Patrick G Érard

AND Thomas Kappeler
email: thomas.kappeler@math.uzh.ch

Petar Topalov
email: p.topalov@northeatsern.edu

ON THE SPECTRUM OF THE LAX OPERATOR OF THE BENJAMIN-ONO EQUATION ON THE TORUS

Keywords: Mathematics Subject Classification. 37K15 primary, 47B35 secondary Lax operator, Benjamin-Ono equation, moment map, spectral properties, complex valued potentials

We investigate the spectrum of the Lax operator L u of the Benjamin-Ono equation on the torus for complex valued potentials u in the Sobolev space H -s (T, C), 0 ≤ s < 1/2, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of L u .

Introduction

In this paper we consider the Lax operator L u of the Benjamin-Ono equation, introduced by Nakamura [START_REF] Nakamura | Backlund transform and conservation laws of the Benjamin-Ono equation[END_REF], on the torus T := R/2πZ. It acts on the Hardy space

H + := {f ∈ L 2 (T, C) : f (n) := 1 2π 2π 0 f (x)e -inx = 0 ∀ n < 0}
and is given by L u = D -T u , where D = -i∂ x is the Fourier multiplier on H + ,

f = n≥0 f (n)e inx → Df (x) = n≥0 n f (n)e inx ,
T u is the Toeplitz operator on H + with potential u, f → Π(uf ) , and where Π : L 2 (T, C) → H + denotes the Szegő projector. The aim of this paper is to analyze the spectrum of L u in the case u is a complex valued potential in a neighborhood of the real Sobolev space H -s r in H -s c with 0 ≤ s < 1/2, where for any σ ∈ R,

H σ c ≡ H σ (T, C) := {f = n∈Z f (n)e inx : f (n) ∈ C ∀ n ∈ Z, f σ < ∞} with f σ := n∈Z n 2σ | f (n)| 2 1/2 , n := max{1, |n|} , and
H σ r ≡ H σ (T, R) := {f ∈ H σ c : f (-n) = f (n) ∀ n ≥ 0} .
Since for any a ∈ C, L u+a = L u -a, there is no loss of generality by restricting our attention to potentials u of mean zero, i.e., potentials in H -s c,0 where for any σ ∈ R H σ c,0 ≡ H σ 0 (T, C) := {u ∈ H σ c : u|1 = 0} . Here f |g denotes the standard L 2 -inner product

f |g := 1 2π 2π 0 f (x)g(x)dx , f, g ∈ L 2 (T, C),
extended by duality to a sesquilinear map H -σ c × H σ c → C. To state our main results, we need to introduce some more notation. For any σ ∈ R, let H σ r,0 := H σ c,0 ∩ H σ r and

H σ + := {f ∈ H σ c : f (n) = 0 ∀ n < 0}
. Furthermore, we denote by 1,σ (N, C) the weighted 1 -sequence space, Our main results are the following ones.

Theorem 1. For any 0 ≤ s < 1/2, there exists a neighborhood U -s of H -s r,0 in H -s c,0 with the following properties: (i) For any u ∈ U -s , the Lax operator L u = D -T u defines an unbounded operator on H -s + with domain H 1-s + . It has compact resolvent and hence its spectrum spec(L u ) is discrete. It consists of a sequence of simple eigenvalues λ n (u), n ≥ 0, satisfying lim n→∞ |λ n -n| = 0 and

|λ n+1 (u) -λ n (u)| > 1/2 , | λ n (u)| < 1/4 , ∀ n ≥ 0 .
Furthermore, λ n : U -s → C is analytic for any n ≥ 0.

(ii) For any u ∈ U -s , the gap lengths,

γ n (u) := λ n (u) -λ n-1 (u) -1, ∀ n ≥ 1 ,
satisfy Γ(u) := (γ n (u)) n≥1 ∈ 1,1-2s (N, C) and the map Γ : U -s → 1,1-2s (N, C) is analytic.

Remark 1. Since for any u in H -s r,0 with 0 ≤ s < 1/2, γ n (u), n ≥ 1, are action variables for the Benjamin-Ono equation (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0), [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2)), we refer to the map Γ as the moment map of the Benjamin-Ono equation. It has been shown in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]Appendix C] that for any u ∈ H 0 r,0 , γ n (u), n ≥ 1, are the lengths of the gaps of the spectrum of L u , when considered as an operator acting on the space of Hardy functions on R.

For any u ∈ U -s , denote by H λ (u) the generating function

H λ (u) := (L u -λ) -1 1|1 , λ ∈ C \ spec(L u ) ,
introduced, up to sign, for real valued potentials in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2). For such potentials, H λ (u) can be thought of as a perturbation determinant of (L u -λ) -1 (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]Section 3]) and encodes spec(L u ) as well as the Benjamin-Ono hierarchy. Theorem 1 allows to extend the product representation for H λ (u), established for potentials in H -s r,0 cf. [3, Section 3] (s=0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2) , to U -s . Corollary 1. For any u in U -s , the following holds: (i) The generating function H λ (u) is a meromorphic function in λ with poles contained in spec(L u ) and admits the product representation

H λ (u) = 1 λ 0 (u) -λ p≥1 1 - γ p (u) λ p (u) -λ
where for any λ ∈ C \ spec(L u ), the infinite product is absolutely convergent.

(ii) For any n ≥ 0, [START_REF] Atiyah | Convexity and commuting Hamiltonians[END_REF] λ n (u) = n -

k≥n+1 γ k (u)
where the infinite sum is absolutely convergent.

(iii) If s = 0, one has

1 2π 2π 0 u 2 dx = 2 k≥1 kγ k (u)
where the infinite sum is absolutely convergent.

Ausgangslage. It follows from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2) that for any u ∈ H -s r,0 , the operator L u is an unbounded operator on H -s + with domain H 1-s + and compact resolvent. Its spectrum spec(L u ) consists of a sequence of simple real eigenvalues λ 0 (u) < λ 1 (u) < • • • , with the property that γ n (u) ≥ 0 for any n ≥ 1. Since spec(L u ) is simple, it is possible to introduce a canonically normalized basis of eigenfunctions f n (•) ≡ f n (•, u), n ≥ 0, corresponding to the eigenvalues λ n (u) (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s=0), [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2)). Furthermore it was established in the latter papers that for any 0 ≤ s < 1/2, the image of the moment map Γ : H -s r,0 → 1,1-2s (N, R), u → (γ n (u)) n≥1 is the positive quadrant Q -s + := 1,1-2s (N, R ≥0 ) of 1,1-2s (N, R) and hence a noncompact infinite-dimensional convex polytope. The convexity of Q -s + is an instance of the convexity of the image of the moment map in an infinite-dimensional setting. The theory of the convexity of the image of the moment map in the finite-dimensional compact case originated in the papers of Atiyah [START_REF] Atiyah | Convexity and commuting Hamiltonians[END_REF] and Guillemin and Sternberg [START_REF] Guillemin | Convexity properties of the moment mapping[END_REF].

Remarkably, the trace formula (1) allows to express the eigenvalues of L u as affine functions of the gap lengths. Hence the positive quadrant Q -s + is a moduli space for the spectrum of the operator L u with u ∈ H -s r,0 . We point out that for the Lax operator of the KdV equation (cf. [START_REF] Kappeler | Ergeb. der Math. und ihrer Grenzgeb[END_REF]) as well as for the one of the defocusing NLS equation (cf. [START_REF] Grébert | The defocusing NLS and its normal form[END_REF]), moduli spaces for their spectra are also given by the sequences of gap lengths, but that the eigenvalues are transcendental functions of the gap lengths.

The moment map is related to the map (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF])

F : H -s r,0 → 1,2-2s (N, R), u → (F n (u)) n≥1 , F n (u) := | 1|f n | 2 , by F n (u) = κ n (u)γ n (u) , ∀ n ≥ 1
, where κ n (u) are scaling factors, admitting product expansions in terms of γ k (u), k ≥ 1. The map F is continuous and is referred to as the quasi-moment map of the Benjamin-Ono equation.

For potentials u with u = 0, the methods developed in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] do not allow to prove that the eigenvalues λ n , n ≥ 0, of L u extend analytically to a neighborhood of H -s r,0 in H -s c,0 , which is independent of n. Furthermore, since the eigenvalues and hence the gap lengths might be complex valued, the trace formula λ 0 (u) = -n≥0 γ n (u) no longer implies that (γ n (u)) n≥0 is an 1 sequence. Therefore new arguments are required to prove decay properties of γ n as n → ∞. Ideas of the proofs. Our analysis of the operator L u is based on perturbation theory. Assume that w is a real valued potential in H -s r,0 with 0 ≤ s < 1/2. For u in a (sufficiently small) neighborhood U -s w ⊂ H -s c,0 of w, we view L u as a perturbation of L w and will use the full strength of the study of the Lax operator with real valued potentials, established in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]. In the case w = 0, the operator L 0 = D is particularly simple and hence we treat this case first. Recall that D is a Fourier multiplier with the property that D is an unbounded operator on H -s + with domain H 1-s + . It has a compact resolvent and its spectrum spec(D) consists of the eigenvalues λ n (0) = n, n ≥ 0, all of them being simple. For any n ≥ 0, the canonically normalized eigenfunction f n (•, 0), corresponding to λ n (0), is given by e inx . In Section 2 we show that for any u in a sufficiently small neighborhood U -s 0 of 0 in H -s c,0 and for any λ ∈ C, L u -λ can be defined as an unbounded linear operator on H -s + with domain dom(L u -λ) = H 1-s + . In addition, L u -λ can be viewed as a bounded linear operator L u -λ :

H 1-s + → H -s + .
To analyze the set of λ s, for which L u -λ is invertible, we write L u -λ as

L u -λ = (Id -T u (D -λ) -1 )(D -λ) , ∀λ ∈ C \ Z ≥0 ,
and estimate the size of the operator T u (D -λ) -1 : H -s + → H -s + . For u in a sufficiently small neighborhood U -s 0 of 0 in H -s c,0 , we show that L u -λ is invertible for any λ with dist(λ, Z ≥0 ) ≥ 1/4 and that L u has compact resolvent with spectrum spec(L u ) consisting of simple eigenvalues λ n (u) ∈ C, n ≥ 0, satisfying |λ n (u) -n| < 1/4. It follows that the eigenvalues λ n , n ≥ 0, and hence the gaps γ n = λ n -λ n-1 -1, n ≥ 1, are analytic functionals on U -s 0 . The key step of the proof of the claimed properties of the moment map on U -s 0 is to show that the quasimoment map F admits an analytic extension, F : U -s 0 → 1,2-2s (N, C) -see Section 3. To this end, we express F n , n ≥ 1, in terms of the Riesz projector P n : H -s + → H 1-s + , n ≥ 0, given by the contour integral

P n (u) = - 1 2πi |λ-n|=1/3 (L u -λ) -1 dλ .
The claimed estimates for F are then obtained by a somewhat involved analysis, using the expansion of the operator (Id -T u (D -λ) -1 ) -1 , appearing in (L u -λ) -1 = (D -λ) -1 (Id -T u (D -λ) -1 ) -1 , in its Neumann series.

In Section 5 we analyze L u for u in a (sufficiently small) neighborhood U -s w ⊂ H -s c,0 of an arbitrary real valued potential w in H -s r,0 with 0 ≤ s < 1/2. Our novel approach is based on the use of finite gap potentials, which were studied in detail in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. A potential w in H 0 r,0 is said to be a finite gap potential if the set {n ≥ 1 : γ n (w) = 0} is finite. Finite gap potentials are contained in ∩ m≥0 H m r,0 . Let U 0 := {0} and for any given N ≥ 1, denote by U N the set of real valued finite gap potentials w with γ N (w) = 0 and γ n (w) = 0 for any n > N . For any 0 ≤ s < 1/2, ∪ N ≥0 U N is dense in H -s r,0 . A key property of finite gap potentials is that for any w ∈ U N , the eigenfunctions f n of L w with n ≥ N are given by [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF]

f n (x, w) = g ∞ (x, w)e inx , ∀ n ≥ N, g ∞ (x, w) := e i∂ -1 x w(x) .
We remark that g ∞ (x, w) is a solution of -i∂ x g -wg = 0 and refer to Appendix A for a discussion of the asymptotics of the eigenfunctions f n (•, w) for arbitrary elements w in H -s r,0 , 0 ≤ s < 1/2. By approximating w ∈ H -s r,0 by an appropriately chosen finite gap potential w 0 with w 0 ∈ U N for some N ≥ 1, the formulas (2) for the eigenfunctions f n (•, w 0 ) with n ≥ N allow us to apply the methods developed to analyze L u for u ∈ H -s c,0 near 0 to obtain corresponding results for L u with u ∈ U -s w .

Subsequent work. Besides being of interest in its own right, our study of the spectrum of the Lax operator L u for complex valued potentials is motivated by our ongoing research project concerning the Birkhoff map Φ of the Benjamin-Ono equation, constructed in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] on L 2 r,0 and then extended in [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] to H -s r,0 for any 0 < s < 1/2. Theorem 1 allows us to prove that Φ is not only a homeomorphism, but a (real analytic) diffeomorphism, Φ : H -s r,0 → h 1/2-s + (cf. [START_REF] Gérard | Analyticity of the Birkhoff map of the Benjamin-Ono equation[END_REF]). The latter result will be applied to further analyze regularity properties of the solution map of the Benjamin-Ono equation and will serve as the starting point for studying (Hamiltonian) perturbations of the Benjamin-Ono equation by KAM methods. Due to the nonlocal nature of the operator L u , to prove that the map Φ is a real analytic diffeomorphism is considerably more difficult than to prove the corresponding result for the Birkhoff map of the KdV equation (cf. [START_REF] Kappeler | Ergeb. der Math. und ihrer Grenzgeb[END_REF]) or of the defocusing NLS equation (cf. [START_REF] Grébert | The defocusing NLS and its normal form[END_REF]). A key ingredient into the proof of the analyticity of Φ is that Γ :

U -s → 1,1-2s + (N, C) is analytic.
Organisation. Section 2 -Section 4 are concerned with the proofs of the results of Theorem 1 and Corollary 1 for complex valued potentials u near zero. In addition, in Section 4 we discuss properties of the scaling factors κ n and the normalizing factors µ n , introduced in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. In Section 5 -Section 7 we analyze the spectrum of the Lax operator L u for u ∈ H -s c,0 with u small and prove Theorem 1 and Corollary 1. In Appendix A we prove asymptotic estimates for canonically normalized eigenfunctions f n (•, u) of L u for real valued potentials in u ∈ H -s r,0 with 0 ≤ s < 1/2. Finally, in Appendix B we review the notion of normally analytic maps.

Notation. By and large, we use the notations introduced in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]. For the convenience of the reader, we list the most frequently used ones in this paragraph.

For any s ∈ R, denote by H s c the Sobolev space H s c = H s (T, C) where T = R/2πZ. For s = 0, we also write L 2 c for H 0 c . Furthermore, H s c,0 denotes the subspace of H s c of elements with average zero. For s = 0, we also write L 2 c,0 for H 0 c,0 . For any s ∈ R, denote by H s r the Sobolev space H s r = H s (T, R). For s = 0, we also write L 2 r for H 0 r . Similarly, H s r,0 is the subspace of H s r of elements in H s r of average zero. For s = 0, we also write L 2 r,0 for H 0 r,0 . The open ball in H -s c,0 of radius r > 0, centered at 0, is denoted by

B -s c,0 (r) ≡ B -s c,0 (0, r). We denote by f |g := 1 2π 2π 0 f gdx the L 2 -inner product on L 2 c
and by f, g the nondegenerate bilinear form on L 2 c × L 2 c , given by f, g := 1 2π 2π 0 f gdx, which is extended to H -s c × H s c for any s ∈ R by duality.

For any f ∈ H s c and k ∈ Z, denote by f (k) = f (x)|e ikx the kth Fourier coefficient of f .

For any s ∈ R, we denote by H s + the subspace of the Sobolev space H s c , consisting of elements f in H s c with f (k) = 0 for any k < 0. For s = 0 we also write H + instead of H 0 + . For any m ∈ Z, let Z ≥m := {k ∈ Z : k ≥ m} and set N 0 := Z ≥0 as well as N := Z ≥1 .

The standard p sequence spaces are denoted as follows: p + ≡ p (N, C) and more generally, p ≥m ≡ p (Z ≥m , C) (1 ≤ p ≤ ∞). The standard weighted sequence spaces, needed in this paper, are

h s + ≡ h s (N, C)(s ∈ R), 1,s + ≡ 1,s (N, C) (s ∈ R).
In case s = 0, we also write 2 + for h 0 + .

