, tz)dt. Z0 As z β(n, z) are analytic functions in z < π/2 , one deduces by 7→ # {| | } Cauchy’s formula

4 sup β#(n, z) sup β0 (n, z) . (2.36) |z|≤π/4 | | ≤ π |z|≤π/2 | |

Summarizing our results of this section gives the following

22 Proposition 2.10 Let δ > 0, M 1, 1 A < and w be a δ-weight. Then there exists C > 0 so that for≥ any ≤1-periodic∞functions ψ , ψ Hw 1 2 ∈ with ψ M (j = 1, 2), k jkw ≤ (i) n 4δ∗ w(2n)2 sup β(n, z) 2 C h i |z|≤π/4 | | ≤ |nX|≥N0 (ii) 6δ∗ 2 2 n w(2n) sup β (n, z) β0 (n) C. h i |z|≤A/hniδ∗ | − | ≤ |nX|≥N0

˜ Proof Notice that β (n, z) = β0 (n) + zβ#(n, z) + β (n, z) and hence (i) is a consequence of Corollary 2.9 and formula (2.36). Statement (ii) is proved in the same fashion. As the supremum of β (n, z) β0 (n) is only taken over z A , the asymptotics of zβ(n, |z) can be−improv|ed by δ to obtain | | ≤ hniδ∗ # ∗ from formula (2.36)

6δ∗ 2 2 n w(2n) sup zβ#(n, z) C. h i |z|≤A/hniδ∗ | | ≤ |nX|≥N0

2.6 z-equation In view of (2.21) - (2.22), and (2.29), the Q-equation leads to the following 2 2 system × z + α(n, z) ψˆ (2n) + β+(n, z) xF 0 − 2 = . (2.37) ψˆ ( 2n) + β−(n, z) z + α(n, z) yF 0 1 − − π Given n N and z 2 , the number λ = nπ + z is a periodic eigenvalue of L iﬀ| there| ≥ exists|a|non≤ trivial solution of (2.37) (xF , yF ) C2 (0, 0), or, equivalently, iﬀ the determinant of the 2 2 matrix in (2.37)∈vanishes,\ × (z α(n, z))2 (ψˆ (2n) + β+(n, z))(ψˆ ( 2n) + β−(n, z)) = 0. (2.38) − − 2 1 −

23 Proceeding similarly as in [KM], equation (2.38) is solved in two steps: For ζ with ζ π given, consider | | ≤ 8

zn = α(n, zn) + ζ. (2.39)

Substituting a solution z(ζ) zn(ζ) of (2.39) into (2.38) leads to an equation for ζ ζ , ≡ ≡ n ζ2 ψˆ (2n) + β+(n, z(ζ) ψˆ ( 2n) + β−(n, z(ζ)) = 0 . (2.40) − 2 1 − Equation (2.39) is referred to as the z-equation and equation (2.40) as the ζ-equation . In this section we deal with the z-equation (2.39). To solve it we use the contractive mapping principle. According to Lemma 2.7 we can choose N1 N (with N given by (2.18)) so that for any 1-periodic functions ψ , ψ H≥w 0 0 1 2 ∈ with ψ M and n N k jkw ≤ | | ≥ 1 sup α(n, z) < π/8. (2.41) π |z|≤ 2 | |

The following result can be proved by the same line of arguments used in the proof of [KM, Proposition 1.6].

Proposition 2.11 Let M 1, 0 < δ 1/2 and w be a δ-weight. Then, ≥ ≤ w there exists N1 N0 so that for any 1-periodic functions ψ1, ψ2 H with ψ M, ζ ≥ π and n N , equation (2.39) has a unique∈ solution k jkw ≤ | | ≤ 8 | | ≥ 1 z = z (ζ) satisfying z < π/4. The solution depends analytically on ζ. n n | n|

2.7 ζ-equation In this section, we improve the existence of solutions of the ζ-equation (2.40)

ζ2 ψˆ (2n) + β+(n, z(ζ) ψˆ ( 2n) + β−(n, z(ζ)) = 0 − 2 1 − using Rouch´e’s Theorem. Introduce

+ − rn := ψˆ2(2n) + sup β (n, z) ψˆ1( 2n) + sup β (n, z) . (2.42) | | π | | ∨ | − | π | | |z|≤ 2 ! |z|≤ 2 !

24 Using the same line of arguments used in the proof of [KM, Proposition 1.15] one obtains the following

1 Proposition 2.12 Let M 1, 0 < δ 2 and w be a δ-weight. Then there exists N N so that,≥ for any 1-perio≤ dic functions ψ , ψ Hw with 2 ≥ 1 1 2 ∈ ψj w M and n N2, equation (2.40) has exactly two (counted with kmultiplicity)k ≤ solutions| | ≥ζ +, ζ− in . n n Drn

2.8 Proof of Theorem 1.1 In this section, Theorem 1.1 is proved.

Proof of Theorem 1.1 (i) Let zn := z(ζn ) = ζn + α(n, zn ) where ζn are the two solutions of the ζ-equation provided by Proposition 2.12 ( n N2). Then, for n N | | ≥ | | ≥ 2 + − + − d + − zn zn ζn ζn + sup α(n, z) zn zn . (2.43) π dz | − | ≤ | − | |z|≤ 4 | || − |

As N2 N1 and n N2 one has by the analyticity of z α(n, z) and (2.41) ≥ | | ≥ 7→ d 1 sup α(n, z) . π dz 2 |z|≤ 4 | | ≤ Together with ζ+ ζ− ζ+ + ζ− 2r equation (2.43) then leads to | n − n | ≤ | n | | n | ≤ n z+ z− 4r . | n − n | ≤ n By the deﬁnition (2.42) of rn, the estimates of βn in Proposition 2.10 (i) and the identity λ+ λ− = z+ z−, the latter equation implies that there exists n − n n − n C 1 such that, for any 1-periodic functions ψ , ψ Hw, ψ M, ≥ 1 2 ∈ k jkw ≤ w(2n)2 λ+ λ− 2 C. | n − n | ≤ |nX|≥N2