Proof of Theorem 1(i) for u small

In this section, we define the Lax operator L u for complex valued potentials u ∈ H -s c,0 with 0 ≤ s < 1/2 and obtain some general results about its spectrum. Our main goal is to study spectral properties of L u for u near the zero potential and to prove Theorem 1(i) for such potentials. In Section 5 we will consider the case of potentials u with u small. We begin with some preliminary considerations. For any 0 ≤ s < 1/2, define σ := (s + 1/2)/2 > 0 . Note that 1/4 ≤ σ < 1/2 Lemma 1. For any 0 ≤ s < 1/2, there exists a constant C s ≥ 1 so that the following holds:

(i) For any f ∈ H 1-σ c , g ∈ H s c , the function f g is in H s c and f g s ≤ C s f 1-σ g s , ∀g ∈ H s c , f ∈ H 1-σ c . (ii) For any u ∈ H -s c and f ∈ H 1-σ c
, the product uf defines an element in H -s c , determined by

g|uf := gf |u , ∀g ∈ H s c . Furthermore, uf -s ≤ C s u -s f 1-σ , ∀u ∈ H -s c , f ∈ H 1-σ c . Since H 1-s c embeds into H 1-σ c
, the estimates in items (i) and (ii) hold in particular when σ is replaced by s.

Proof. (i) Since 1 -σ > 1/2, H 1-σ c
acts by multiplication on itself and on L 2 c . By interpolation it then follows that it acts on H s c and that there exists C s ≥ 1 so that

f g s ≤ C s f 1-σ g s , ∀g ∈ H s c , f ∈ H 1-σ c .
(ii) By item (i) it follows that for any u ∈ H -s c , g ∈ H s c , and f ∈ H

uf -s ≤ C s u -s f 1-σ .
Lemma 1 implies that for any 0 ≤ s < 1/2, the Toeplitz operator T u with potential u ∈ H -s c is a well defined bounded linear operator,

T u : H 1-s + → H -s + , f → Π(uf ) and so is L u , L u : H 1-s + → H -s + , f → Df -T u f where D = -i∂ x and where Π is the Szegő projector, Π : H -s c → H -s + : f → k≥0 f (k)e ikx .
Alternatively, we view L u as an unbounded operator L u :

H -s + → H -s + with domain H 1-s + .
In a first step, we determine half planes in C consisting of complex numbers λ ∈ C with the property that the operator L u -λ :

H 1-s + → H -s
+ is invertible for any u in a given bounded subset of H -s c,0 with 0 ≤ s < 1/2. Note that the spectrum of the operator D :

H + → H + with domain H 1 + is discrete, consisting of the sequence of simple eigenvalues λ n = n, n ≥ 0. It follows that for any λ ∈ C \ N 0 , D -λ : H 1-s + → H -s + is a bounded invertible operator with bounded inverse, (D -λ) -1 : H -s + → H 1-s +
and we can write

L u -λ = Id -T u (D -λ) -1 (D -λ).
Here and in the sequel, N 0 denotes the set of nonnegative integers. To show that L u -λ :

H 1-s + → H -s
+ is invertible, it thus suffices to show that the operator norm of T u (D -λ) -1 : H -s + → H -s + is smaller than 1 where the latter norm is defined with respect to a conveniently chosen norm on H -s + which might depend on λ. For any 0 ≤ s < 1/2 and M > 0, define

(3) K M := (2C s M ) 2/( 1 2 -s) + 1
where C s is the constant given in Lemma 1.

Lemma 2. Let M > 0 and 0 ≤ s < 1/2. Then the following holds:

(i) For any λ ∈ C in the half plane { λ ≤ -K M } and any u ∈ H -s c,0 with u -s ≤ M, the operator L u -λ : H 1-s + → H -s + is invertible. (ii) For any u ∈ H -s c,0 with u -s ≤ M and any λ ∈ C in the half planes { λ ≥ λ + 2K M }, { λ ≤ -λ -2K M }, the operator L u -λ : H 1-s + → H -s + is invertible. Proof. (i) For any λ ∈ C \ N 0 and f ∈ H -s + , we have by Lemma 1 T u (D -λ) -1 f -s = Π(u(D -λ) -1 f ) -s ≤ C s u -s (D -λ) -1 f 1-σ . Note that (D -λ) -1 f 2 1-σ = k≥0 k 2(1-σ) |k -λ| 2 | f (k)| 2 .
Since for any λ with λ ≤ -K M , one has

|k -λ| ≥ K M , |k -λ| ≥ k , ∀ k ∈ N 0 , it follows that (D -λ) -1 f 2 1-σ ≤ 1 K 2(σ-s) M k≥0 1 k 2s | f (k)| 2 ≤ 1 K 2(σ-s) M f 2 -s .
Combining the estimates obtained so far and using that σ

-s = (1/2 -s)/2 one infers from the definition of K M that for any u ∈ H -s c,0 with u -s ≤ M T u (D -λ) -1 f -s ≤ C s M K ( 1 2 -s)/2 M f -s ≤ 1 2 f -s , implying that L u -λ is invertible for any λ ∈ C with λ ≤ -K M .
(ii) In view of item (i) it suffices to show that L u -λ is invertible for any λ ∈ C with µ := λ ≥ -K M and ±ν := λ ≥ µ + 2K M . Since for such λ,

|k -λ| 2 = (k -µ) 2 + ν 2 ≥ (k -µ) 2 + (µ + 2K M ) 2 , -2K M -K M 2K M -2K M K M -K M µ = λ ν = λ ν = µ + 2K M ν = -(µ + 2K M ) Figure 2.1.
For λ in the grey area, L u -λ is invertible.

arguing as in item (i), it suffices to verify that for any k ∈ N 0 and µ

≥ -K M , (k -µ) 2 + (µ + 2K M ) 2 ≥ k 2 /2, (k -µ) 2 + (µ + 2K M ) 2 ≥ K 2 M .
First note that the latter inequalities clearly holds for such k and µ and that the first one holds for k = 0. Finally, or any k ∈ N, one has

(k -µ) 2 + (µ + 2K M ) 2 -k 2 /2 = 1 2 k -2µ 2 + 4K M (µ + K M ) ≥ 0 .
It then follows that for any u ∈ H -s c,0 with u -s ≤ M and λ as above

T u (D -λ) -1 f -s ≤ √ 2C s M K ( 1 2 -s)/2 M f -s ≤ √ 2 2 f -s ,
implying that L u -λ is invertible for any λ ∈ C in either of the two half planes.

In the remainder of this section we study the operator L u for u ∈ H -s c,0 near zero. First we need to introduce some more notation. For any m ∈ Z and s ∈ R, define the sequence space

h s ≥m ≡ h s (Z ≥m , C) := {z = (z k ) k≥m ⊂ C : z s < ∞}, z s := ∞ k=m k 2s |z k | 2 1/2 ,
and for any n ∈ Z, the n-shift operator S n :

h s ≥0 → h s ≥-n , given by (4) z = (z k ) k≥0 → (S n z) k k ≥-n , (S n z) k := z k +n . Note that (5) S n z s = ∞ k =-n k 2s |(S n z) k | 2 1/2 = ∞ k=0 k -n 2s |z k | 2 1/2 .
Denote by H s;n + the C-vector space H s + endowed with the inner product, induced by the norm [START_REF] Grébert | The defocusing NLS and its normal form[END_REF] f s;n := S n f s , f := ( f (k)) k≥0 .

Lemma 3. For any s ∈ R ≥0 , n ∈ N 0 , and f ∈ H -s + , the following estimates hold:

(7) f -s;n ≤ 2 s n s f -s , f -s ≤ 2 s n s f -s;n .
Proof. To verify the first estimate of [START_REF] Guillemin | Convexity properties of the moment mapping[END_REF], note that k ≤ 2 k -n n for any k, n ≥ 0 and hence

1 k-n ≤ 2 n k , implying that 1 k -n 2s ≤ 2 2s n 2s k 2s , ∀s ≥ 0 , k, n ≥ 0 .
Hence in view of ( 5) and ( 6) one has for any f ∈ H -s

+ f -s;n ≤ 2 s n s ∞ k=0 1 k 2s | f (k)| 2 1/2 = 2 s n s f -s .
To establish the second estimate of (7), we argue similarly. For any k, n ≥ 0, one has k -n ≤ 2 k n or

1 k 2s ≤ 2 2s n 2s k -n 2s , ∀s ≥ 0 , k, n ≥ 0 ,
and hence for any f ∈ H -s + with s ≥ 0,

f -s ≤ 2 s n s ∞ k=0 1 k -n 2s | f (k)| 2 1/2 = 2 s n s f -s;n .
For any 0 < ρ ≤ 1/3 and n ≥ 0, denote by

D n (ρ) the open disk in C D n (ρ) := {λ ∈ C : |λ -n| < ρ}
and by ∂D n (ρ) its boundary,

∂D n (ρ) = {λ = n + ρe iθ : θ ∈ R/2πZ} .
Note that the distance between different discs is at least 1/3. Furthermore, for any n ≥ 1, we introduce the closed vertical strips V ert n (ρ) of width one with the disc D n (ρ) removed, We remark that the union of Vert n (ρ) ∪ D n (ρ), n ≥ 0, covers the complex plane.

Lemma 4. For any k, n ∈ N 0 , 0 < ρ ≤ 1/3, and λ ∈ Vert n (ρ),

|k -λ| ≥ ρ k -n . Proof. Let λ ∈ Vert n (ρ) with n ≥ 0. Then |n -λ| ≥ ρ = ρ • 0 whereas for k = n, |k -λ| ≥ |k -n| -1/2 ≥ 1 2 k -n .
For u ∈ H -s c,0 with 0 ≤ s < 1/2 and λ ∈ Vert n (ρ) with n ≥ 0 and 0 < ρ ≤ 1/3, we write L u -λ in the form

(8) L u -λ = (D -λ) -T u = Id -T u (D -λ) -1 D -λ .
In order to show that L u -λ :

H 1-s + → H -s
+ is invertible for u near 0, we prove the following Lemma 5. Let 0 ≤ s < 1/2 and 0 < ρ ≤ 1/3. Then for any u ∈ H -s c,0 and n ≥ 0,

T u (D -λ) -1 H -s;n + →H -s;n + ≤ 1 ρ C s u -s , ∀λ ∈ Vert n (ρ) ,
where C s > 0 is the constant of Lemma 1.

Proof. Let f ∈ H -s + and λ ∈ Vert n (ρ). Expressing T u ((D -λ) -1 f ) as a Fourier series yields

T u ((D -λ) -1 f ) = ∞ m=0 ∞ k=0 u(m -k) f (k) k -λ e imx .
Hence T u ((D -λ) -1 f ) 2 -s;n can be estimated by

∞ m=0 1 m -n 2s ∞ k=0 | u(m -k)| | f (k)| |k -λ| 2 .
With m := m -n, k := k -n and using that f

(k + n) = S n f (k ), Lemma 4 implies that T u ((D -λ) -1 f ) 2 -s,;n ≤ ∞ m =-n 1 m 2s ∞ k =-n | u(m -k )| |S n f (k )| ρ k 2 .
Applying Lemma 1 one concludes that

T u ((D -λ) -1 f ) -s;n ≤ 1 ρ C s u -s S n f -s = 1 ρ C s u -s f -s;n .
As an immediate consequence of Lemma 5, one obtains the following Corollary 2. Let 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, and let C s be the constant of Lemma 1. Then for any u ∈ H -s c,0 with u -s ≤ ρ/4C s the following holds: (i) For any n ≥ 0

T u (D -λ) -1 H -s;n + →H -s;n + ≤ 1 4 , ∀λ ∈ Vert n (ρ).
(ii) For any λ ∈ n≥0 Vert n (ρ), the operator L u -λ :

H 1-s + → H -s + is invertible with inverse given by (9) (L u -λ) -1 = (D -λ) -1 (Id -T u (D -λ) -1 ) -1 .
Proof. Item (i) follows from Lemma 5. It implies that for any λ ∈ Vert n (ρ) with n ≥ 0, the Neumann series

∞ k=0 (T u (D -λ) -1 ) k ab- solutely converges to a bounded operator on H -s;n + , hence in turn Id -T u (D -λ) -1 : H -s;n + → H -s;n + is invertible and by Lemma 3, so is Id -T u (D -λ) -1 : H -s + → H -s + .
By [START_REF] Kappeler | Ergeb. der Math. und ihrer Grenzgeb[END_REF] it then follows that for any λ ∈ n≥0 Vert n (ρ), the operator L u -λ :

H 1-s + → H -s
+ is invertible with inverse given by (9).

To summarize our findings, denote by B s c,0 (r) ≡ B s c,0 (0, r) the open ball in H s c,0 , s ∈ R, of radius r > 0, centered at 0, B s c,0 (r) := {u ∈ H s c,0 : u s < r}. and by Vert 0 n (ρ) the interior of Vert n (ρ). So e.g. for any n ≥ 1 Vert 0 n (ρ) = {λ ∈ C : |λ -n| > ρ; | λ -n| < 1/2} . We have proved the following Proposition 1. Let 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, and let C s be the constant of Lemma 1. Then for any u ∈ H -s c,0 with u -s ≤ ρ/4C s , L u defines an unbounded operator on H -s + with domain H 1-s + . It has compact resolvent, hence has discrete spectrum, and its resolvent set contains n≥0 Vert n (ρ). Furthermore, the map

B -s c,0 (r s ) × n≥0 Vert 0 n (1/4) → B(H -s + , H 1-s + ), (u, λ) → (L u -λ) -1
is analytic where r s is defined as r s := 1/16C s .

Proposition 1 allows to introduce the Riesz projectors.

Corollary 3. Let 0 ≤ s < 1/2, ρ = 1/4, and let C s be the constant of Lemma 1. Then for any u ∈ H -s c,0 with u -s ≤ 1/16C s and n ≥ 0, the Riesz projector [START_REF] Nakamura | Backlund transform and conservation laws of the Benjamin-Ono equation[END_REF] P n (u) := -

1 2πi ∂Dn(1/3) (L u -λ) -1 dλ ∈ B(H -s + , H 1-s + )
is well defined (with ∂D n (1/3) being counterclockwise oriented) and

B -s c,0 (r s ) → B(H -s + , H 1-s + ), u → P n (u) is analytic where r s = 1/16C s .
We remark that for any λ ∈ ∪ n≥0 Vert n (ρ), the inverse (L u -λ) -1 is given by the (absolutely convergent) series

∞ k=0 (D -λ) -1 (T u (D -λ) -1 ) k = (D -λ) -1 +(D -λ) -1 T u (D -λ) -1 +. . . .
We now have all the ingredients to prove Theorem 1(i) for u ∈ H -s c,0 near 0, which in this case reads as follows.

Theorem 2. Let u ∈ H -s c,0 , 0 ≤ s < 1/2, with u -s ≤ ρ/4C s , 0 < ρ ≤ 1/4, and let C s be the constant of Lemma 1. Then the spectrum of L u consists of a sequence of eigenvalues, denoted by λ n (u), n ≥ 0. For any n ≥ 0, λ n (u) has algebraic multiplicity one and is contained in D n (ρ). Furthermore, the eigenvalues

B -s c,0 (r s ) → C , u → λ n (u) , ∀n ≥ 0,
and hence the gap lengths

B -s c,0 (r s ) → C , u → γ n (u) = λ n (u) -λ n-1 (u) -1 , ∀n ≥ 1,
are analytic maps. with u -s < ρ/4C s , it follows that

λ n (u) + 1 ∈ D n+1 (ρ) , ∀n ≥ 0 .
(ii) Since for any u ∈ H -s c,0 with u -s ≤ 1/16C s and n ≥ 0, the eigenvalue of

λ n (u) of L u is contained in D n (1/4) one has dist(λ n (u), ∂D n (1/3)) ≥ 1/12 , ∀n ≥ 0 .
Proof of Theorem 2. Let u ∈ B -s c,0 (r s ). By Proposition 1, L u has compact resolvent and hence for any n ≥ 0, the projection P n (u) has finite rank and hence is of trace class. The value of TrP n at u is given by the rank of P n (u), hence is an integer. Since by Corollary 3, for any n ≥ 0, the map B -s c,0 (r s ) → C, u → TrP n (u) is analytic (and hence continuous), TrP n is constant. Noting that the spectrum of D consists of the sequence of simple eigenvalues λ n = n, n ≥ 0, it then follows that for any n ≥ 0, L u has precisely one eigenvalue in D n (1/4) and that this eigenvalue is of algebraic multiplicity one. We denote it by λ n (u). Going through the above arguments one sees that for any u ∈ H -s c,0 with u -s ≤ ρ/4C s , one has λ n (u) ∈ D n (ρ). It remains to prove that for any n ≥ 0, B -s c,0 (r s ) → C , u → λ n (u) is analytic. This follows by observing that λ n (u) equals the trace of the operator -

1 2πi ∂Dn(1/3) λ(L u -λ) -1 dλ, λ n (u) = -Tr 1 2πi ∂Dn(1/3) λ(L u -λ) -1 dλ .
Furthermore, λ n (u) is real valued for u real valued (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]).

We finish this section with discussing sufficient conditions as well as a necessary condition for a potential u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2 to satisfy γ n (u) = 0 for a given n ≥ 1. We refer to [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]Lemma 2.5] for corresponding results in the case u is in L 2 r . First we make some preliminary considerations. Arguing as in [3, Section 2], one shows that for any u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2 one has (cf. [3, (2.7)]) ( 11)

L u S = SL u + S -uS • |1 1
where for any σ ∈ R, we denote by S ≡ S 1 :

H σ + → H σ + , f → e ix f the one shift operator on H σ + .
The following lemma is a version of [3, Lemma 2.6], adapted to the situation at hand. Lemma 6. For any u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2 and any set h n , n ≥ 0, of eigenfunctions of L u , corresponding to the eigenvalues λ n ≡ λ n (u), one has

(λ p -λ k -1)P p Sh k = -uSh k |1 P p 1 , ∀ p, k ≥ 0.
Proof. Let u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2 and let h n , n ≥ 0, be a set of eigenfunctions of L u , corresponding to the eigenvalues λ n ≡ λ n (u). Note that h n ∈ H 1-s + for any n ≥ 0. Recall that by [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF]

(L u -λ k -1)Sh k = -uSh k |1 1 .
We now apply the projection P p ≡ P p (u), p ≥ 0, to both sides of the latter identity. Using that L u and P p commute and that L u P p = λ p P p one concludes that

(λ p -λ k -1)P p Sh k = -uSh k |1 P p 1 . Lemma 7. Let u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2.
Then the following holds: (i) For any set h k , k ≥ 0, of eigenfunctions of L u , corresponding to the eigenvalues λ k (u), one has h 0 |1 = 0 and for any n ≥ 1, the following statements are equivalent:

(G1) h n |1 = 0; (G2) L u Sh n-1 = λ n Sh n-1 ; (G3) uSh n-1 |1 = 0.
Furthermore, any of these three conditions implies (G4)

γ n (u) = 0. (ii) If γ n (u) = 0, then P n (u)1|1 = 0. Proof. Let u ∈ B -s
c,0 (r s ) with 0 ≤ s < 1/2 and let h k , k ≥ 0, be a set of eigenfunctions of L u , corresponding to the eigenvalues λ k ≡ λ k (u). (i) Assume that for some n ≥ 0, h n |1 = 0. Then h n = Sg n where g n ∈ H 1-s + , g n = 0, and by ( 11)

λ n Sg n (= L u Sg n ) = SL u g n + Sg n -uSg n |1 1 .
Applying S * = T e -ix to both sides of the latter identity and using that S * 1 = 0 one concludes that λ n g n = (L u + 1)g n or

L u g n = (λ n -1)g n , implying that λ n -1 is an eigenvalue of L u . In the case n = 0, this contradicts that λ 0 -1 ∈ Vert 0 (1/4) is contained in the resolvent set of L u . Hence we conclude that h 0 |1 = 0. If n ≥ 1, it follows that λ n -1 is in D n-1 (1/4
) and hence by Theorem 2 one has λ n -1 = λ n-1 , implying that Sh n-1 and h n are collinear. This shows that (G1) implies (G2). Applying [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF] to h n-1 , one gets

L u Sh n-1 = SL u h n-1 + Sh n-1 -uSh n-1 |1 1 = (λ n-1 + 1)Sh n-1 -uSh n-1 |1 1 ,
from which we infer that the statements (G2) and (G3) are equivalent.

Assuming that (G2) holds, one has

L u Sh n-1 = (λ n-1 +1)Sh n-1 . Hence λ n-1 +1 is an eigenvalue of L u . Since it is contained in D n (ρ), it follows from Theorem 2 that λ n-1 + 1 = λ n ,
showing that γ n = 0, i.e., (G4) holds. Furthermore, it implies that h n and Sh n-1 are collinear and hence h n |1 = 0, meaning that (G1) holds.

(ii) Assume that γ n (u) = 0 for some n ≥ 1. Since by Lemma 6 with p = n and k = n -1,

γ n (u)P n (u)Sh n-1 = -uSh n-1 |1 P n (u)1 ,
it then follows that uSh n-1 |1 P n (u)1 = 0. If P n (u)1 = 0, then clearly, P n (u)1|1 = 0. On the other hand if uSh n-1 |1 = 0, then by item (i), h n |1 = 0. Since P n (u)1 is a scalar mutiple of h n , this implies that P n 1|1 = 0.

Quasi-moment map for u small

In this section we introduce the so called quasi-moment map for u in a neighborhood of 0 in H -s c,0 , 0 ≤ s < 1/2, and prove that it is analytic with values in a weighted 1 space. It is a key ingredient into the proof of Theorem 1(ii) for potentials near zero. Furthermore, the results obtained in this section will be used in Section 5 to derive corresponding results for the quasi-moment map in a neighborhood of H -s r,0 in H -s c,0 . By Corollary 2 with 0 ≤ s < 1/2 and ρ = 1/4, it follows that for any n ≥ 0 and u ∈ B -s c,0 (r s )

T u (D -λ) -1 H -s;n + →H -s;n + ≤ 1/4 , ∀λ ∈ Vert n (1/4) ,
where r s = 1/16C s and C s > 0 is the constant of Lemma 1. As a consequence, for any λ ∈ n≥0 Vert n (1/4), the inverse of the operator

L u -λ : H 1-s + → H -s + is given by the Neumann series (12) (L u -λ) -1 = ∞ m=0 (D -λ) -1 (T u (D -λ) -1 ) m .
For any n ≥ 0, introduce the map (13)

F n : B -s c,0 (r s ) → C , u → 1 2πi ∂Dn(1/3) (L u -λ) -1 1|1 dλ ,
where we recall that the circle ∂D n (1/3) is counterclockwise oriented. By Corollary 3, the map F n is analytic and by ( 12) it can be written as a series,

F n (u) = ∞ m=0 1 2πi ∂Dn(1/3) (D -λ) -1 (T u (D -λ) -1 ) m 1|1 dλ .
We note that for a real valued potential u ∈ B -s r,0 (r s ) ⊂ H -s r,0 with 0 ≤ s < 1/2, the value F n (u), n ≥ 0, can then be computed as follows: let (f j ) j≥0 be the L 2 -orthonormal basis of eigenfunctions, corresponding to the eigenvalues (λ j (u)) j≥0 of L u , introduced in [3] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2). Expressing the constant function 1 with respect to the basis (f j ) j≥0 , one has 1 = ∞ j=0 1|f j f j and hence

(L u -λ) -1 1 = ∞ j=0 1|f j (L u -λ) -1 f j = ∞ j=0 1|f j f j λ j -λ .
By Cauchy's theorem it then follows that

(14) F n (u) = ∞ j=0 1|f j f j |1 1 2πi ∂Dn(1/3) 1 λ j -λ dλ = -| 1|f n | 2 .
Therefore, for any n ≥ 0, F n is the analytic extension of -| 1|f n | 2 to B -s c,0 (r s ). Note that for any n ≥ 0, | 1|f n | 2 does not depend on the normalization of the phases of (f j ) j≥0 . Hence any other choice of an L 2 -orthonormal basis of eigenfunctions of L u would yield the same result.

Proposition 2. There exists 0 < r s, * ≤ r s so that the map

F : B -s c,0 (r s, * ) ⊂ H -s c,0 → 1,2-2s + , u → F (u) := (F n (u)) n≥1 is analytic. Furthermore ∞ n=1 ∞ m=0 n 2-2s 1 2πi ∂Dn(1/3) (D -λ) -1 (T u (D -λ) -1 ) m 1|1 dλ
converges uniformly with respect to u.

Remark 3. (i) It follows from [3, Corollary 3.4] (s = 0), [4, (27)] (0 < s < 1/2) that for any n ≥ 1 and u ∈ H -s r,0 with 0 ≤ s < 1/2, ( 15
) | 1|f n (•, u) | 2 = γ n (u)κ n (u)
where κ n (u) > 0 is a scaling factor of the action variable γ n (u). Since

κ n (u) = O(n -1 ) (cf. [3, Corollary 3.4](s = 0), [4, (3.2)](0 < s < 1/2)) it follows that for any u ∈ H -s r,0 , F (u) = (-κ n (u)γ n (u)) n≥1 ∈ 1,2-2s (N, R).
In view of (15), we refer to F as the quasi-moment map.

(ii) Our proof shows that for any 0 ≤ s < 1/2, the map

F : B -s c,0 (r s, * ) → 1,2-2s +
is normally analytic. We refer to Appendix B for a review of the notion of normally analytic maps.

Before proving Proposition 2 we make some preliminary considera-

tions. Let u ∈ B -s c,0 (r s ) with 0 ≤ s < 1/2. Since for any n ≥ 1, the integral ∂Dn(1/3) (D-λ) -1 1|1 dλ = ∂Dn(1/3) -λ -1 dλ vanishes, one has F n (u) = ∞ m=1 1 2πi ∂Dn(1/3) (D -λ) -1 (T u (D -λ) -1 ) m 1|1 dλ and since (D -λ) -1 T u (D -λ) -1 1|1 = 1 λ 2 Πu|1 = 0, also ∂Dn(1/3) (D -λ) -1 T u (D -λ) -1 1|1 dλ vanishes for any n ≥ 1.
Changing the index of summation from m to m -1 then yields

(16) F n (u) = ∞ m=1 1 2πi ∂Dn(1/3) (T u (D -λ) -1 ) m Πu|1 1 λ 2 dλ , ∀n ≥ 1.
The terms for m = 1 and m = 2 in the latter series can be computed as follows: For m = 1 one has

T u (D -λ) -1 Πu|1 = ∞ k=0 Π(u u(k) k -λ e ikx )|1 = ∞ k=0 ∞ j=0 u(j) u(k) k -λ e i(k+j)x |1 = ∞ k=0 u(-k) u(k) k -λ , implying that 1 2πi ∂Dn(1/3) T u (D -λ) -1 Πu|1 1 λ 2 dλ = - 1 n 2 u(n) u(-n), ∀n ≥ 1.
Similarly, one computes the term for m = 2,

(T u (D -λ) -1 ) 2 Πu = T u (D -λ) -1 ∞ k 2 =0 ∞ k 1 =0 u(k 1 ) k 1 -λ u(k 2 -k 1 )e ik 2 x = Π ∞ j=-∞ u(j) ∞ k 2 =0 ∞ k 1 =0 u(k 1 ) k 1 -λ u(k 2 -k 1 ) k 2 -λ e i(k 2 +j)x = ∞ k 3 =0 ∞ k 2 =0 ∞ k 1 =0 u(k 3 -k 2 ) u(k 2 -k 1 ) u(k 1 ) (k 2 -λ)(k 1 -λ) e ik 3 x
and hence 1 2πi

∂Dn(1/3) (T u (D -λ) -1 ) 2 Πu|1 1 λ 2 dλ = ∞ k 2 =0 ∞ k 1 =0 u(0 -k 2 ) u(k 2 -k 1 ) u(k 1 ) 1 2πi ∂Dn(1/3) 1 (k 2 -λ)(k 1 -λ)λ 2 dλ = - 1 n 2 u(-n) k 1 ≥0,k 1 =n u(n -k 1 ) u(k 1 ) k 1 -n - 1 n 2 k 2 ≥0,k 2 =n u(-k 2 ) u(k 2 -n) k 2 -n u(n) + u(-n) u(0) u(n) ∂Dn(1/3) 1 (n -λ) 2 λ 2 dλ .
Since by assumption u(0) = 0, the latter term vanishes. With j := k j -n for j = 1, 2 and using that

(17) u * (x) := u(-x) = j∈Z u(-j)e ijx
one then gets

1 2πi ∂Dn(1/3) (T u (D -λ) -1 ) 2 Πu|1 1 λ 2 dλ = - 1 n 2 u(-n) 1 ≥-n, 1 =0 u * (-n -1 ) u * (k 1 ) 1 - 1 n 2 2 ≥-n, 2 =0 u(-n -2 ) u( 2 ) 2 u * (-n) .
It is convenient to introduce for any n ≥ 1 and any u ∈ H -s c,0 with 0 ≤ s < 1/2 the operators

Q u,n : z = (z( )) ∈Z → (Q u,n [z](k)) k∈Z and Q u : z = (z( )) ∈Z → (Q u [z](k)) k∈Z where (18) Q u,n [z](k) := ≥-n, =0 | u(k -)| z( ) | | , (19) Q u [z](k) := =0 | u(k -)| z( ) | | .
By (the proof of) Lemma 1(ii) , for any 0 ≤ s < 1/2,

Q u,n , Q u : h -s (Z, C) → h -s (Z, C)
are bounded linear operators. More precisely, by Lemma 1(ii) the following holds:

Lemma 8. For any 0 ≤ s < 1/2, there exists a constant M s ≥ 1 so that for any n ≥ 1 and z ∈ h -s (Z, C)

Q u,n [z] -s ≤ Q u [z] -s ≤ M s u -s z -s .
With this notation introduced, one obtains the following estimates

1 2πi ∂Dn(1/3) T u (D -λ) -1 Πu|1 1 λ 2 dλ = 1 n 2 | u * (-n)|| u(-n)| and 1 2πi ∂Dn(1/3) (T u (D -λ) -1 ) 2 Πu|1 1 λ 2 dλ ≤ 1 n 2 Q u * ,n [| u * |](-n) | u(-n)| + 1 n 2 Q u,n [| u|](-n) | u * (-n)| where u := ( u(k)) k∈Z , | u| := (| u(k)|) k∈Z .
Proof of Proposition 2. Consider the integrand in (16) with m ≥ 1 arbitrary. Arguing as for the terms with m = 1 and m = 2, the term (T u (D -λ) -1 ) m Πu|1 can be written as

k m+1 ≥0 k j ≥0 u(k m+1 -k m ) u(k m -k m-1 ) • • • u(k 2 -k 1 ) u(k 1 ) (k m -λ) • • • (k 1 -λ) e ik m+1 x |1 = k j ≥0 u(-k m ) u(k m -k m-1 ) • • • u(k 2 -k 1 ) u(k 1 ) (k m -λ) • • • (k 1 -λ)
where in the sums above 1 ≤ j ≤ m. Using that by Cauchy's formula for derivatives of an analytic function f ,

1 (p -1)! d p-1 dµ p-1 f (µ) = 1 2πi ∂Dn(1/3) f (λ) (λ -µ) p dλ , one obtains | 1 2πi ∂Dn(1/3) (T u (D -λ) -1 ) m Πu|1 1 λ 2 dλ| ≤ m p=1 |J|=p k j ≥0 k j =n ∀j∈J k j =n ∀j∈J c | u(-k m )| m j=2 | u(k j -k j-1 )| | u(k 1 )|g k 1 ,...,km;p (n)
where in the latter sum, J runs over all subsets of {1, . . . , m} with |J| = p, J c := {1, . . . , m} \ J, and

g k 1 ,...,km;p (λ) := 1 (p -1)! d p-1 dλ p-1 ( 1 λ 2 j∈J c 1 k j -λ
) .