Towards the proof of Theorem 1.1 (ii), rewrite equation (2.40), (ζ)2 ρ2 = η(n, z(ζ)) (2.44) n − n n 25 where 1/2 ρ = (ψˆ (2n) + β+(n))(ψˆ ( 2n) + β−(n)) n 2 0 1 − 0 with an arbitrary but fixed choice of the square root and η(n, z) = ψˆ (2n)(β−(n, z) β−(n)) 2 − 0 + ψˆ ( 2n)(β+(n, z) β+(n)) (2.45) 1 − − 0 + (β−(n, z) β−(n))(β+(n, z) β+(n)). − 0 − 0 In view of the definition (2.42) and as w is assumed to be a δ-weight, we have for some constant C 1 depending on δ and M 1 ≥ C1 rn ( n N0). (2.46) ≤ n δ∗ ∀| | ≥ h i By Lemma 2.7 there exists C 1 depending on δ and M such that for 2 ≥ n N0 and z π/2 | | ≥ | | ≤ C α(n, z) 2 . (2.47) | | ≤ n δ∗ h i Let A = C1 + C2 and define

sn := sup η(n, z) . (2.48) |z|≤2A/hniδ∗ | | Notice that by Proposition 2.10 (ii), there exists C > 0 so that

n 3δ∗ w(2n)2s C. (2.49) h i n ≤ |nX|≥N2 Choose N N , depending on δ and M, so that 3 ≥ 2 n δ∗ 1 h i √s < , n N . (2.50) A n 2 ∀| | ≥ 3

Lemma 2.13 Let M 1, 0 < δ and w be a δ-weight. For 1-periodic func- ≥ tions ψ , ψ in Hw with ψ M(j = 1, 2) and n N , 1 2 k jkw ≤ | | ≥ 3 ζ+ ρ + ζ− + ρ 6√s | n − n| | n n| ≤ n or ζ+ ρ + ζ− ρ 6√s . | n − n| | n − n| ≤ n

26 Proof W.l.o.g. assume that δ 1/2 and hence δ = δ∗. By (2.44) we have for n N ≤ | | ≥ 3 (ζ ρ )(ζ + ρ ) = η(n, z(ζ)). (2.51) n − n n n n By deﬁnition, z(ζn ) = ζn +α(n, z(ζn )) and therefore from (2.46), (2.47) and Proposition 2.12, we conclude for n N , | | ≥ 3 A z(ζ) . (2.52) | n | ≤ n δ h i

From the deﬁnition of sn (see (2.48)) and (2.51) we deduce

ζ ρ ζ + ρ s . (2.53) | n − n|| n n| ≤ n + − 1/2 Thus min ζ ρn √sn and min ζ ρn sn . We distinguish two | n | ≤ | n | ≤ cases: case 1 ρ 2√s . In this case ζ ρ √s implies | n| ≤ n | n − n| ≤ n ζ + ρ ζ ρ + 2 ρ 5√s | n n| ≤ | n − n| | n| ≤ n and, similarly, ζn + ρn √sn implies ζn ρn 5√sn, thus Lemma 2.13 is proves in case| 1. | ≤ | − | ≤ case 2 ρn > 2√sn. It suﬃces to show that it is impossible to have | | max ζn ρn √sn, or max ζn + ρn √sn. To the contrary, assume that | − | ≤ | | ≤ max ζn ρn √sn. (2.54) | − | ≤ (The other case is treated in the same way.) By (2.54), ζn + ρn 2 ρn 3 | | ≥ | | − √sn > ρn , hence 2 | | ζ+ + ζ− ζ+ + ρ ζ− ρ > ρ . (2.55) | n n | ≥ | n n| − | n − n| | n| Divide (ζ+)2 (ζ−)2 = η(n, z(ζ+)) η(n, z(ζ−)) n − n n − n + − by ζn + ζn and use (2.55) and (2.52) to deduce

+ − 1 dη + − ζn ζn sup (n, z) z(ζn ) z(ζn ) . (2.56) | − | ≤ ρ δ |dz || − | | n| |z|≤A/hni

27 + − + To arrive at a contradiction we ﬁrst show that ζn ζn = 0. As z(ζn ) z(ζ−) ζ+ ζ− + sup d α(n, z) z(ζ+) −z(ζ−) , (2.41) |leads to− n | ≤ | n − n | |z|≤π/2 | dz || n − n | ( n N2) | | ≥ z(ζ+) z(ζ−) 2 ζ+ ζ− . (2.57) | n − n | ≤ | n − n | On the other hand, as z η(n, z) is analytic in z, z < π/2 , we have by Cauchy’s inequality, 7→ { | | } d n δ sup η(n, z) h i sup η(n, z) |z|≤ A |dz | ≤ A |z|≤ 2A | | hniδ hniδ (2.58) n δ h i s . ≤ A n Combining (2.56) - (2.57) with (2.50) we obtain,

2 n δ ζ+ ζ− h i s ζ+ ζ− | n − n | ≤ ρ A n| n − n | | n| n δ 1 h i √s ζ+ ζ− ζ+ ζ− ≤ A n| n − n | ≤ 2| n − n | and we conclude that ζ + = ζ− ζ . This contradicts the assumption n n ≡ n ρn > 2√sn as one can see in the following way: By the equation (2.44), | | d d d d 2ζn = dζ η(n, z(ζn)) = dz η(n, z(ζn)) dζ z(ζn). By (2.58), dz η(n, z(ζn)) δ · | | ≤ hni s and by (2.41), d z(ζ) = d (ζ + α(n, z(ζ))) 1 + 1 2, hence A n | dζ | | dζ | ≤ 2 ≤ n δ ζ h i s (2.59) | n| ≤ A n and, by (2.55), n δ ρ < 2 ζ 2h i s < √s . | n| | n| ≤ A n n where for the last inequality we used (2.50).