Using that |k j -n| ≥ 1 for any j ∈ J c one obtains the estimate

g k 1 ,...,km;p (n) ≤ 1 n 2 j∈J c 1 |k j -n| 1 (p -1)! # m,p
where # m,p is the number of terms of d p-1 dλ p-1 ( 1

λ 2 j∈J c 1 k j -λ ), counted with their multiplicities, # m,p = (|J c | + 2) • • • (|J c | + 2 + p -2) = (m -p + 2) • • • m . Hence 1 (p -1)! # m,p = m! (p -1)!(m -(p -1))! = m p -1 , yielding g k 1 ,...,km;p (n) ≤ 1 n 2 j∈J c 1 |k j -n| m p -1 .
Since by Lemma 9 below, for any J ⊂ {1, . . . , m} with |J| = p,

n≥1 n 2-2s k j ≥0 k j =n ∀j∈J k j =n ∀j∈J c 1 n 2 | u(-k m )|| u(k m -k m-1 )| • • • | u(k 2 -k 1 )|| u(k 1 )| j∈J c |k j -n| ≤ M s u -s m+1 and m p=0 m p = 2 m we then conclude that ∞ n=1 n 2-2s 1 2πi ∂Dn(1/3) (T u (D -λ) -1 ) m Πu|1 1 λ 2 dλ ≤ m p=1 m p M s u -s m+1 sup 1≤p≤m m p -1 ≤ 4M s u -s m+1 . It implies that for any u ∈ H -s c,0 with u -s ≤ r s, * := min(r s , 1/8M s ) , one has ∞ n=1 ∞ m=0 n 2-2s 1 2πi ∂Dn(1/3) (D -λ) -1 (T u (D -λ) -1 ) m 1|1 dλ ≤ 4M s u -s 1 -4M s u -s ≤ 8M s u -s ≤ 1 .
This finishes the proof of Proposition 2. It remains to prove the following lemma, used in the proof of Proposition 2. With the notation established in the proof of that proposition it reads as follows:

Lemma 9. For any m ≥ 1 and any nonempty subset J ⊂ {1, . . . , m},

n≥1 n -2s S n,m,J ≤ M s u -s m+1 ,
where

S n,m,J := k 1 ,...,km≥0 k j =n ∀j∈J k j =n ∀j∈J c | u(-k m )|| u(k m -k m-1 )| • • • | u(k 2 -k 1 )|| u(k 1 )| j∈J c |k j -n| .
Proof. Let m ≥ 1 and J ⊂ {1, . . . , m} = ∅. First observe that we can assume that J ⊂ {1, . . . , m} does not contain consecutive integers j, j + 1, since otherwise S n,m,J vanishes for any n ≥ 1 due to the assumption that u(0) = 0. We decompose J c into pairwise disjoint intervals of integers of maximal lengths, J c a , 1 ≤ a ≤ A,

J c = A a=1
J c a where J c 1 , . . . , J c A are listed in decreasing order. Note that

A a=1 |J c a | = |J c | = m -|J|
and that A equals |J|-1, |J|, or |J|+1, depending on whether |{1, m}∩ J| equals 2, 1, or 0. For any r ≥ 1, introduce

S n,r := k 1 ,...,kr≥0 k j =n ∀j | u(n -k r )|| u(k r -k r-1 )| • • • | u(k 2 -k 1 )|| u(k 1 -n)| r j=1 |k j -n| .
Setting j := k j -n in the latter sum, we obtain

S n,r = 1 ,..., r ≥-n j =0 ∀j | u(-r )|| u( r -r-1 )| • • • | u( 2 -1 )|| u( 1 )| r j=1 | j | = (Q u,n ) r [|û|](0) .
We recall that Q u,n is defined by (18) and that for any sequence z = (z( )) of complex numbers, |z| denotes the sequence (|z( )|) . Since

|z(0)| ≤ z -s , Lemma 8 implies that (20) S n,r ≤ (M s u -s ) r u -s ≤ (M s u -s ) r+1 ,
where M s ≥ 1 is the constant given by Lemma 8. Let us first consider the case where 1 ∈ J and m ∈ J. Setting r a := |J c a |, one has

S n,m,J = | u(-n)| A a=1 S n,ra | u(n)| ≤ (M s u -s ) |J c |+A | u(-n)|| u(n)| ≤ (M s u -s ) m-1 | u(-n)|| u(n)|
where we used that A = |J| -1 in the case at hand. Consequently

∞ n=1 n -2s S n,m,J ≤ (M s u -s ) m-1 ∞ n=1 n -2s | u(-n)|| u(n)| ≤ (M s u -s ) m+1 .
In order to deal with the other cases, we introduce * S n,r :=

k 1 ,...,kr≥0 k j =n ∀j | u(-k r )|| u(k r -k r-1 )| • • • | u(k 2 -k 1 )|| u(k 1 -n)| r j=1 |k j -n| = 1 ,..., r ≥-n j =0 ∀j | u(-n -r )|| u( r -r-1 )| • • • | u( 2 -1 )|| u( 1 )| r j=1 | j | = (Q u,n ) r [| u|](-n) and S * n,r := k 1 ,...,kr≥0 k j =n ∀j | u(n -k r )|| u(k r -k r-1 )| • • • | u(k 2 -k 1 )|| u(k 1 )| r j=1 |k j -n| = 1 ,..., r ≥-n j =0 ∀j | u(-r )|| u( r -r-1 )| • • • | u( 2 -1 )|| u( 1 + n)| r j=1 | j | .
Using u * (x) = u(-x), defined in (17), S * n,r can be written as

S * n,r = 1 ,..., r ≥-n j =0 ∀j | u * (-n -1 )|| u * ( 1 -2 )| • • • | u * ( r-1 -r )|| u * ( r )| r j=1 | j | = (Q u * ,n ) r [| u * |](-n) .
With these notations introduced, S n,m,J can be expressed as

S n,m,J = * S n,r 1 A a=2 S n,ra | u(n)| , if 1 ∈ J, m ∈ J , S n,m,J = | u(-n)| A-1 a=1 S n,ra S * n,r A , if 1 ∈ J, m ∈ J , S n,m,J = * S n,r 1 A-1 a=2 S n,ra S * n,r A , if 1 ∈ J, m ∈ J .
Note that by the definitions (18

) -(19), 0 ≤ Q u,n [|z|]( ) ≤ Q u [|z|]( ).
Hence in the case where 1 ∈ J and m ∈ J,

S n,m,J ≤ (Q u ) r 1 [| u|](-n)| u(n)| (M s u -s ) A a=2 (ra+1)
and therefore

∞ n=1 n -2s S n,m,J ≤ (M s u -s ) A a=2 (ra+1) ∞ n=1 n -2s (Q u ) r 1 [| u|](-n)|û(n)| ≤ (M s u -s ) A a=2 (ra+1) (Q u ) r 1 [| u|] -s u -s ≤ (M s u -s ) m+1 ,
since, in the case at hand, A = |J|. Using that u * -s = u -s , the remaining two cases can be dealt with in a similar way.

Proof of Theorem 1(ii) and Corollary 1 for u small

In this section we prove Theorem 1(ii) and Corollary 1 for potentials in H -s c,0 , 0 ≤ s < 1/2, near zero. In addition we discuss properties of the scaling factors κ n (u), n ≥ 1, and the normalizing constants µ n (u), n ≥ 1.

As a first step, we study the analytic extension of the generating function, introduced in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] and then further analyzed in [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]. This function is then used to define the scaling factors κ n , n ≥ 1, for the components of the quasi-moment map F , needed for the proof of Theorem 1(ii) for u small.

For u ∈ B -s c,0 (r s, * ) with r s, * > 0 given by Proposition 2, and λ in the resolvent set of L u , define

(21) H λ (u) := (L u -λ) -1 1|1 .
Up to a sign, the function H λ (u) equals the generating function, introduced in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] for u ∈ H 0 r . In view of Proposition 1 and Theorem 2 it follows that H λ (u) is a meromorphic function of λ with the set of poles contained in the spectrum of L u , all of them being simple. By the definition (13), for any n ≥ 0, F n (u) is the residue of H λ (u) at λ = λ n (u). We point out that F n (u) might vanish. Our first result says that H λ (u) can be written as an infinite sum: Lemma 10. For any u in B -s c,0 (r s, * ), the following identities hold:

(i) F n (u) = -P n (u)1|1 , ∀ n ≥ 0; (ii) H λ (u) = n≥0 Fn(u) λ-λn(u) .
Proof. Item (i) follows from the definition (13) of F n (u) and definition [START_REF] Nakamura | Backlund transform and conservation laws of the Benjamin-Ono equation[END_REF] of the projection P n (u). Towards (ii), note that by Proposition 2, for any u ∈ B -s c,0 (r s, * ), n≥0

Fn(u) λ-λn(u) is a meromorphic function, having the same poles as the meromorphic function H λ (u). For any λ with λ ≤ -1/3, the functions n≥0 Fn λ-λn and H λ are both analytic functions on B -s c,0 (r s, * ). It then suffices to show that they coincide for u real valued: for any real valued potential u in B -s c,0 (r s, * ), let f n ≡ f n (•, u), n ≥ 0, be the L 2 -orthonormal basis of H + , introduced in [3] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2), where for any n ≥ 0, f n is an eigenfunction of L u , corresponding to the eigenvalue λ n (u). Then 1 = n≥0 1|f n f n and hence for any λ with λ ≤ -1/3,

(22) H λ (u) = n≥0 1|f n (L u -λ) -1 f n |1 = n≥0 1|f n f n |1 λ n -λ . Since F n (u) is the residue of H λ (u) at λ = λ n (u), it then follows that 1|f n • f n |1 = -F n (u),
showing that the claimed identity holds for u real valued.

Note that by the definition (21), one has H λ (0) = -1/λ. Hence F n (0) = 0 for any n ≥ 1 and F 0 (0) = -1.

Lemma 11. Assume that u ∈ B -s c,0 (r s, * ) with 0 ≤ s < 1/2. Then for any n ≥ 1 with γ n (u) = 0 one has

H λ n-1 (u)+1 (u) = 0 . Proof. Let u ∈ B -s
c,0 (r s ) with 0 ≤ s < 1/2 and n ≥ 1. Assume that γ n (u) = 0 and let h k , k ≥ 0, be a set of eigenfunctions of L u , corresponding to the eigenvalues λ k ≡ λ k (u). By [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF] one has

(23) L u Sh n-1 = (λ n-1 + 1)Sh n-1 -uSh n-1 |1 1 .
Since by assumption, γ n (u) = 0 and hence λ n-1 + 1 is in the resolvent set of L u , it follows that

Sh n-1 = -uSh n-1 |1 (L u -λ n-1 -1) -1 1 .
Taking the inner product with the constant function 1 of both sides of the latter identity and noting that Sh n-1 |1 = 0 yields

0 = -uSh n-1 |1 (L u -λ n-1 -1) -1 1|1 = -uSh n-1 |1 H λ n-1 +1 (u) .
By Lemma 7(i), (G3) implies (G4). Since by assumption γ n (u) = 0, it means that uSh n-1 |1 = 0, implying that H λ n-1 +1 (u) = 0.

Lemma 12. For any u ∈ B -s c,0 (r s, * ) and any n ≥ 0,

|H λ (u)| ≥ 1 |λ| • 2 3 , ∀λ ∈ Vert n (1/4) , and
|H λ (u)| ≤ 1 |λ| • 4 3 , ∀λ ∈ Vert n (1/4) .
Proof. Let u ∈ B -s c,0 (r s, * ) and n ≥ 0. Expanding (L u -λ) -1 = (Dλ) -1 (Id -T u (D -λ) -1 ) -1 in its Neumann series one gets

H λ (u) = - 1 λ 1 + k≥1 (T u (D -λ) -1 ) k 1|1 .
Since by Corollary 2,

T u (D -λ) -1 H -s;n + →H -s;n + ≤ 1 4 , ∀λ ∈ Vert n (1/4),
and by the definition ( 6), 1 -s;n = n -s as well as 1 s;n = n s , it follows that

| (T u (D -λ) -1 ) k 1|1 | ≤ ( 1 4 ) k 1 -s;n 1 s;n ≤ ( 1 4
)

k
and hence

|H λ (u)| ≥ 1 |λ| 1 - k≥1 ( 1 4 ) k = 1 |λ| • 2 3 , ∀λ ∈ Vert n (1/4) .
Similarly, one gets

|H λ (u)| ≤ 1 |λ| 1 + k≥1 ( 1 4 ) k = 1 |λ| • 4 3 , ∀λ ∈ Vert n (1/4) .
Lemma 13. Let u ∈ B -s c,0 (r s, * ). Then for any n ≥ 1, the difference ZP u;n of the number of zeroes of H λ (u) in D n (1/3) and the number of its poles in D n (1/3) vanishes. For n = 0, the difference ZP u;0 equals -1. As a consequence, H λ (u) has no zeroes in D 0 (1/3) and for any n ≥ 1, at most one zero in D n (1/3).

Proof. Let u be in B -s c,0 (r s, * ) with 0 ≤ s < 1/2 and n ≥ 0. By Lemma 12, H λ (u) has no poles and no zeroes on the circle ∂D n (1/3). Since H λ (u) is meromorphic, it then follows by the argument principle that ZP u;n is constant on B -s c,0 (r s, * ). Recall that for u = 0, one has H λ (0) = -1/λ, implying that ZP 0;0 = -1 and for any n ≥ 1, ZP 0;n = 0.

Corollary 4. Assume that u ∈ B -s c,0 (r s, * ) with 0 ≤ s < 1/2. For any n ≥ 1, F n (u) = 0 if and only if γ n (u) = 0.
Proof. Let u be in B -s c,0 (r s, * ) with 0 ≤ s < 1/2 and n ≥ 1. Assume that γ n (u) = 0. By Lemma 11, the generating function vanishes at λ = λ n-1 (u) + 1 and hence by Lemma 13, λ n (u) is a pole of H λ , implying together with Lemma 10(ii) that F n (u) = 0. To prove the converse, assume that γ n (u) = 0. Then by Lemma 7(ii) we conclude that P n (u)1|1 = 0 and hence by Lemma 10(i) that F n (u) = 0.

For any u ∈ B -s c,0 (r s, * ), define for n ≥ 1,

η n (λ, u) := - λ -λ n (u) λ -λ n-1 (u) -1 (λ -λ 0 (u))H λ (u) , ∀λ ∈ Vert n (1/4) ,
and for n = 0,

η 0 (λ, u) := -(λ -λ 0 (u))H λ (u) , ∀λ ∈ Vert 0 (1/4).
The functions η n , n ≥ 0, are analytic in λ on their domains of definition. By Lemma 11 and Lemma 13, for any n ≥ 0, the function η n (λ, u) extends analytically to D n (1/4) and hence for any λ ∈ D n (1/4) one has by Cauchy's formula

(24) η n (λ, u) = 1 2πi ∂Dn(1/3) η n (µ, u) µ -λ dµ.
As a consequence, for any n ≥ 0,

η n : D n (1/4) ∪ Vert n (1/4) × B -s c,0 (r s, *
) → C is analytic and by Lemma 13 vanishes nowhere. Since H λ (0) = -1/λ one infers that for any n ≥ 0, η n (λ, 0) = 1 for λ in Vert n (1/4). Lemma 14. For any u ∈ B -s c,0 (r s, * ) and n ≥ 0,

1 C ≤ |η n (λ, u)| ≤ C , ∀λ ∈ D n (1/4) ,
where

C := 5 • 7 • 4.
Proof. Let u ∈ B -s c,0 (r s, * ) and n ≥ 0. By (57)

η n (λ, u) = 1 2πi ∂Dn(1/3) η n (µ, u) µ -λ dµ , ∀λ ∈ D n (1/4) .
We

|µ -λ n (u)| |µ -λ n-1 (u) -1| 1 |µ -λ| ≤ 1 3 + 1 4 ( 1 3 -1 4 ) 2 = 7
• 12 and by Lemma 12

|(µ -λ 0 )H µ (u)| ≤ n + 1 3 + 1 4 n -1 3 4 3 ≤ 4 .
Hence for any µ ∈ ∂D n (1/3) and λ ∈ D n (1/4), one has by the definition of

η n |η n (µ, u)| |µ -λ| = |µ -λ n (u)| |µ -λ n-1 (u) -1| |(µ -λ 0 (u))H µ (u)| |µ -λ| ≤ 4 • 7 • 12 .
Formula (57) then yields

|η n (λ, u)| ≤ C , ∀λ ∈ D n (1/4) ,
where C = 5 • 7 • 4. In case n = 0, one argues similarly and obtains

|η 0 (µ, u)| |µ -λ| = |(µ -λ 0 (u))H µ (u)| |µ -λ| ≤ 1 3 + 1 4 1 3 • 4 3 • 12 ≤ C , yielding also |η 0 (λ, u)| ≤ C for any λ ∈ D n (1/4
∈ D n (1/4) (25) 1 η n (λ, u) = 1 2πi ∂Dn(1/3) 1 η n (µ, u) 1 µ -λ dµ.
Let us again first consider the case n ≥ 1. For any µ ∈ ∂D n (1/3) and