Proof of Theorem 1.1 (ii): Let N3 be given by (2.50). Recall that

λ+ λ− = z+ z− = ζ+ ζ− + α(n, z(ζ+)) α(n, z(ζ−)). n − n n − n n − n n − n By Lemma 2.13, for n N , | | ≥ 3 + − min (ζn ζn ) 2ρn 6√sn. | − | ≤

28 By the analyticity of α(n, z) and Lemma 2.7, for n N , | | ≥ 0 d C sup α(n, z) . |dz | ≤ n 2δ |z|≤π/4 h i Combining these two estimates, we get for n N , | | ≥ 3 + − + − min (λn λn ) 2ρn min (ζn ζn ) 2ρn | − | ≤ | − |

d + − + sup α(n, z) λn λn |z|≤π/2 |dz |! | − | λ+ λ− 6√s + C | n − n | . ≤ n n 2δ h i Hence, by (2.49) and Theorem 1.1 (i),

3δ 2 + − 2 n w(2n) min (λn λn ) 2ρn C. h i | − | ≤ |nX|≥N3

2.9 Improvement of Theorem 1.1 for L selfadjoint For ψ a 1-periodic functions in H w, the operator L(ψ, ψ) is selfadjoint. In this section we show that in this case the decay rate of the asymptotics in Theorem 1.1 (ii) can be improved as follows :

Theorem 2.14 Let M 1, δ > 0 and w be a δ-weight. Then there exist ≥ constants 1 C < and 1 N < so that for any n N and any 1-periodic function≤ ψ∞ H w with≤ ψ ∞M, | | ≥ ∈ k kw ≤

4δ∗ 2 + − 2 n w(2n) min (λn λn ) 2ρn C. h i | − | ≤ |nX|≥N

(1) Proof Using the deﬁnition (2.13) with ψ1 = ψ and ψ2 = ψ we get Rn (k, j)(z) = (2) − + Rn (k, j)(z) and thus β (n, z) = β (n, z). As the eigenvalues λn = nπ +

29 z(ζn ) of L(ψ, ψ) are real, equation (2.40) then reads (with n N2 and N2 as in Proposition 2.12) | | ≥

(ζ)2 = ψˆ(2n) + β+(n, z(ζ)) 2 n | n | (2.60) = ψˆ(2n) + β+(n) + β˜+(n, z(ζ)) 2 | 0 n | and ρn is given by (with an appropriate choice of the square root)

Deﬁne N N such that 4 ≥ 2 12 n δt < A n N . h i n ∀| | ≥ 4

Lemma 2.15 Let M 1, δ > 0 and w be a δ-weight. For any 1-periodic ≥ function ψ in Hw with ψ M and n N , k kw ≤ | | ≥ 4 ζ+ ρ + ζ− + ρ 6t | n − n| | n n| ≤ n or ζ+ + ρ + ζ− ρ 6t . | n n| | n − n| ≤ n

Proof The proof is similar to the one of Lemma 2.13.

30 3 Riesz spaces and normal form of L

3.1 Riesz spaces Let M 1, δ > 0 and w be a δ-weight. By Theorem 1.1, there exists 1 N < ≥so that for any 1-periodic functions ψ , ψ in Hw with ψ M≤, ∞ 1 2 k jkw ≤ the operator L = L(ψ1, ψ2) has two (counted with multiplicity) periodic + − eigenvalues λn , λn near nπ. In Appendix A we introduce the periodic and antiperiodic boundary conditions bc P er+ and bc P er−. We point out that

specL = specL + specL − P er ∪ P er 2 2 2 2 and introduce the Riesz projectors Π2n : L ([0, 1]; C ) L ([0, 1]; C ), corre- + 2 2 2 → 2 sponding to bcP er and Π2n−1 : L ([0, 1]; C ) L ([0, 1]; C ) , corresponding to bcP er− (n Z). Further denote by E the→ C-vector spaces ∈ n E := Π (L2([0, 1]; C2)) ( n N). n n | | ≥ + − + − Notice that dimC E = trΠ = 2 n N. If λ = λ or λ = λ is of geo- n n ∀| | ≥ n 6 n n n metric multiplicity two, there exists a basis of En consisting of eigenfunctions + − + − F and F corresponding to the eigenvalues λn . If λn = λn is of geometric + 2 multiplicity 1, En is the root space of λn . Denote by F a L -normalized eigenfunction of L corresponding to the eigenvalue λ = nπ + z, (L λ)F = 0, F = 1 − k k where denotes the L2-norm in L2([0, 1]; C2). Then k · k F (x) =xF e+(x) + yF e−(x) + (F˘ (k)e+(x) + F˘ ( k)e−(x)) n n 2 k 1 − k Xk6=n where F F˘2 V11 V12 y = F JF˘ V21 V22 x 1 with V = (z D )−1(Id P )−1Snψˆ 11 − n − n 2 V = (z D )−1R(2)(Id Q )−1SnJψˆ 12 − n n − n 1 V = (z D )−1R(1)(Id P )−1Snψˆ 21 − n n − n 2 V = (z D )−1(Id Q )−1SnJψˆ . 22 − n − n 1

31 Proposition 3.1 Let 0 < δ 1, M 1 and w be a δ-weight. Then there exist C C(δ, M) 1 and≤N N≥(M, δ) 1 such that for 1-periodic ≡ ≥ ≡ ≥ functions ψ , ψ Hw with ψ M and n N 1 2 ∈ k jkw ≤ | | ≥ (i) 1 xF 2 + yF 2 1 2 ≤ | | | | ≤ (ii) F˘ 2 C ; JF˘ 2 C k 2k ≤ hniδ k 1k ≤ hniδ where stands for the `2-norm. k · k Proof As F = 1, we have k k F 2 = xF 2 + yF 2 + F˘ 2 + JF˘ 2 = 1. k k | | | | k 2k k 1k Hence xF 2 + yF 2 1. | | | | ≤ Further, by Proposition 2.4, for n N | | ≥ 0 −1 −1 (Id Pn) L(`2 ) 2; (Id Qn) L(`2 ) 2. k − k Snw ≤ k − k Snw ≤ By Corollary 2.3, there exists C > 1 such that

(j) C Rn L(`2 ,`2) k k Snw ≤ n δ∧1 h i and by the deﬁnition of D , for some 1 C < , n ≤ ∞ −1 C (z Dn) L(`2 ,`2) k − k Snw ≤ n 1∧δ −1 h i (z D ) 2 2 1. k − n kL(` ,` ) ≤ Hence for n N | | ≥ 0 C V + V k 11k k 22k ≤ n δ∧1 h i C V + V k 12k k 21k ≤ n δ∧1 h i for some 1 < C < and one concludes that ∞ C C F˘ ( xF + yF ) 2 k 2k ≤ n δ∧1 | | | | ≤ n δ∧1 h i h i C C JF˘ ( xF + yF ) 2 . k 1k ≤ n δ∧1 | | | | ≤ n δ∧1 h i h i

32 By choosing N N suﬃciently large we have, for n N, ≥ 0 | | ≥ 1 F˘ 2 + JF˘ 2 k 2k k 1k ≤ 2 and hence 1 xF 2 + yF 2. 2 ≤ | | | |

3.2 Normal form of L In this section we want to derive a normal form of the restriction of L to the Riesz spaces En. For this purpose introduce an orthonormal basis of + En as follows: Choose F F to be an L2-normalized eigenfunction of L corresponding to the eigen≡value λ+ λ+ and Φ E with ≡ n ∈ n (Φ, F ) = 0; Φ = 1 k k 1 + where, as usual, (Φ, F ) = 0 Φ(x)F (x)dx. In case λ is a double eigenvalue, LRΦ λ+ ξ Φ = (3.1) LF 0 λ+ F + where ξ ξn vanishes iﬀ λ is of geometric multiplicity two. ≡− + − − − In case λn = λn , choose an L2-normalized eigenfunction F of λ λn . Then 6 ≡ F − = aF + bΦ; a 2 + b 2 = 1; b = 0. | | | | 6 With Φ = 1 F − a F , b − b 1 a LΦ = λ− F − + λ+ F b b 1 a a = λ−( F − F ) γ F b − b − b where γ γ := λ+ λ−. Hence ≡ n − LΦ λ− ξ Φ = (3.2) LF 0 λ+ F with ξ ξ := γ a . Notice that (3.1) and (3.2) have the same form. We ≡ n − b refer to this form as the normal form of the restriction of L to the Riesz space En.

33 In the remaining part of this section we want to estimate the size of (ξn)|n|≥N . To this end, we write the equation (L λ−)Φ = ξF in the basis e+, e−(k Z). − k k ∈ With Φ = xΦe+ +yΦe− + Φ˘ (k)e+ +Φ˘ ( k)e− and F = xF e+ +yF e− + n n k6=n 2 k 1 − k n n F˘ (k)e+ + F˘ ( k)e−, we then obtain the following inhomogeneous sys- k6=n 2 k 1 − kP tem (cf. (2.8) - (2.10)) P z−xΦ + ψˆ (2n)yΦ + Snψˆ , JΦ˘ = ξxF (3.3) − 2 h 2 1i ψˆ ( 2n)xΦ z−yΦ + SnJψˆ , Φ˘ = ξyF (3.4) 1 − − h 1 2i Φ n ˆ ˘ ˘ y (S ψ2)Z\n − Φ2 F2 Φ n + (An z ) = ξ (3.5) x (S Jψˆ )Z − JΦ˘ JF˘ 1 \n 1 1 where, as usual, λ− λ− = nπ + z−. We use the above system to obtain an n ≡ estimate for ξ ξ . ≡ n Write Φ˘ = (Φ˘ 2, JΦ˘ 1) and F˘ = (F˘2, JF˘1). Recall that w is assumed to be a δ-weight and hence by Corollary 2.5, equation (3.5) (with n N ) can be | | ≥ 0 solved for Φ,˘

Φ n ˆ ˘ − −1 y (S ψ2)Z\n − −1 ˘ Φ = (z An) Φ n ξ(z An) F . − x (S Jψˆ )Z − − 1 \n In this form, Φ˘ is substituted into (3.3) - (3.4) to obtain (cf. Corollary 2.5)

− − + − Φ z + α(n, z ) ψˆ2(2n) + β (n, z ) x − − − − − Φ ψˆ1( 2n) + β (n, z ) z + α(n, z ) y − − (3.6) xF Snψˆ = ξ + ξ 2 , (z− A )−1F˘ . yF h SnJψˆ − n i 1 − −1 Denote the right side of (3.6) by RS. By Corollary 2.5 (z An) is uniformly bounded for n suﬃciently large and by Proposition 3.1,− for n | | | | ≥ N, 1 C xF 2 + yF 2; F˘ . 2 ≤ | | | | k k ≤ n δ h i Hence RS can be estimated from below: There exists 1 C C < so ≤ ≡ δ,M ∞ that for n N (N as in Proposition 3.1) | | ≥ 1 C RS ξ ( ). ≥ | | √2 − n δ h i 34 By choosing N larger if necessary, we can assume that 1 C 1 n N (3.7) √2 − n δ ≥ 2 ∀| | ≥ h i and (3.6) leads to