λ ∈ D n (1/4), |µ -λ n-1 (u) -1| |µ -λ n (u)| 1 |µ -λ| ≤ 1 3 + 1 4 ( 1 3 -1 4 ) 2 = 7 • 12 and by Lemma 12 1 |(µ -λ 0 )H µ (u)| ≤ |µ| |λ 0 -µ| 3 2 ≤ n + 1 3 n -1 3 -1 4 3 2 ≤ 5 .
Hence for any µ ∈ ∂D n (1/3) and λ ∈ D n (1/4), one has by the definition of η n

1 |η n (µ, u)| 1 |µ -λ| = |µ -λ n-1 (u) -1| |µ -λ n (u)| 1 |(µ -λ 0 (u))H µ (u)| 1 |µ -λ| ≤ 7 • 12 • 5 . Formula (25) then yields 1 |η n (λ, u)| ≤ C , ∀λ ∈ D n (1/4) ,
where C is again the constant 5•7•4. In case n = 0, one argues similarly and obtains

1 |η 0 (µ, u)| 1 |µ -λ| = 1 |(µ -λ 0 (u))H µ (u)| 1 |µ -λ| ≤ |µ| |µ -λ 0 | 3 2 ≤ 1 3 • 12 • 3 2 ≤ C ,
yielding also

1 |η 0 (λ,u)| ≤ C for any λ ∈ D n (1/4). Define for any n ≥ 1 κ n : B -s c,0 (r s, * ) → C, u → 1 λ n (u) -λ 0 (u) η n (λ n (u), u)
and for n = 0, κ 0 : B -s c,0 (r s, * ) → C, u → η 0 (λ 0 (u), u) . Being a composition of analytic functions, the κ n are analytic maps. They satisfy the following estimates. Proposition 3. For any n ≥ 0, the map κ n : B -s c,0 (r s, * ) → C is analytic and satisfies for any u ∈ B -s c,0 (r s, * ) the estimate

|κ n (u)| ≤ 2C 1 n , 1 |κ n (u)| ≤ 2C n , ∀ n ≥ 0 ,
where C = 5 • 7 • 4 is the constant of Lemma 14. In particular, the map

B -s c,0 (r s, * ) → ∞ + , u → ( 1 nκn(u) ) n≥1 is analytic. Proof.
Let n ≥ 0. We have already seen that κ n is analytic. In the case u real valued, κ n (u) is real valued as well (cf. also [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF](s = 0), [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]

(0 < s < 1/2)). Let u ∈ B -s c,0 (r s, * )
. By the definition of κ n (u) and Lemma 14, the claimed estimates hold for n = 0. If n ≥ 1, then

|κ n (u)| = 1 |λ n (u) -λ 0 (u)| |η n (λ n (u), u)| ≤ 1 n -1 2 C ≤ 2C n and |κ n (u)| = 1 |λ n (u) -λ 0 (u)| |η n (λ n (u), u)| ≥ 1 n + 1 2 1 C ≥ 1 2C 1 n .
By [START_REF] Kappeler | Ergeb. der Math. und ihrer Grenzgeb[END_REF]Theorem A.3], the latter estimate implies that the map

B -s c,0 (r s, * ) → ∞ + , u → ( 1 nκ n (u) ) n≥1
is analytic.

When combined with our results on the residues F n (u) of the generating function and the gap lengths γ n (u), Proposition 3 yields the following

Corollary 5. Let 0 ≤ s < 1/2. For any u in B -s c,0 (r s, * ), (26) F n (u) = -κ n (u)γ n (u) , ∀n ≥ 1 .
Proof. Let n ≥ 1. The identity (58) holds for u real valued in B -s c,0 (r s, * ) (cf. Remark 14 and Remark 3). By Proposition 2, Proposition 3, and Theorem 2, the functions F n , κ n , and γ n are analytic on B -s c,0 (r s, * ) and hence the identity continues to hold on B -s c,0 (r s, * ). We now have all the ingredients to prove Theorem 1(ii) for potentials in a neighborhood of 0 in H -s c,0 . It reads in this case as follows. Theorem 3. Let 0 ≤ s < 1/2 and let r s, * > 0 be the radius, given by Proposition 2. Then the moment map Theorem 2 and Theorem 3 are used to prove Corollary 1 for potentials near zero. We reformulate it slightly and include a result on the product representation of κ n (u), n ≥ 0: Corollary 6. Let 0 ≤ s < 1/2 and let r s, * > 0 be the radius, given by Proposition 2. For any u in B -s c,0 (r s, * ), the following holds: (i) The generating function H λ (u) admits the product representation

Γ : B -s c,0 (r s, * ) → 1,1-2s + , u → (γ n (u)) n≥1 is
H λ (u) = 1 λ 0 (u) -λ p≥1 1 - γ p (u) λ p (u) -λ
where the infinite product is absolutely convergent. (ii) For any n ≥ 0,

λ n (u) = n - k≥n+1 γ k (u)
where the infinite sum is absolutely convergent. (iii) If s = 0, one has

1 2π 2π 0 u 2 dx = 2 k≥1 kγ k
where the infinite sum is absolutely convergent. (iv) For any n ≥ 1, the function κ n (u) admits the product representation

κ n (u) = 1 λ n (u) -λ 0 (u) p =n 1 - γ p (u) λ p (u) -λ n (u)
whereas for n = 0 one has

κ 0 (u) = p≥1 1 - γ p (u) λ p (u) -λ 0 (u)
.

All these infinite products are absolutely convergent. For u = 0, one has κ 0 (0) = 1 and κ n (0) = 1/n for any n ≥ 1.

Proof. (i) First note that the claimed identity holds for real valued poentials u ∈ B -s c,0 (r s, * ) cf. [3, Proposition 3.1] (s = 0), [4, (32)] (0 < s < 1/2) and that for any λ ∈ C with λ < -1/3, H λ (u) is analytic on B -s c,0 (r s, * ). Since λ p (u) ∈ D n (1/4) is analytic for any p ≥ 0 and by Theorem 3, Γ : B -s c,0 (r s, * ) → 1,1-2s + is analytic, it then follows that for any such λ, the infinite product p≥1 1 -γp(u) λp(u)-λ is absolutely convergent and induces an analytic map B -s c,0 (r s, * ) → C. Altogether one concludes that the claimed product representation holds on B -s c,0 (r s, * ). Item (ii), (iii), and (iv) are proved by similar arguments, using that the stated identities hold for real valued u: For item (ii) cf. [3, (3.13)] (s = 0), [4, (29)] (0 < s < 1/2), for item (iii) cf. [3, Proposition 3.1], and for item (iv) cf. [3, Corollary 3.4] (s = 0), [4, (26), ( 27)](0 < s < 1/2). Theorem 3 and Corollary 6 lead to estimates of the maps κ n , n ≥ 1, which will be used in subsequent work for the analytic extension of the Birkhoff map, constructed on H -s r,0 in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [4] (0 < s < 1/2). By Theorem 3, for any 0 ≤ s < 1/2, there exists 0 < r s, * * ≤ r s, * so that (27)

k≥1 |γ k (u)| ≤ 1 5 , ∀u ∈ B -s c,0 (r s, * * ) .
Proposition 4. Let 0 ≤ s < 1/2. For any u in B -s c,0 (r s, * * ),

|nκ n (u) -1| ≤ 7 12 e 1/3 < 1 , ∀n ≥ 1 .
Hence for any n ≥ 1, the principal branch of the square root of nκ n is well defined on B -s c,0 (r s, * * ) and it follows that

√ nκ n n≥1 : B -s c,0 (r s, * * ) → ∞ + , u → nκ n (u) n≥1
is analytic.

Proof. Let 0 ≤ s < 1/2, n ≥ 1, and u ∈ B -s c,0 (r s, * * ). To simplify notation, we do not indicate the dependence on u in the course of the proof. By the product representation of κ n , n ≥ 1, of Corollary 1(iv), nκ n -1 = I n + II n where

I n := n λ n -λ 0 -1 p =n 1- γ p λ p -λ n , II n := p =n 1- γ p λ p -λ n -1 .
We begin with estimating the term I n . By Corollary 1(ii), λ n -

λ 0 = n + n k=1 γ k and hence n λ n -λ 0 -1 = - 1 n n k=1 γ k 1 + 1 n n k=1 γ k . Since by assumption, k≥1 |γ k | ≤ 1 5 , it follows that | n λ n -λ 0 -1| ≤ 1/5 1 -1/5 ≤ 1 4 .
Next we estimate the infinite product p =n 1 -γp λp-λn . One has

| p =n 1 - γ p λ p -λ n | ≤ p =n 1 + |γ p | |λ p -λ n | ≤ exp p =n log(1 + |γ p | |λ p -λ n |
) .

Since log(1 + |x|) ≤ |x| and

|λ p -λ n | ≥ |p -n| - n k=1 |γ k | ≥ 1 -1/5 , one infers that | p =n 1 - γ p λ p -λ n | ≤ exp 5 4 p =n |γ p | ≤ e 1/4 .
Altogether we have shown that

|I n | ≤ 1 4 e 1/4 , ∀ n ≥ 1 .
It remains to estimate II n = p =n 1 -γp λp-λn -1. First note that since for any p = n,

| γ p λ p -λ n | ≤ |γ p | 4/5 ≤ 1/4 ,
the principal branch of the logarithm of 1 -γp λp-λn is well defined and hence one has

p =n 1 - γ p λ p -λ n = exp p =n log 1 - γ p λ p -λ n .
Using that for any x ∈ C, We finish this section with a discussion of the spectral invariants µ n ≡ µ n (u), n ≥ 1, introduced in [3, (4.9)] for potentials u ∈ L 2 r,0 ,

|e x -1| = |x
µ n = 1 - γ n λ n -λ 0 p =n 1 -γp λp-λn 1 - γp λp-λ n-1 -1
.

By algebraic transformations one obtains

µ n = 1 - γ n λ n -λ 0 p =n 1 - γ n λ p-1 -λ n-1 1 + γ n λ p -λ n , or, by writing λ p-1 -λ n-1 = λ p -λ n + γ n -γ p , ( 28
) µ n = 1 - γ n λ n -λ 0 p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n )
.

By Theorem 3 one sees that µ n , n ≥ 1, extend analytically to B -s c,0 (r s, * ) for any 0 ≤ s < 1/2. Note that for any u ∈ B -s c,0 (r s, * ) and n ≥ 1, µ n (u) > 0 and in case γ n (u) = 0, µ n (u) = 1. Theorem 3 and Corollary 6 lead to estimates of the maps µ n , n ≥ 1, which will be used in subsequent work to show that for any 0 ≤ s < 1/2, the Birkhoff map analytically extends to a neighborhood of zero in H -s c,0 . Proposition 5. Let 0 ≤ s < 1/2 and let r s, * * be given as in (27). Then for any u in B -s c,0 (r s, * * ),

|µ n (u) -1| ≤ 2e 1/16 |γ n (u)| < 2 5 e 1/16 ≤ 1 2 , ∀n ≥ 1 .
Hence for any n ≥ 1, the principal branch of the square root of µ n is well defined on B -s c,0 (r s, * * ) and it follows that

√ µ n n≥1 : B -s c,0 (r s, * * ) → ∞ + , u → µ n (u) n≥1
is analytic.

Proof. We argue as in the proof of Proposition 4. Let 0 ≤ s < 1/2, n ≥ 1, and u ∈ B -s c,0 (r s, * * ). Again, to simplify notation, we do not indicate the dependence on u in the course of the proof. By the definition of µ n one has µ n -1 = I n + II n where

I n := 1 - γ n λ n -λ 0 -1 p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) , II n := p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) -1 .
We begin with estimating the term I n . Since by Corollary 1(ii), λ nλ 0 = n + n k=1 γ k and by asssumption n k=1 |γ k | ≤ 1/5 one has

| 1 - γ n λ n -λ 0 -1| = | - γ n n + n k=1 γ n | ≤ |γ n | n -1/5 ≤ 5 4 |γ n | .
Next we estimate the infinite product p =n 1 -γnγp (λ p-1 -λ n-1 )(λp-λn) . One has

| p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) | ≤ p =n 1 + |γ n | |γ p | |λ p-1 -λ n-1 ||λ p -λ n | ≤ exp p =n log(1 + |γ n | |γ p | |λ p-1 -λ n-1 ||λ p -λ n |
) .

Since log(1 + |x|) ≤ |x| and for any p = n, It remains to estimate II n = p =n 1 -γnγp (λ p-1 -λ n-1 )(λp-λn) -1. First note that since for any p = n,

|λ p-1 -λ n-1 |, |λ p -λ n | ≥ |p -n| - ∞ k=1 |γ k | ≥ 1 -1/5 , one infers that | p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) | ≤ exp |γ n | 5
| γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) | ≤ |γ n | 4/5 |γ p | 4/5 ≤ 1/16 ,
the principal branch of the logarithm of 1 -γnγp (λ p-1 -λ n-1 )(λp-λn) is well defined and hence one has

p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) = exp p =n log 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) .
Using that for any x ∈ C,

|e x -1| = |x 1 0 e tx dt| ≤ |x|e |x|
and that for any y ∈ C with |y| ≤ a < 1,

| log(1 -y)| = | 1 0 -y 1 -ty dt| ≤ 1 1 -a |y| , one concludes that | p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) -1| ≤ 5
| p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) -1| ≤ 5 16 |γ n |e 1/16 .
Combining all the estimates derived yields the desired bound,

|µ n (u) -1| ≤ 5 4 |γ n |e 1/16 + 5 16 |γ n |e 1/16 ≤ 2|γ n |e 1/16 .
The claimed analyticity of the map u → µ n (u) n≥1 then follows from [START_REF] Kappeler | Ergeb. der Math. und ihrer Grenzgeb[END_REF]Theorem A.3]. Remark 4. Going through the proof of Proposition 5 one verifies that for any 0 ≤ s < 1/2, there exists a constant C > 0 so that for any u ∈ B -s c,0 (r s, * * ),

|µ n (u) -1| ≤ C |γ n (u)| n + |γ n (u)| n 1-2s 1 n , ∀n ≥ 1 ,
where the constant C > 0 only depends on s. Indeed, one verifies that for any 0 ≤ s < 1/2, there exists a constant C so that the term I in the proof of Proposition 5 can be bounded by C |γn(u)| n , whereas

|II| ≤ C |γ n | n 1-2s A n , A n := n 1-2s p =n |γ p | (p -n) 2 ,
and n≥1 A n is estimated by splitting it into the following three parts,

n≥1 n 1-2s p>n |γ p | (p -n) 2 ≤ p≥1 p 1-2s |γ p | n =p 1 (p -n) 2 < ∞ , n≥1 n 1-2s n/2≤p<n |γ p | (p -n) 2 p≥1 p 1-2s |γ p | n≥1 1 n 2 < ∞ , n≥1 n 1-2s p<n/2 |γ p | (p -n) 2 p≥1 |γ p | n =p n 1-2s n 2 < ∞ .
Combining these estimates yields the claimed one.

Proof of Theorem 1(i)

In this section we analyze the spectrum spec(L u ) of the Lax operator L u for u ∈ H -s c,0 with imaginary part u which is small in H -s c,0 . Whereas the part of spec(L u ) in any half plane { λ ≤ c} can be analyzed by standard perturbation theory, the part in the half space { λ ≥ c} relies on the approximation of u by finite gap potentials.