ξ 4( ζ− + ψˆ ( 2n) + ψˆ (2n) + β+(n, z−) + β−(n, z−) ) (3.8) | n| ≤ | n | | 1 − | | 2 | | | | | where we used that xΦ 2 + yΦ 2 1 and ζ− = z− α(n, z−) with z− z(ζ−). | | | | ≤ n − ≡ n In view of Proposition 2.10 and Lemma 2.13, one then concludes from (3.8) the following

Proposition 3.2 Let M 1, 0 < δ, and w be a δ-weight. Then there exist ≥ 1 N < , 1 C = Cδ < such that for any 1-periodic functions ψ ≤, ψ Hw∞with ≤ψ M ∞ 1 2 ∈ k jkw ≤ w(2n)2 ξ 2 C. | n| ≤ |nX|≥N

4 Dirichlet eigenvalues

4.1 Dirichlet boundary value problem Consider the Zakharov-Shabat operator L L(ψ , ψ ) on the interval [0, 1]. ≡ 1 2

1 2 Definition 4.1 F = (F1, F2) H ([0, 1]; C ) satisfies Dirichlet boundary conditions if ∈ F (0) F (0) = 0; F (1) F (1) = 0. (4.1) 1 − 2 1 − 2 We mention that the Dirichlet boundary conditions take a more familiar form when the operator L is written as an AKNS operator LAKNS 0 1 d q p LAKNS = − + − (4.2) 1 0 dx p q 35 where (ψ1, ψ2) and (p, q) are related by ψ = q + ip; ψ = q ip. 1 − 2 − − If F = (F , F ) H1([0, 1]; C2) satisfies LF = λF , then LF˜ = λF˜ where 1 2 ∈ F˜ = (F˜1, F˜2) is given by 1 1 F˜1 = (F1 + F2); F˜2 = (F2 F1). √2i √2 − The Dirichlet boundary conditions (4.1) then take the familiar form

F˜2(0) = 0; F˜2(1) = 0. For the remaining part of section 4, let M 1, δ > 0, and a δ-weight w be ≥ w given as well as arbitrary 1-periodic functions ψ1, ψ2 H with ψj w < M. In Appendix A we have introduced, for n N with N∈ given by Theoremk k 1.1, | | ≥ the Riesz projectors Π2n, Π2n−1 corresponding to periodic resp. antiperiodic boundary value problem on [0, 1] for L and the two dimensional subspaces En = Range(Πn). The following proposition assures that there exists a 1-dimensional subspace of En which satisﬁes Dirichlet boundary conditions. Let (F, Φ) denote the orthonormal basis of E L2([0, 1]; C2), introduced in section 3.2. n ⊆ Proposition 4.2 For any n N, there exists G = (G , G ) E | | ≥ 1 2 ∈ n G = αF + βΦ; α 2 + β 2 = 1 | | | | which satisﬁes Dirichlet boundary conditions

G (0) G (0) = 0; G (1) G (1) = 0. 1 − 2 1 − 2

Proof First consider the case where F satisfies F1(0) F2(0) = 0. As F is either periodic or antiperiodic we conclude that F (1)− F (1) = 0 as well 1 − 2 and thus G := F has the required properties. If F1(0) F2(0) = 0, notice that − 6 G˜(x) := (F (0) F (0)) Φ(x) (Φ (0) Φ (0)) F (x) 1 − 2 − 1 − 2 satisfies Dirichlet boundary conditions. As G˜ 0, we may define 6≡ G˜ G := . G˜ k k 36

− + By (3.1) - (3.2), LΦ = λ Φ + ξF and LF = λ F , hence, with γ γn = λ+ λ− and λ λ+ ≡ − ≡ LG = αλF + βLΦ (4.3) = λG βγΦ + βξF. −

For n N suﬃciently large, ξ ξn and γ γn are small and G is almost a Diric| |hlet≥ eigenfunction. In the≡next sections≡ we prove that λ λ+ and ≡ n G are good approximations of the Dirichlet eigenvalue µ µn respectively Dirichlet eigenfunction H. ≡

4.2 Decomposition

Let LDir denote the closed operator LDir = L(ψ1, ψ2) with domain

domL := F H1[0, 1] F (0) F (0) = 0; F (1) F (1) = 0 . Dir { ∈ | 1 − 2 1 − 2 } Let us ﬁx n with n N (N as in Theorem 1.1). Π Π denotes the | | ≥ Dir ≡ n,Dir Riesz projector 1 −1 ΠDir := (z LDir) dz 2πi π − Z|z−nπ|= 2 acting on L2([0, 1]; C2) (cf. Appendix A). Let Ω := Id Π . Dir − Dir Notice that RangeΠ = aH a C Dir { | ∈ } where H domL is an L2[0, 1]-normalized eigenfunction for the Dirichlet ∈ Dir eigenvalue µ µn, ≡ L H = µH; H = 1. Dir k k Let χ C with χ 1 deﬁned by Π G = χH where G is given by ∈ | | ≤ Dir Proposition 4.2. We have

G = χH + ΩDirG.

As G and H are in domL , Ω G domL and Dir Dir ∈ Dir

LDirG = χµH + LDirΩDirG = χµH + ΩDirLDirΩDirG . (4.4)

37 Where for the last equality we have used, ΠDirLDirΩDirG = 0, as LDir and ΠDir commute on domLDir and ΠDirΩDir = 0. On the other hand by (4.3),

LG = λG + R ; R = βγΦ + βξF − and thus, with G = χH + ΩDirG,

LDirG = λχH + λΩDirG + (ΠDir + ΩDir)R. (4.5)

Comparing the decompositions of the right sides of (4.4) and (4.5) leads to the following

Lemma 4.3

χ(µ λ)H = Π R; (4.6) − Dir (L λ)(Ω G) = Ω R (4.7) Dir − Dir Dir where R is given by R = βγΦ + βξF. (4.8) −

4.3 Proof of Theorem 1.2

The equations (4.6) - (4.8) are now used to obtain estimates for µn λ+ ( n N). For this we need to establish that χ 1 is bounded|awa−y n | | | ≥ | | ≤ from 0 and that ΠDirR is small. The latter is easily seen as R γ + ξ . To verify that χk is boundedk away from 0 we show that Ω kG =k ≤G | |χG| is| | | Dir − small. This is proved by using equation (4.7).

Lemma 4.4 There exists N 1 so that ≥ 1 χ n N. | n| ≥ 2 ∀| | ≥

Proof As G = χH + ΩDirG,

χ H = G Ω G = 1 Ω G . | |k k k k − k Dir k − k Dir k 38 By Lemma 4.3 and Lemma A.2 (for (4.9)), Proposition 3.2 (for (4.10)), and Theorem 1.1 (i) (for (4.11)) there exist 1 N < and 1 C < so that ≤ ∞ ≤ ∞ for n N | | ≥ Ω G = (L λ)−1(Ω R) C R C( ξ + γ ) (4.9) k Dir k k Dir − Dir k ≤ k k ≤ | n| | n| C ξ (4.10) | n| ≤ n δ h i C γ , (4.11) | n| ≤ n δ h i (where for the last two inequalities we used that w is a δ-weight). Combining the above inequalities shows that for n large enough | | 1 χ . | n| ≥ 2

Proof of Theorem 1.2 By (4.6),

χ µ λ H = Π R . | || − |k k k Dir k By Lemma 4.4 there exists N 1 so that for n N ≥ | | ≥ µ λ+ 2C ( ξ + γ ) | n − n | ≤ | n| | n| where we have used that Π C (cf. Lemma A.2). The claimed estimate k Dirk ≤ then follows from the estimates of ξn (Proposition 3.2) and of γn (Theorem 1.1 (i)).

A Appendix A: Spectral properties of L(ψ1, ψ2)

In this appendix we consider the operator L(ψ1, ψ2) (ψ1, ψ2 1-periodic functions in L2([0, 2], C2)) with various boundary conditions. For bc Dir, P er, P er ∈ { }

39 denote by Lbc the Zakharov-Shabat operator Lbc = L(ψ1, ψ2) with the following domains:

1 domLDir := F H [0, 1] F1(0) F2(0) = 0; F1(1) F2(1) = 0 ; { ∈ 1 | − − } domLP er+ := F H [0, 1] F (0) = F (1) ; { ∈ 1 | } domL − := F H [0, 1] F (0) = F (1) . P er { ∈ | − }

The operator L LP er is deﬁned on the interval [0, 2] and has the following domain, ≡ domL := F H1[0, 2] F (0) = F (2) . P er { ∈ | } Let specbc spec(Lbc) be the spectrum of Lbc. For potentials ψ1 = ψ2 0, ≡ 1 0 ≡ i.e. L := L(0, 0) = i d , spec (L ) can be given explicitely: 0 0 1 dx bc 0 − spec (L ) = kπ k Z ; (A.1) Dir 0 { | ∈ } spec + (L ) = 2kπ k Z ; (A.2) P er 0 { | ∈ } spec − (L ) = 2(k + 1)π k Z ; (A.3) P er 0 { | ∈ } spec (L ) = kπ k Z . (A.4) P er 0 { | ∈ }

Proposition A.1 Let δ > 0, M 1 and w be a δ-weight. There exists ≥ an even integer N such that for any 1-periodic functions ψ1 and ψ2 in Hw, ψ M, the following statements hold: k jkw ≤ (i) for bc Dir, P er, P er , ∈ { }

spec λ C λ < Nπ π/2 λ C λ kπ < π/2 ; bc ⊂ { ∈ | | | − }∪ { ∈ | | − | } |k[|≥N   (ii) for k N, spec λ C λ kπ < π/2 contains exactly one | | ≥ P er ∩ { ∈ | | − | } isolated pair of eigenvalues; (iii) for k N and bc := P er+ (k even) and bc := P er− (k odd), | | ≥ specbc λ C λ kπ < π/2 contains exactly one isolated pair of eigenvalues;∩ { ∈ | | − | }

(iv) for k N, specDir λ C λ kπ < π/2 contains exactly one eigenvalue;| | ≥ ∩ { ∈ | | − | }

40 (v) the cardinality Nbc of specbc λ C λ < Nπ π/2 is equal to 4N 2 for bc = P er, 2N 1∩for{ bc∈ = D| ir| ,|2N 2 −for bc }= P er+ and − − − 2N for bc = P er−.