We first need to make some preliminary considerations and introduce some additional notation. Recall that for any s ∈ R, h s ≡ h s (Z, C) denotes the weighted 2 -sequence space of sequences z = (z(k)) k∈Z with values in C. It is convenient to extend the n-shift operator S n , introduced in (4), to all of h s ,

S n : h s → h s , (S n z)(k) := z(k + n) ,
and correspondingly, to define the shifted norm f s;n for any

f ∈ H s c by f s;n := S n f s , f := ( f (k)) k∈Z .
It is straightforward to verify that for any two sequences z (j) = (z (j) (k)) k∈Z , j = 1, 2, for which the convolution z (1) * z (2) is well defined, one has for any n ≥ 1, S n (z (1) * z (2) ) = (S n z (1) ) * z (2) . We claim that for any 0 ≤ s < 1/2, (29)

f g -s;n ≤ C s f -s;n g 1-s , ∀ f ∈ H -s c , g ∈ H 1-s c ,
where C s > 0 is the constant of Lemma 1. Indeed, since

S n f g = S n ( f * g) = (S n f ) * g = F F -1 (S n f ) g ,
where F denotes the Fourier transform and F -1 its inverse, one has by Lemma 1,

f g -s;n = S n f g -s ≤ C s F -1 (S n f ) -s g 1-s ≤ C s f -s;n g 1-s ,
which is the claimed estimate. We will need the following Lemma 15. For any

0 ≤ s < 1/2, f, v ∈ H -s c , g ∈ H 1-s c
, N ≥ 1, n ≥ N , and 0 < ρ ≤ 1/3, the function

I(λ)(x) := k≥N f | ge ikx k -λ Π(vge ikx ) , λ ∈ Vert n (ρ) ,
satisfies the following estimate

I(λ) -s;n ≤ C 3 s ρ v -s g 2 1-s f -s;n , ∀ λ ∈ Vert n (ρ), ∀ n ≥ N ,
where C s > 0 is the constant of Lemma 1.

Proof. By Lemma 1, f ḡ and vg are in H -s c . Since f ḡ(k +n) = S n ( f ḡ)(k ) one has by the definition of the shifted norm that for any λ ∈ Vert n (ρ)

I(λ) 2 -s;n = ≥-n 1 2s k ≥N -n | vg( -k )| |S n ( f ḡ)(k )| |k -(λ -n)| 2 .
Applying Lemma 1 and Lemma 4 one concludes that

I(λ) -s;n ≤ C s ρ vg -s S n ( f ḡ) -s , ∀ λ ∈ Vert n (ρ) .
Using Lemma 1 once more it then follows from (29) that

I(λ) -s;n ≤ C 3 s ρ v -s g 2 1-s f -s;n .
We will apply Lemma 15 with g being of the form g(x) = e ih(x) where h is a real valued function in H 1-s r,0 . By Moser's composition estimates one has

(30) e ih(•) 1-s ≤ C s,1 (1 + h 1-s ) where C s,1 ≥ 1 is a constant, depending only on 0 ≤ s < 1/2. In order to show that for any λ ∈ Vert n (ρ), L u -λ : H 1-s + → H -s + is invertible for u ∈ H -s
c,0 near the real valued potential w ∈ H -s r,0 , we argue as follows: From [3, Theorem 1.1] it follows that for any 0 ≤ s < 1/2, the set ∪ N ≥1 U N is dense in H -s r,0 , where according to [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]Section 7] for any N ≥ 1, U N denotes the set of potentials w ∈ H 0 r,0 so that

γ N (w) > 0 , γ n (w) = 0 , ∀ n ≥ N + 1 .
Furthermore, for any N ≥ 1, U N ⊂ ∩ m≥1 H m r,0 and for any w

∈ U N λ n (w) = n , f n (•, w) = e i∂ -1
x w e inx , ∀ n ≥ N , where we recall that for any k ≥ 0, f k (•, w) denotes the eigenfunction, corresponding to the eigenvalue λ k (w) of L w , canonically normalized as in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. We remark that ∂ -1 x w(x) is in H 1-s r,0 . For any M > 0, 0 ≤ s < 1/2, and 0 < ρ ≤ 1/3, one has

(31) C M,s,ρ := 1 ρ C 3 s C 2 s,1 (2 + M ) 2 > 12 .
Given any w ∈ H -s r,0 with w = 0 and w -s ≤ M , it follows from [3, Theorem 1.1] that there exist N ≥ 1 and a potential w 0 ∈ U N so that

(32) w -w 0 -s ≤ 1/8C M,s,ρ , γ n (w 0 ) = γ n (w), ∀n ≤ N.
Note that by the definition of U N , one has γ n (w 0 ) = 0 for any n > N .

If w is itself a finite gap potential, we might choose w 0 = w.

Lemma 16. For any M > 0, 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, w ∈ H -s r,0 with w -s ≤ M , and w 0 ∈ U N with N ≥ 1, satisfying (32), there exists n 0 > N so that for any v ∈ H -s c,0 and n ≥ n 0 ,

T v (L w 0 -λ) -1 H -s;n + →H -s;n + ≤ 2C M,s,ρ v -s , ∀λ ∈ Vert n (ρ)
, where H -s;n + is the Hilbert space H -s + , endowed with the inner product, associated with the shifted norm • -s;n defined in [START_REF] Grébert | The defocusing NLS and its normal form[END_REF].

Proof. Let f k ≡ f k (•, w 0 ), k ≥ 0,
be the eigenfunctions of L w 0 , canonically normalized as in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. Since w 0 ∈ U N , one has for any n ≥ N , λ n ≡ λ n (w 0 ) = n and f n = g ∞ e inx where g ∞ := e i∂ -1

x w 0 . Furthermore, for any λ ∈ Vert n (ρ), L w 0 -λ :

H 1-s + → H -s + is invertible. For any v ∈ H -s c,0 and f ∈ H -s + , write T v (L w 0 -λ) -1 f = I(λ) + II(λ)
where

I(λ) := k≥N f | f k λ k -λ T v (f k ) , II(λ) := N -1 k=0 f | f k λ k -λ T v (f k ) .
Then

I(λ) = k≥N f | g ∞ e ikx k -λ Π(vg ∞ e ikx )
and since by (30),

g ∞ 1-s ≤ C s,1 (1 + ∂ -1 x w 0 1-s ) ≤ C s,1 (1 + w 0 -s ) ≤ C s,1
(2 + M ) , it then follows from Lemma 15 that for any n ≥ N,

I(λ) -s;n ≤ C 3 s ρ v -s g ∞ 2 1-s f -s;n ≤ C M,s,ρ v -s f -s;n .
To obtain the required estimate of II(λ) we will choose n 0 ≥ N sufficiently large. Note that

II(λ) -s;n ≤ N -1 k=0 | f | f k | |λ k -λ| T v (f k ) -s;n .
Taking into account Lemma 3, one has

N -1 k=0 | f | f k | ≤ N -1 k=0 f -s f k s ≤ N max 0≤k≤N -1 f k s f -s;n 2 s n s
and by Lemma 1 (with σ := (s + 1/2)/2), for any 0

≤ k ≤ N -1, T v (f k ) -s;n ≤ C s v -s f k 1-σ;n ≤ C s v -s max 0≤k≤N -1 f k 1-σ;n . Furthermore, f k 1-σ;n ≤ 2 s n 1-σ f k 1-σ (cf. Lemma 3) and for any n ≥ N, λ ∈ Vert n (ρ) (cf. by Lemma 4), |λ -λ k | ≥ |n -k| - 1 2 ≥ n -N + 1 2 , ∀ 0 ≤ k ≤ N -1 ,
where we used that λ k (w 0 ) = k ->k γ (w 0 ) ≤ k. Combining the above estimates yields

II(λ) -s;n ≤ n n -N + 1/2 N C s max 0≤k≤N f k 1-σ 2 4 s n σ-s v -s f -s;n .
By choosing n 0 > N sufficiently large it follows that

II(λ) -s;n ≤ C M,s,ρ v -s f -s;n ,
yielding the claimed estimate when combined with the one obtained for I(λ) -s;n .

Lemma 16 yields the following Corollary 7. Let M > 0, 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, w ∈ H -s r,0 with w -s ≤ M , w 0 ∈ U N with N ≥ 1, satisfying (32), and n 0 > N so that Lemma 16 holds. Then for any n ≥ n 0 and λ ∈ Vert n (ρ),

T v (L w 0 -λ) -1 H -s;n + →H -s;n + ≤ 1/2 , ∀ v ∈ B -s c,0 (w 0 , 1/4C M,s,ρ ) ,
where C M,s,ρ is given by (31).

Remark 5. Note that by (32), B -s c,0 (w, 1/8C M,s,ρ ) ⊂ B -s c,0 (w 0 , 1/4C M,s,ρ ). As a last ingredient of the proof of the Counting Lemma for potentials in H -s c,0 near H -s r,0 we need an extension of [3, Lemma 5.1] to H -s r,0 . To state it denote for any w ∈ H -s r,0 , n ≥ 1, and K ≥ 1, by Box K,n (w) the closed rectangle in C, given by the set of complex numbers, satisfying

-K + λ 0 (w) ≤ (λ) ≤ λ n (w) + 1/2 , | (λ)| ≤ K.
Let r 0 := 0, τ 0 := λ 0 (w) and for any k ≥ 1, Lemma 17. Let 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, K ≥ 1, and n ≥ 0. For any w ∈ H -s r,0 , there exists ε ≡ ε s,ρ,K,n > 0 so that for any u ∈ B c,0 (w, ε)

r k := γ k (w)/2 , τ k := λ k (w) -γ k (w)/2 .
and 0 ≤ k ≤ n, # spec(L u ) ∩ D τ k (r k + ρ) = 1 and spec(L u ) ∩ Box K,n (w) ⊂ 0≤k≤n D τ k (r k + ρ). The unique eigenvalue of L u in D τ k (r k + ρ) is denoted by λ k (u).
Proof. The proof of Lemma 5.1 in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] can easily be adapted to the setup at hand and hence is omitted. Now we have all the ingredients to prove the Counting Lemma. To state it, we define for any τ ∈ C, ν, ν > 0, and r > 0,

Vert τ (r; ν, ν ) := {λ ∈ C : |λ -τ | ≥ r; τ -ν ≤ λ ≤ τ + ν }. It is convenient to also define Vert τ (r; ∞, ν ) := {λ ∈ C : |λ -τ | ≥ r; λ ≤ τ + ν } .
Lemma 2, Corollary 7, and Lemma 17 yield a proof of Theorem 1(i), which we reformulate slightly as follows.

Theorem 4. (Counting Lemma) Let 0 ≤ s < 1/2, 0 < ρ ≤ 1/3, and M > 0. For any w ∈ H -s r,0 with w -s ≤ M , choose w 0 ∈ U N with N ≥ 1, satisfying (32), and n 0 > N so that Lemma 16 holds. Then there exists a neighborhood U -s w of w in H -s c,0 , contained in B -s c,0 (w, 1/8C M,s,ρ ), with the following properties: for any u ∈ U -s w , the spectrum spec(L u ) of the operator L u is discrete, consists of simple eigenvalues only, and for any n ≥ 0, one has

# spec(L u ) ∩ D τn (r n + ρ) = 1 and spec(L u ) ∩ Vert τn (r n + ρ; ν n , ν n+1 ) = ∅.
Here τ 0 := λ 0 (w), ν 0 = ∞, r 0 := 0 and for any 1 ≤ n < n 0 ,

r n := γ n (w)/2, τ n := λ n (w) -γ n (w)/2 , ν n := (τ n -τ n-1 )/2 , whereas r n 0 := 0 , τ n 0 := n 0 , ν n 0 := 1 2 1 + γ n 0 -1 (w)/2 + k≥n 0 γ k (w) ,
and for any n ≥ n 0 + 1,

r n := 0 , τ n := n , ν n := 1/2 .
In particular, D τn (r n + ρ) = D n (ρ) for any n ≥ n 0 and Vert n 0 (ρ) ⊂ Vert τn (r n 0 + ρ; ν n 0 , ν n 0 +1 ) whereas

Vert τn (r n + ρ; ν n , ν n+1 ) = Vert n (ρ) , ∀ n ≥ n 0 + 1 . • λ n (w) + 1/2 • (λ n (w) + 1/2, K) • K • -K + λ 0 (w) µ = λ ν = λ • τ 0 • τ 1 • τ 2 • τ n Box K,n (w)

Quasi-moment map

In this section, we study the quasi-moment map

F (u) = (F n (u)) n≥1 . Recall that for any u ∈ H -s r,0 and n ≥ 1, the component F n (u) of F (u) is given by -| 1|f n (•, u) | 2 = -γ n κ n and hence F (u) ∈ 1,2-2s (N, R) (cf.
Remark 3) and that F admits an analytic extension to a neighborhood of 0 ∈ H -s c,0 for any 0 ≤ s < 1/2 (cf. Proposition 2). Our aim is to show that for any given w ∈ H -s r,0 with 0 ≤ s < 1/2, F : H -s r,0 → 1,2-2s (N, R) analytically extends to a neighborhood of w in H -s c,0 . The proof relies on the approximation of w by finite gap potentials, which allows us to use similar methods of proofs as in the case where u -s is small. So we will be as brief as possible and only discuss in detail the additional arguments involved.

First note that it follows by Theorem 4 that for any given w ∈ H -s r,0 with 0 ≤ s < 1/2 and n ≥ 1, F n admits an analytic extension to U -s w , (33)

F n : U -s w → C , u → 1 2πi ∂Dτ n (rn+1/3) (L u -λ) -1 1|1 dλ ,
where the circle ∂D τn (r n +1/3) is counterclockwise oriented. Here U -s w , τ n , r n , and D τn (r n + 1/3) are given by Theorem 4 with ρ = 1/4 and M = w -s .

Proposition 6. Let w ∈ H -s r,0 with 0 ≤ s < 1/2. Then for any u ∈ U -s w , F (u) ∈ 1,2-2s + and F : U -s w → 1,2-2s + is analytic. In particular, by shrinking U -s w if needed, F (U -s w ) is bounded in 1,2-2s + .
Before proving Proposition 6, we make some preliminary considerations. Let 0 ≤ s < 1/2, ρ = 1/4, and M > 0 be given. For any w ∈ H -s r,0 with w -s ≤ M , choose w 0 ∈ U N with N ≥ 1, satisfying (32), and n 0 > N so that Lemma 16 holds. Then there exists a neighborhood U -s w of w in H -s c,0 , contained in B -s c,0 (w, 1/8C M,s,1/4 ), so that Proposition 4 holds. Together with Corollary 7 it then follows that for any u = w 0 + v ∈ U -s w and n ≥ n 0 , (34)

T v (L w 0 -λ) -1 H -s;n + →H -s;n + ≤ 1/2 , ∀λ ∈ Vert n (1/4).
It implies that for any λ ∈ n≥n 0 Vert n (1/4), the inverse of the operator L u -λ :

H 1-s + → H -s + can be written as (L w 0 -λ) -1 (Id -T v (L w 0 -λ) -1 ) -1 .
Denote by f n ≡ f n (•, w 0 ), n ≥ 0, the eigenfunctions of L w 0 , corresponding to the eigenvalues λ n ≡ λ n (w 0 ), n ≥ 0, normalized as in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. Then λ n = n and f n = g ∞ e inx for any n ≥ N where g ∞ = e i∂ -1

x w 0 . Furthermore, let

E 1 := span{f α : 0 ≤ α ≤ N } , E 2 ≡ E -s 2 := span{f k : k > N } . Here E 1 is a subspace of ∩ m≥0 H m + of dimension N + 1, spanned by f α , 0 ≤ α ≤ N ,
and endowed with the L 2 -norm, whereas E 2 is the closed subspace of H -s + , spanned by f k , k > N , endowed with the norm • -s . We denote by E -s;n 2 the space E 2 , when endowed with the norm • -s;n . Since for any n > N , γ n (w 0 ) = 0 and hence 1|f n = 0 it follows that the constant function

1 is in E 1 , (35) 1 = α≤N 1|f α f α . It implies that (L u -λ) -1 1|1 is given by α,β≤N 1|f α f β |1 (L w 0 -λ) -1 (Id -T v (L w 0 -λ) -1 ) -1 f α |f β which equals (36) α,β≤N 1|f α f β |1 λ β -λ (Id -T v (L w 0 -λ) -1 ) -1 f α |f β .
We now introduce for 1 ≤ j ≤ 2 the canonical L 2 -projection Π j and inclusion ι j , associated to the subspace E j ,

Π j : H -s + → E j , ι j : E j → H -s + ,
and define for any λ ∈ Vert n (1/4), n ≥ n 0 ,

A ij ≡ A(λ) ij := Π i T v (L w 0 -λ) -1 ι j : E j → E i , 1 ≤ i, j ≤ 2 .
A straightforward computation shows that

Π 1 Id -T v (L w 0 -λ) -1 -1 ι 1 = (Id -A 11 ) -1 (Id -B) -1 , where B ≡ B(λ) : E 1 → E 1 is given by B := A 12 (Id -A 22 ) -1 A 21 (Id -A 11 ) -1 .
To estimate the quasi-moment map F (u), we expand

(Id -A 11 ) -1 = Id + (Id -A 11 ) -1 A 11 = Id + A 11 + (Id -A 11 ) -1 A 2 11 and (Id -B) -1 = Id + (Id -B) -1 B = Id + B + B 2 + (Id -B) -1 B 3 , yielding (37) Π 1 (Id -T v (L w 0 -λ) -1 ) -1 ι 1 = (Id -A 11 ) -1 + A , where A ≡ A(λ) := (Id -A 11 ) -1 (Id -B) -1 B : E 1 → E 1 is given by A = B + A 11 B + B 2 + R and the remainder R ≡ R(λ) : E 1 → E 1 by R := (Id-A 11 ) -1 A 2 11 B +(Id-A 11 ) -1 A 11 B 2 +(Id-A 11 ) -1 (Id-B) -1 B 3 .
By (36) and (37) one has for any n ≥ n 0

(38) F n (u) = α,β≤N 1|f α f β |1 (F 0 n;α,β (u) + F n;α,β (u))
where

(39) F 0 n;α,β (u) := 1 2πi ∂Dn(1/3) 1 λ β -λ (Id -A 11 (λ)) -1 f α |f β dλ and
F n;α,β (u) := 1 2πi ∂Dn(1/3) 1 λ β -λ A(λ)f α |f β dλ = 1 2πi ∂Dn(1/3) 1 λ β -λ B(λ) + A 11 (λ)B(λ) + B 2 (λ) f α | f β dλ + 1 2πi ∂Dn(1/3) 1 λ β -λ R(λ)f α |f β dλ. (40)
To estimate the remainder term, we analyze the operator norms of the various operators involved. For any bounded linear operator G : E i → E j , we denote by G the usual operator norm. For a bounded linear operator G : E 2 → E 2 where E 2 is endowed with the norm E -s;n 2 , the corresponding operator norm is denoted by G -s;n .

Estimate of

A 11 (λ) . For any f ∈ E 1 and λ ∈ Vert n (1/4) ∪ D n (1/4), n ≥ n 0 , one has with f = α≤N f |f α f α , A 11 (λ)f = α≤N f |f α Π 1 T v (L w 0 -λ) -1 f α = α,β≤N f |f α vf α |f β λ α -λ f β . Since sup λ∈Vertn( 1 4 )∪Dn( 1 4 ) 1 |λ -λ α | ≤ 1 n -1/2 -N , ∀ n ≥ n 0 , one has with f = α≤N | f |f α | 2 1/2 , sup Vertn( 1 4 )∪Dn( 1 4 ) A 11 (λ)f ≤ 1 n -1/2 -N β≤N α≤N f |f α vf α |f β 2 1/2 ≤ N + 1 n -1/2 -N max α,β vf α -s f β s f . Using that vf α -s ≤ C s v -s f α 1-s (cf. Lemma 1) one can find a constant C N,s 11 > 0, independent of n ≥ n 0 , so that for any n ≥ n 0 and u = w 0 + v ∈ U -s w , (41) sup λ∈Vertn
( 1 4 )∪Dn( 1 4 ) A 11 (λ) ≤ C N,s 11 1 n .
As a consequence, there exists n 1 ≥ n 0 so that sup

λ∈Vertn( 1 4 )∪Dn( 1 4 ) A 11 (λ) ≤ 1/2 , ∀ n ≥ n 1
and hence for any u = w 0 + v ∈ U -s w , and any n ≥ n 1 , (42) sup

λ∈Vertn( 1 4 )∪Dn( 1 4 ) (Id -A 11 (λ)) -1 ≤ 2
In particular it follows that λ → (Id -A 11 (λ)) -1 is analytic on D n (1/3) for any n ≥ n 1 , hence by definition (39), (43)

F 0 n;α,β (u) = 0 , ∀ n ≥ n 1 , ∀ u = w 0 + v ∈ U -s w . Estimate of A 21 (λ) . For any f ∈ E 1 and any λ ∈ Vert n (1/4), n ≥ n 0 , one has with f = α≤N f |f α f α , A 21 (λ)f = α≤N f |f α Π 2 T v (L w 0 -λ) -1 f α = α≤N,k>N f |f α vf α |f k λ α -λ f k .
Since by [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]Lemma 6], there exists a constant C > 0 so that for any

g ∈ H -s + , g -s ≤ C k≥0 1 k 2s | g|f k | 2 1/2
one has by Cauchy-Schwarz, sup λ∈Vertn(1/4)

A 21 (λ)f -s ≤ C n -1/2 -N k>N 1 k 2s α≤N f |f α vf α |f k 2 1/2 ≤ C • (N + 1) n -1/2 -N max α≤N k>N 1 k 2s | vf α |f k | 2 1/2 f . Using that f k = g ∞ e ikx one infers that k>N 1 k 2s | vf α |f k | 2 1/2 ≤ vf α g ∞ -s .
Recall that g ∞ = e i∂ -1 x w 0 is C ∞ -smooth and that H 1-s c is an algebra, implying that there exists a constant C > 0 so that

h 1 h 2 1-s ≤ C h 1 1-s h 2 1-s , ∀ h 1 , h 2 ∈ H 1-s c .
It then follows from Lemma 1 and (30) that

vf α g ∞ -s ≤ C s v -s • C f α 1-s C s,1 (1 + w 0 -s ) .
Altogether we have shown that there exists a constant C N,s 21 > 0 so that for any n ≥ n 0 and u

= w 0 + v ∈ U -s w , (44) sup λ∈Vertn
(1/4) A 21 (λ) ≤ C N,s 21 1 n .
Estimate of (Id -A 22 (λ)) -1 -s;n . It follows from (34) that for any n ≥ n 0 and any

u = w 0 + v ∈ U -s w (45) sup λ∈Vertn(1/4) A 22 (λ) -s;n ≤ 1/2
and hence, writing (Id -A 22 (λ)) -1 as a Neumann series,

(1/4) (Id -A 22 (λ)) -1 -s;n ≤ 2 , (46) sup λ∈Vertn
Estimate of A 12 (λ) -s;n . It follows from (34) that for any n ≥ n 0 and

any u = w 0 + v ∈ U -s w (47) sup λ∈Vertn(1/4) A 12 (λ) -s;n ≤ 1 2 .
The estimates (41) -(47) imply that there exists C N,s B > 0 so that for any n ≥ n 1 (48) sup

λ∈Vertn(1/4) B(λ) ≤ C N,s B 1 n .
Proof of Proposition 6. Since F n , n ≥ 1, defined by (33), is analytic on

U -s w , it remains to show that n≥n 1 n 2-2s |F n (u)| < ∞ locally uniformly on U -s
w where n 1 is given by (42). By (38) -( 40) and ( 43), it suffices to show that for any 0 ≤ α, β ≤ N and G ∈ {B, A 11 B, BA 11 , B 2 , R},

sup u∈U -s w n≥n 1 n 2-2s 1 2πi ∂Dn(1/3) 1 λ β -λ G(λ)f α | f β dλ < ∞.
Estimate of R. In view of the equivalence of the norms (cf. Lemma 7),

f -s;n ≤ 2 s n s f -s , f -s ≤ 2 s n s f -s;n ,
and the estimates (41), (42), and (48) one concludes that there exists a constant C N,s R > 0 so that for any 0 ≤ α, β ≤ N and n ≥ n 1 ,

| 1 λ β -λ R(λ)f α | f β | ≤ 1 n -1/2 -N sup λ∈Vertn(1/4) R(λ) ≤ C N,s R 1 n n 2s n 3 implying that for any 0 ≤ α, β ≤ N , sup u∈U -s w n≥n 1 n 2-2s 1 2πi ∂Dn(1/3) 1 λ β -λ R(λ)f α | f β dλ < ∞ . (49) Estimate of G ∈ {B, A 11 B, BA 11 , B 2 }. First note that B = B 1 + B 1 A 11 + B 1 (Id -A 11 ) -1 A 2 11 , B 1 := A 12 (Id -A 22 ) -1 A 21 ,
implying that

A 11 B = A 11 B 1 + A 11 B 1 (Id -A 11 ) -1 A 11 , BA 11 = B 1 A 11 + B 1 (Id -A 11 ) -1 A 2 11
, and

B 2 = B 2 1 + B 1 (Id -A 11 ) -1 A 11 B 1 + BB 1 (Id -A 11 ) -1 A 11 .
The estimates (41) -(47) imply (cf. (48)) that there exists C N,s B 1 > 0 so that for any n ≥ n 1 (50) sup

λ∈Vertn(1/4) B 1 (λ) ≤ C N,s B 1 1 n .
It then follows that

B 1 (Id -A 11 ) -1 A 2 11 , A 11 B 1 (Id -A 11 ) -1 A 11 , B 1 (Id -A 11 ) -1 A 11 B 1 , BB 1 (Id -A 11 ) -1 A 11
can be treated in the same way as R and it remains to consider the operators B 1 , A 11 B 1 , B 1 A 11 , and B 2 1 . Estimate of B 1 . We claim that for any given 0 ≤ α, β ≤ N , (51) sup

u∈U -s w n≥n 1 n 2-2s | 1 2πi ∂Dn(1/3) 1 λ β -λ B 1 (λ)f α | f β dλ | < ∞ .
One computes, using that f k = g ∞ e ikx and λ k = k for any k > N,

B 1 f α | f β = k, >N vf α |f k λ α -λ vf β |f -λ (Id -A 22 ) -1 f k |f = k, >N vf α g ∞ |e ikx λ α -λ vf β g ∞ |e -i x -λ (Id -A 22 ) -1 f k |f .
By (45) it follows that for any λ ∈ Vert n (1/4) with n ≥ n 0 , one can write (Id -A 22 ) -1 as a Neumann series, m≥0 (A 22 ) m . We are thus lead to introduce for any m ≥ 0 and any k, > N, G n,k, ,m := 1 2πi

∂Dn(1/3)

1 (λ β -λ)(λ α -λ)( -λ) A 22 (λ) m f k | f dλ so that 1 2πi ∂Dn(1/3) B 1 (λ)f α | f β = k, >N,m≥0 vf α g ∞ (k) vf β g ∞ (-)G n,k, ,m .
For m = 0, we get by Cauchy's theorem

G n,k, ,0 = 1 2πi ∂Dn(1/3) 1 (λ β -λ)(λ α -λ)( -λ) f k | f dλ = δ k δ kn 1 (λ β -n)(λ α -n) , implying that n≥n 1 n 2-2s k, >N vf α g ∞ (k) vf β g ∞ (-)G n,k, ,0 | ≤ n≥n 1 n 2 (n -λ α )(n -λ β ) | vf α g ∞ (n)| n s | vf β g ∞ (-n)| n s vf α g ∞ -s vf β g ∞ -s .
We thus have proved that (52)

n≥n 1 n 2-2s k, >N vf α g ∞ (k) vf β g ∞ (-)G n,k, ,0 | v 2 -s .
To continue, note that for any k > N, one has

A 22 f k = Π 2 T v (L w 0 -λ) -1 f k = Π 2 (vf k ) k -λ .
Since |g ∞ (x)| = 1 one then infers that for any > N

A 22 f k |f = vg ∞ e ikx |g ∞ e i x k -λ = v( -k) k -λ .
Hence using that v(0) = 0, G n,k, ,1 can be computed as

G n,k, ,1 = 1 2πi ∂Dn(1/3) v( -k) (λ β -λ)(λ α -λ)( -λ)(k -λ) dλ = v(n -k)δ n (λ β -n)(λ α -n)(n -k) + v( -n)δ kn (λ β -n)(λ α -n)(n -) , implying that n≥n 1 n 2-2s | k, >N vf α g ∞ (k) vf β g ∞ (-)G n,k, ,1 | n≥n 1 | vf β g ∞ (-n)| n s 1 n s k>N | vf α g ∞ (k)| | v(n -k)| |n -k| + n≥n 1 | vf α g ∞ (n)| n s 1 n s >N | vf β g ∞ (-)| | v( -n)| | -n| vf α g ∞ -s vf β g ∞ -s v -s .
Hence

n≥n 1 n 2-2s | k, >N vf α g ∞ (k) vf β g ∞ (-)G n,k, ,1 | v 3 -s .
Similarly, one computes

(A 22 ) 2 f k |f = k 1 >N vf k |f k 1 vf k 1 |f (k -λ)(k 1 -λ) = k 1 >N v(k 1 -k) v( -k 1 ) (k -λ)(k 1 -λ) .
More generally, for any m ≥ 1, the expression (A 22 ) m f k |f can be expanded as

k j >N v(k 1 -k) v(k 2 -k 1 ) • • • v(k m-1 -k m-2 ) v( -k m-1 ) (k -λ)(k 1 -λ) • • • (k m-1 -λ) ,
leading to the following formula for G n,k, ,m ,

k j >N 1 2πi ∂Dn(1/3) v(k 1 -k) v(k 2 -k 1 ) • • • v(k m-1 -k m-2 ) v( -k m-1 ) (λ β -λ)(λ α -λ)( -λ)(k -λ)(k 1 -λ) • • • (k m-1 -λ) dλ .
One then shows by the arguments of the proof of Proposition 2 that sup u∈U -s w n≥n 1 n 2-2s 1 2πi

∂Dn(1/3)

1 λ β -λ B 1 (λ)f α | f β dλ < ∞ . Estimate of A 11 (λ)B 1 (λ). For any 0 ≤ α, β ≤ N , 1 λ β -λ A 11 B 1 f α |f β = β 1 ≤N vf β |f β 1 1 λ β -λ 1 λ β 1 -λ B 1 f α | f β 1
and hence A 11 B 1 f α |f β can be treated in the same way as

B 1 f α | f β . Estimate of B 1 (λ)A 11 (λ). For any 0 ≤ α, β ≤ N , 1 λ β -λ B 1 A 11 f α |f β = α 1 ≤N vf α |f α 1 1 λ β -λ 1 λ α -λ B 1 f α 1 | f β
and hence B 1 A 11 f α |f β can be treated in the same way as

B 1 f α | f β . Estimate of B 1 (λ) 2 . The expression B 2 1 f α | f β can be expanded as k, >N vf α |f k λ α -λ vf β |f -λ (Id -A 22 ) -1 A 21 A 12 (Id -A 22 ) -1 f k |f = k, >N vf α g ∞ (k) λ α -λ vf β g ∞ (-) -λ (Id -A 22 ) -1 A 21 A 12 (Id -A 22 ) -1 f k |f . Since A 12 f k 1 = α 1 ≤N vf k 1 |fα 1 k 1 -λ f α 1 one has A 21 A 12 f k 1 = α 1 ≤N,k 2 >N vf k 1 |f α 1 k 1 -λ vf α 1 |f k 2 λ α 1 -λ f k 2 = α 1 ≤N,k 2 >N vf α 1 g ∞ (-k 1 ) k 1 -λ vf α 1 g ∞ (k 2 ) λ α 1 -λ f k 2 and hence B 2 1 f α | f β can be written as B 2 1 f α | f β = α 1 ≤N H 1,α 1 H 2,α 1
where H j,α 1 ≡ H j,α 1 (λ), j = 1, 2, are given by

H 1,α 1 := k,k 1 >N vf α g ∞ (k) λ α -λ vf α 1 g ∞ (-k 1 ) k 1 -λ (Id -A 22 ) -1 f k |f k 1 and H 2,α 1 := ,k 2 >N vf β g ∞ (-) -λ vf α 1 g ∞ (k 2 ) λ α 1 -λ (Id -A 22 ) -1 f k 2 |f .
Writing (Id-A 22 ) -1 as a Neumann series, m≥0 (A 22 ) m , and using that (A 22 ) m f k |f can be expanded as

k j >N v(k 1 -k) v(k 2 -k 1 ) • • • v(k m-1 -k m-2 ) v( -k m-1 ) (k -λ)(k 1 -λ) • • • (k m-1 -λ) ,
one then shows again by the arguments of the proof of Proposition 2 that sup

u∈U -s w n≥n 1 n 2-2s 1 2πi ∂Dn(1/3) 1 λ β -λ B 1 (λ) 2 f α | f β dλ < ∞ .

Proof of Theorem 1(ii) and Corollary 1

In this section we prove Theorem 1(ii), saying that for any 0 ≤ s < 1/2, the moment map Γ : H -s r,0 → 1,1-2s (N, R) : u → (γ n (u)) n≥1 analytically extends to a neighborhood of H -s r,0 in H -s c,0 . By Theorem 3, we already know that such an extension exists in a neighborhood of 0 in H -s c,0 . Since by Theorem 4, the gaps γ n (u), n ≥ 1, analytically extend to a neighborhood H -s r,0 , which is independent of n, it remains to show that for any w ∈ H -s r,0 , there exists a neighborhood U -s w of w in H -s c,0 so that sup u∈U -s w n≥n * n 1-2s |γ n (u)| < ∞ for some n * ≥ 1. Since we only need to consider γ n (u) for n large, we again approximate u by a finite gap potential, which allows to use arguments of the proof of Theorem 3.