As spec + spec − spec , Proposition A.1 implies P er ∪ P er ⊆ P er

spec = spec + spec − . P er P er ∪ P er

Proof Deﬁne for n 1, the union of contours, ≥

= λ C λ = nπ π/2 λ C λ kπ = π/2 . Rn { ∈ | | | − } ∪  { ∈ | | − | } |k[|>n   By (A.1) - (A.4), (L λ) : dom(L ) L2 is invertible for any λ , 0 − bc → ∈ Rn hence (L λ) = (L λ)(Id + Q ) (A.5) − 0 − λ where −1 0 ψ1 Qλ = (L0 λ) . − ψ2 0 2 Using the orthogonal decomposition of L by the eigenfunctions of (L0)bc and the assumption that w is a δ-weight, one gets (with (L2)) L ≡ L 1/2 1 Qλ L M 2δ 2 . (A.6) k k ≤ Z k kπ λ ! Xk∈ | | | − | 1/2 1 1 As max Z 1 2 1 4 , one deduces from (A.6) that, for λ k∈ |k|2δhk−ni / ≤ hniδ∧ / ∈ n, R 1/2 M 1 Qλ L δ∧1/4 3/2 . (A.7) k k ≤ n Z k ! h i Xk∈ h i Let N be an even integer such that

1/2 M 1 δ∧1/4 3/2 1/2. n Z k ! ≤ h i Xk∈ h i

41 Then, for λ with n N ∈ Rn ≥ Q 1/2. (A.8) k λkL ≤ 2 Combining (A.5) and (A.8), one deduces that (L λ) : dom(Lbc) L is invertible for any λ (n N) and any 1-periodic− functions ψ , ψ→in Hw ∈ Rn ≥ 1 2 with ψ M. In particular, (n N) is contained in the resolvent k jkw ≤ Rn ≥ set of Lbc(tψ1, tψ2) for any 0 t 1. Hence the number of eigenvalues of L (tψ , tψ ) in each connected≤ comp≤ onent of the interior of stays the bc 1 2 Rn same for any 0 t 1. To see that all eigenvalues are inside n one chooses n bigger and bigger.≤ ≤ R

It follows from Proposition A.1 that the Riesz projectors Πn and Πn,Dir are well deﬁned (for any n N and 1-periodic functions ψ1, ψ2 with ψj w M) | | ≥ k k ≤

1 −1 Π := (z L + ) dz (n even , n N), n 2πi − P er | | ≥ Z|λ−nπ|=π/2 1 −1 Π := (z L − ) dz (n odd , n N) n 2πi − P er | | ≥ Z|λ−nπ|=π/2 and 1 Π := (z L )−1dz ( n N), n,Dir 2πi − Dir | | ≥ Z|λ−nπ|=π/2 where the contours λ λ nπ = π/2 in the integrals above are counter- clockwise oriented. F{urthermore,| | − | using }(A.5) and (A.8), one deduces

Lemma A.2 Assume that the assumptions of Proposition A.1 hold. Then there exists a constant 1 C such that for any n N (with N as in Proposition A.1) ≤ ≤ ∞ | | ≥ Π 2 C k nkL(L [0,1]) ≤ and Π 2 C. k n,DirkL(L [0,1]) ≤

42 B Appendix B: Proof of Lemma 2.8

W.l.o.g. we may assume that δ∗ = δ.

(i) As w∗ is submultiplicative, one has

w(2n) = 2n δw (2n) 2δ n δw (n + k)w (k + j)w (j + n) (B.1) h i ∗ ≤ h i ∗ ∗ ∗ and, by assumption, n αd (k) d(k) ( k). This leads to h i n ≤ ∀ n 2δ+αw(2n)Ψ (a, b, d )(2n) h i n n a(k + n) b(k + j) n 2δw(2n) d(j + n) ≤ h i k n j n j (B.2) Xk h − i X h − i K (k, j)a˜(k + n)˜b(k, j)d˜(j + n) ≤ n Xk,j where for any u `2 we denote by u˜ the `2-sequence u˜(j) := w(j)u(j) and ∈ w Kn(k, j) is given by

2δ n 3δ K (k, j) := h i . n k n j n k + n δ k + j δ j + n δ h − ih − ih i h i h i

Notice that Kn(k, j) is symmetric in k and j. To estimate Kn(k, j) we need to consider four diﬀerent regions: Estimate of K (k, j) in k n < |n| , j n < |n| : In this case n | − | 2 | − | 2 k + n n ; j + n n ; k + j 2 n k n j n n , | | ≥ | | | | ≥ | | | | ≥ | | − | − | − | − | ≥ | | hence 2δ 1 1 K (k, j) + . n ≤ k n j n ≤ k n 2 j n 2 h − ih − i h − i h − i Estimate of K (k, j) in k n |n| , j n < |n|: In this case n | − | ≥ 2 | − | 2 n k n | |; j + n > n , | − | ≥ 2 | | | | hence 2δ K (k, j) . n ≤ j n k + j δ h − ih i 43 Estimate of K (k, j) in k n < |n| , j n |n| : Using the symmetry of n | − | 2 | − | ≥ 2 Kn(k, j) in k and j, the latter estimate leads to

2δ K (k, j) . n ≤ k n k + j δ h − ih i

Estimate of K (k, j) in k n |n| , j n |n| : We get n | − | ≥ 2 | − | ≥ 2 16δ K (k, j) . n ≤ k n 1−δ k + j δ j + n δ h − i h i h i Combining the above estimates one obtains for k, j, n Z, ∈ 1 1 2 1 4 K (k, j) + + + . n ≤ k n 2 j n 2 k + j δ k n k n 1−δ k + j δ j + n δ h − i h − i h i h − i h − i h i h i Therefore ˜ ˜ Kn(k, j)a˜(k + n)b(k + j)d(j + n) Xk,j 1 1 a˜ (J˜b d˜) (2n) + d˜ (J˜b a˜) (2n)+ (B.3) ≤ ∗ k 2 ∗ ∗ k 2 ∗ h i h i 1 J˜b 1 J˜b d˜ + 2 a˜ ( d˜) (2n) + 4 a˜ ( ) (2n) ∗ k k δ ∗ ∗ k 1−δ k δ ∗ k δ h i h i ! h i h i h i ! where for u `2(Z) and η 0, u denotes the sequence given by u (j) := ∈ ≥ hkiη hkiη u(j) ( j). Using the standard convolution estimates u v 2 u 1 v 2 hjiη ∀ k ∗ k` ≤ k k` k k` and u v ∞ u 2 v 2 for the ﬁrst two terms on the right side of (B.3), k ∗ k` ≤ k k` k k` Corollary B.2 (i) for the third term and Corollary B.2 (ii) for the last term on the right side of (B.3), one obtains from (B.2)

2 n 2δ+αw(2n)Ψ (a, b, d )(2n) C a b d h i n n ≤ k kwk kwk kw n X for a constant 1 C C < only depending on δ. ≤ ≤ δ ∞

44 (ii) Using (B.1) and the assumption n αd (k) d(k) ( k) we get h i n ≤ ∀ n δ+αw(n + `)Ψ (a, b, d )(` + n) h i n n ≤ a(k + `) b(k + j) n δw(n + `) d(j + n) ≤ h i k n k j Xk,j h − i h − i H (`, k, j)a˜(k + `)˜b(k + j)d˜(j + n) ≤ n Xk,j where Hn(`, k, j) is given by n δ ` + n δ H (`, k, j) := h i h i . n k n j n k + ` δ k + j δ j + n δ h − ih − ih i h i h i To estimate Hn(`, k, j) we need to consider two diﬀerent regions: Estimate of H (`, k, j) in j n < |n| : In this case n | − | 2 j + n > n ; ` + n δ ` + k δ k + n δ, | | | | h i ≤ h i h− i hence 1 H (`, k, j) n ≤ k n 1−δ j n k + j δ j + n δ h − i h − ih i h i 1 . ≤ k n 1−δ k + j δ j + n δ h − i h i h i Estimate of H (`, k, j) in j n > |n| : In this case n | − | 2 2 j n > n ; ` + n δ ` + k δ k + n δ, | − | | | h i ≤ h i h− i hence 2δ H (`, k, j) n ≤ k n 1−δ j n 1−δ k + j δ j + n δ h − i h − i h i h i 2δ . ≤ k n 1−δ k + j δ j + n δ h − i h i h i Hence in both cases we obtain the same estimate. Deﬁne e˜(` + n) w(` + ≡ n)e(` + n) by 1 e˜(` + n) := a˜(k + `)˜b(` + j)d˜(j + n). k n j + n δ k + j δ Xk,j h − ih i h i 45 Then we have

n δ+αw(n + `)Ψ (a, b, d )(` + n) w(` + n)e(` + n) h i n n ≤ and 1 J˜b d˜ e˜(`) = a˜ ( ) (`). ∗ k 1−δ k δ ∗ k δ h i h i h i ! By Corollary B.2 (ii),

e˜ 2 C a b d k k` ≤ k kwk kwk kw for some constant 1 C C < . ≤ ≡ δ ∞ It remains to establish the auxilary results used in the proof of Lemma 2.8. First we need the following

Lemma B.1 Let 0 < η 1. Then ≤ a 2 2 (i) p C a 2 a ` and p > k hkiη k` ≤ p,ηk k` ∀ ∈ 2η+1 a q 1 (ii) 1 C a q a ` with 1 q < . k hkiη k` ≤ q,ηk k` ∀ ∈ ≤ 1−η

2 2 Proof (i) follows from H¨older’s inequality with α = p and β = 2−p ,

1/p 1/p a(k) 1 ( )p = a(k)p k η k ηp k h i ! k h i ! X X 1/βp 1/2 2 1 a(k) ηpβ ≤ | | k ! X Xk h i where ηpβ = ηp 2 > 1 or 2ηp > 2 p as (2η + 1)p > 2 by assumption. 2−p − (ii) follows from H¨older’s inequality with α = q and 1 = 1 1 = q−1 β − q q q−1 1/q q a(k) 1 q q q−1 | η | a(k) ( η ) k ≤ ! k ! Xk h i Xk Xk h i where η q > 1 or ηq > q 1 as 1 > (1 η)q by assumption. q−1 − − 46 Recall Young’s inequality

u v C u v k ∗ kq ≤ r,p,qk kpk kr where r, p, q 1 with 1 + 1 = 1 + 1 . ≥ p r q

Corollary B.2 Let 0 < δ 1 ≤ 2

1 a 2 (i) b 1 C a 2 b 2 a, b ` k hki hkiδ ∗ k` ≤ δk k` k k` ∀ ∈ 1 a b 2 (ii) 1 1 C a 2 b 2 a, b ` k hki −δ hkiδ ∗ hkiδ k` ≤ δk k` k k` ∀ ∈ 1 1 δ 1 δ 1 1 δ 1 Proof (i) Let p := 2 + 2 and q := 2 . Then p + 2 = 1 + 2 = 1 + q and hence by Young’s inequality a a b q C p b 2 . k k δ ∗ k` ≤ k k δ k` k k` h i h i 2 2 As p = 1+δ > 1+2δ , Lemma B.1 (i) can be applied, a p C a 2 , k k δ k` ≤ k k` h i and as q = 2 < , Lemma B.1 (ii) gives δ ∞ 1 a b 1 C a 2 b 2 k k k δ ∗ k` ≤ k k` k k` h i h i as claimed. 1 1 2δ 1 (ii) By Lemma B.1, for p := 2 + 3 (< 2 + δ) a p C a 2 . k k δ k` ≤ k k` h i By Young’s inequality with 2 1 = 1 + 1 or 1 = 4δ < 1 (as 0 < δ 1 ) · p q q 3 ≤ 2 a b q C a 2 b 2 . k k δ ∗ k δ k` ≤ k k` k k` h i h i

47 By Lemma B.1 (ii) with η = 1 δ (hence 1 q = 3 < 1 = 1 ) − ≤ 4δ δ 1−η 1 a b 1 C a 2 b 2 k k 1−δ k δ ∗ k δ k` ≤ k k` k k` h i h i h i where 1 C C < depends only on δ. ≤ ≡ δ ∞

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