Theorem 1(ii) will then be applied to prove Corollary 1 and to define for any n ≥ 1 and u ∈ U -s w the scaling factor κ n (u) and the normalizing constant µ n (u) by extending product formulas of these quantities, established in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] for u ∈ H -s r,0 . In addition, we prove properties of κ n (u) and µ n (u), n ≥ 1, needed in subsequent work.

As a first step, we study the extension of the generating function H λ (u). For w ∈ H -s r,0 with 0 ≤ s < 1/2 and M := w -s , choose w 0 ∈ U N with N ≥ 1, satisfying (32), and n 0 > N , so that Lemma 16 holds. Denote by U -s w the neighborhood of w in H -s c,0 , given by Theorem 4 with ρ = 1/4 and M = w -s . Then for any u ∈ U -s w , λ → H λ (u) = (L u -λ) -1 1|1 defines a meromorphic function on C with the set of poles contained in spec(L u ). By the definition (33), it follows that for any n ≥ 0 and u ∈ U -s w , the residue of H λ (u) at λ = λ n (u) is given by F n (u) and arguing as in the proof of Lemma 10, one sees that

H λ (u) = n≥0 F n (u) λ -λ n (u)
.

Furthermore, the arguments in the proof of Lemma 11 show that for any u ∈ U -s w and n ≥ 1 with γ n (u) = 0, (53)

H λ n-1 (u)+1 (u) = 0 .
Denote by f n ≡ f n (•, w 0 ), n ≥ 0, the eigenfunctions of L w 0 , corresponding to the eigenvalues λ n ≡ λ n (w 0 ), n ≥ 0, normalized as in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. Then λ n = n and f n = g ∞ e inx for any n ≥ N where g ∞ = e i∂ -1 x w 0 . By (35), one has 1 = α≤N 1|f α f α and by (36)

H λ (u) = α,β≤N 1|f α f β |1 λ β -λ (Id -T v (L w 0 -λ) -1 ) -1 f α |f β .
We need the following estimates for H λ (u)

Lemma 18. There exist n 2 ≥ n 1 with n 1 given by (42), so that for any n ≥ n 2 and u ∈ U -s w ,

1 4n ≤ |H λ (u)| ≤ 4 n , ∀ λ ∈ Vert n (1/4) .
Proof. We adapt the proof of Lemma 12 to the more general situation at hand. For any u = w 0 + v ∈ U -s w and n ≥ n 0 , we use

(Id -T v (L w 0 -λ) -1 ) -1 = Id + (Id -T v (L w 0 -λ) -1 ) -1 T v (L w 0 -λ) -1 , to compute (54) H λ (u) = α≤N | 1|f α | 2 λ α -λ + α,β≤N 1|f α f β |1 R α,β
where

R α,β := 1 (λ α -λ)(λ β -λ) (Id -T v (L w 0 -λ) -1 ) -1 Π(vf α )|f β .
We first estimate the principal term H λ,0 :=

α≤N | 1|fα | 2 λα-λ in (54), which is independent of u. Since α≤N | 1|f α | 2 = 1 and λ α ≤ λ N = N , ∀ 0 ≤ α ≤ N , λ 0 = - N 1 γ n , γ n ≡ γ n (w 0 ) ,
one infers that for any λ ∈ Vert n (1/4) with n ≥ n 0 , (55) 1

n + 1/2 + N 1 γ n ≤ |H λ,0 | ≤ 1 n -1/2 -N .
Now let us estimate the remainder terms R α,β . By Corollary 7 with M = w -s , ρ = 1/4, one has for any n ≥ n 0 and any u

= w + v ∈ U -s w (Id -T v (L w 0 -λ) -1 ) -1 H -s;n + →H -s;n + ≤ 2 , ∀ λ ∈ Vert n (1/4) .
One then infers from Lemma 3 and Lemma 1

(Id-T v (L w 0 -λ) -1 ) -1 Π(vf α ) -s ≤ 2 s n s (Id -T v (L w 0 -λ) -1 ) -1 Π(vf α ) -s;n ≤ 2 s n s 2 Π(vf α ) -s;n ≤ 2 1+s n s 2 s n s vf α ) -s ≤ 2 1+2s n 2s C s v -s f α 1-s
with C s given as in Lemma 1. It then follows that for any 0 ≤ α, β ≤ N 55) -( 56) yield the claimed estimates.

(56) |R α,β | ≤ 2 1+2s C s n 2s (n -1/2 -N ) 2 v -s max α≤N f α 2 1-s Since 0 ≤ 2s < 1, (
Arguing is in the proof of Lemma 13 one sees that Lemma 18 yields the following lemma and its corollary.

Lemma 19. For any u ∈ U -s w and n ≥ n 2 (with n 2 as in Lemma 18), the difference ZP u;n of the number of zeroes of H λ (u) in D n (1/3) and the number of its poles in D n (1/3) vanishes. As a consequence, for any n ≥ n 2 , H λ (u) has at most one zero in D n (1/3).

Corollary 8. Assume that u ∈ U -s w with 0 ≤ s < 1/2. For any n ≥ n 2 , F n (u) = 0 if and only if γ n (u) = 0.

For any u ∈ U -s w and n ≥ n 2 , define

η n (λ, u) := - λ -λ n (u) λ -λ n-1 (u) -1 (λ -λ 0 (u))H λ (u) , ∀λ ∈ Vert n (1/4
η n (λ, u) = 1 2πi ∂Dn(1/3) η n (µ, u) µ -λ dµ.
As a consequence, for any n ≥ n 2 ,

η n : D n (1/4) ∪ Vert n (1/4) × U -s
w → C is analytic and by Lemma 13 vanishes nowhere. Arguing as in the proof of Lemma 14 and using that |λ 0 (u) -λ 0 (w)| ≤ 1/4 for any u ∈ U -s w (cf. Theorem 4) and λ 0 (w) ≤ 0 ( [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF](s = 0), [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2)) one obtains the following Lemma 20. For any u ∈ U -s w and n ≥ n 2 (with n 2 as in Lemma 18),

1 C w ≤ |η n (λ, u)| ≤ C w , ∀λ ∈ D n (1/4) ,
where C w ≥ 1 is a constant, only depending on λ 0 (w).

For any n ≥ n 2 , define

κ n : U -s w → C, u → 1 λ n (u) -λ 0 (u) η n (λ n (u), u)
Being a composition of analytic functions, κ n is analytic as well. Arguing as in the proof of Proposition 3 one obtains the following Proposition 7. For any n ≥ n 2 (with n 2 as in Lemma 18), the map κ n : U -s w → C is analytic. For any u ∈ U -s w , one has

|κ n (u)| ≤ 2C w 1 n , 1 |κ n (u)| ≤ 2C w n , ∀ n ≥ n 2 ,
where C w ≥ 1 is a constant, only depending on λ 0 (w). In particular, the map

U -s w → ∞ (Z ≥n 2 , C), u → (1/nκ n (u)) n≥n 2 is analytic.
Arguing as in the proof of Corollary 5 one sees, using our results on F n (u) and the gap lengths γ n (u), that Proposition 7 yields the following Corollary 9. Let 0 ≤ s < 1/2. For any u in U -s w , (58)

F n (u) = -κ n (u)γ n (u) , ∀n ≥ n 2 .
With all these preliminary results established, we now can prove Theorem 1(ii) and Corollary 1. We reformulate Theorem 1(ii) as follows: Theorem 5. Let 0 ≤ s < 1/2, w ∈ H -s r,0 , and let U -s w be the neighborhood of w in H -s c,0 of Proposition 6. Then the moment map Γ :

U -s w → 1,1-2s + , u → (γ n (u)) n≥1 is analytic. In addition, Γ(U -s w ) is bounded in 1,1-2s + .
Proof. Let 0 ≤ s < 1/2 and w ∈ H -s r,0 and denote by U -s w the neighborhood of w in H -s c,0 of Theorem 4. Since by Proposition 6, U -s w → 1,2-2s (Z ≥n 2 , C), u → (F n (u)) n≥n 2 and by Proposition 7,

U -s w → ∞ (Z ≥n 2 , C), u → (1/nκ n (u)) n≥n 2 are analytic, it follows from (58) that U -s w → 1,1-2s (Z ≥n 2 , C) , u → (γ n (u)) n≥n 2
(ii) There exists a neighborhood Ũ -s w ⊂ U -s w of w in H -s c,0 so that for any 1 ≤ n < n 3 and any u ∈ Ũ -s

w |κ n (u) -κ n (w)| < 1 2 κ n (w) , |µ n (u) -µ n (w)| < 1 2 µ n (w) .
(iii) For any n ≥ 1, the principal branch of the square root of nκ n and the one of µ n are well defined on Ũ -s w and

√ nκ n n≥1 : Ũ -s w → ∞ + , u → nκ n (u) n≥1 and √ µ n n≥1 : Ũ -s w → ∞ + , u → µ n (u) n≥1
are analytic.

Proof. By Theorem 5, there exists n 3 ≥ max(n 2 , 3) so that (60)

p =1 |γ p (u)| λ p (u) -λ n (u) ≤ 1 4 , ∀ n ≥ n 3 , ∀ u ∈ U -s w .
(i) To simplify notation, whenever possible, we do not indicate the dependence of the quantities involved on u.

First we estimate |κ n (u) -1| for n ≥ n 3 . We argue similarly as in the proof of Proposition 4. For any n ≥ n 3 and u ∈ U -s w , write nκ n -1 = I n + II n where

I n := n λ n -λ 0 -1 p =n 1- γ p λ p -λ n , II n := p =n 1- γ p λ p -λ n -1 .
We begin by estimating the term I n . Since for any n ≥ n 3 , |λ n -n| ≤ 1/4 (cf. Theorem 4) and λ 0 (w) ≤ 0 (w real valued) it follows that (61)

|λ n -λ 0 | ≥ n -1/4 -λ 0 (w) -1/4 ≥ n -1/2 .
Since by assumption n 3 ≥ 3, it follows that

| n λ n -λ 0 -1| ≤ 1 4 . Furthermore, since log(1 + |x|) ≤ |x| one has | p =n 1 - γ p λ p -λ n | ≤ p =n 1 + |γ p | |λ p -λ n | ≤ exp p =n log(1 + |γ p | |λ p -λ n | ) ≤ exp p =n |γ p | |λ p -λ n | .
From (60) one then infers that altogether one has

|I n (u)| ≤ 1 4 e 1/4 , ∀ n ≥ n 3 , ∀ u ∈ U -s w .
To estimate II n , first note that by (60), the principal branch of the logarithm of 1 -γp λp-λn is well defined for any p = n and n ≥ n 3 . Thus one has

p =n 1 - γ p λ p -λ n = exp p =n log 1 - γ p λ p -λ n .
Arguing as in the proof of Proposition 4 one then infers from (60) that

|II n (u)| ≤ 1 3 e 1/3 , ∀ n ≥ n 3 , ∀ u ∈ U -s w .
Combining all the estimates derived yields the desired bound,

|nκ n (u) -1| ≤ 1 4 e 1/4 + 1 3 e 1/3 = 7 12 e 1/3 , ∀ n ≥ n 3 , ∀ u ∈ U -s w .
Now let us turn to the estimate for |µ n (u) -1| for n ≥ n 3 . We argue as in the proof of Proposition 5. By the definition of µ n one has µ n -1 = I n + II n where

I n := 1 - γ n λ n -λ 0 -1 p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) , II n := p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) -1 .
We begin with estimating the term I n . By (61) one has for any n ≥ n 3 ,

| 1 - γ n λ n -λ 0 -1| ≤ 1 n -1/2 |γ n | ≤ 1 2 |γ n | .
Using that for any p = n and n ≥ n 3 , |λ p-1 -λ n-1 | ≥ 1/2 and |γ n | ≤ 1/2 (cf. Theorem 4), it then follows by (60) that

| p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) | ≤ exp 2|γ n | 1 4 ≤ e 1/4
and hence

|I n (u)| ≤ 1 2 |γ n (u)|e 1/4 , ∀ n ≥ n 3 , ∀ u ∈ U -s w .
It remains to estimate

II n = p =n 1 - γnγp (λ p-1 -λ n-1 )(λp-λn) -1. First note that since for any p = n, | γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) | ≤ 2|γ n | |γ p | |λ p -λ n | ≤ 1/4 ,
the principal branch of the logarithm of 1 -γnγp (λ p-1 -λ n-1 )(λp-λn) is well defined and hence one has

p =n 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) = exp p =n log 1 - γ n γ p (λ p-1 -λ n-1 )(λ p -λ n ) .
In particular, |g ∞ (x)| = 1 for any x ∈ T and g

∞ = 1. (iii) For any u ∈ H -s r,0 , k≥1 g k (•, u) -g k-1 (•, u) < ∞ and g k (•, u) -g k-1 (•, u) ≤ C 1 k τ , ∀ k ≥ 0 ,
where τ := (1/2 -s)/2 and where the constant C > 0 can be chosen locally uniformly with respect to u ∈ H -s r,0 . Remark 6. For u = 0, one has f n (x) = e inx and hence g n ≡ 1 for any n ≥ 0.

The proof of Proposition 9 is based on the following lemmas. First we compute lim n→∞ g n for a finite gap potential. Recall from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] that for any integer N ≥ 1, we denote by O N the subset of functions u ∈ H 0 r,0 with

γ n (u) > 0 , ∀1 ≤ n ≤ N, γ n (u) = 0, ∀ n ≥ N + 1 .
We remark that by [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF],

N ≥1 O N is dense in L 2 r,0 . Lemma 21. For any integer N ≥ 1 and any u ∈ O N , one has in H 1 c lim n→∞ g n (•, u) = e D -1 u .
Proof. We use the notation and the results established in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. For any u ∈ O N and n ≥ N, the eigenfunction f n ≡ f n (•, u) of L u , corresponding to the eigenvalue λ n ≡ λ n (u), is given by f n (x) = e i(n-N )x f N (x) and therefore, g n ≡ g n (•, u) satisfies

g n (x) = e -inx f n (x) = g N (x), ∀ n ≥ N.
As a consequence, the sequence (g n ) n≥1 converges and its limit g ∞ is given by (64) g ∞ (x) = e -iN x f N (x) .

By [3, Theorem 7.1] and its proof, Πu(x) is of the form

(65) Πu(x) = -e ix Q (e ix ) Q(e ix )
where

Q(z) is a polynomial in z ∈ C of degree N of the form Q(z) = N -1 j=0 (1 -q j z), q j ∈ C, 0 < |q j | < 1 , and
Q (z) = ∂ z Q(z). In addition, f N is given by (66) f N (x) = a N e iN x Q(e ix )/Q(e ix )
where

a N ∈ C \ 0 is constant. Since D log Q(x) = 1 Q(e ix )
Q (e ix )e ix = -Πu(x) , one concludes that D -1 Πu(x) = log(1/Q(e ix )) and hence

e D -1 Πu(x) = 1 Q(e ix ) .
Taking the complex conjugate of the latter identity one obtains

e -D -1 Πu(x) = 1 Q(e ix ) , or e D -1 Πu(x) = Q(e ix ) .
Combining the latter two formulas one concludes that

e D -1 u(x) = Q(e ix ) Q(e ix ) .
The identity (66) then becomes

f N (x) = a N e iN x e D -1 u(x) .
Since u is assumed to be real valued it then follows by the normalization condition

f N = 1 that |a N | = 1. Hence (64) becomes (67) g ∞ (x) = a N e D -1 u(x) , |a N | = 1 .
It remains to show that a N = 1. To this end we compute 1|f N . Since by (66)

f N (x) = a N N -1 j=0 e ix -q j 1 -q j e ix = a N N -1 j=0 (e ix -q j ) ∞ n=0
(q j e ix ) n , one concludes that f N (x) = a N (-1) N N -1 j=0 q j + e ix fN (x) and hence

(68) 1|f N = a N (-1) N N -1 j=0 q j , or a N 1|f N = (-1) N N -1 j=0 q j .
On the other hand, as z → ∞,

Q(z) z N = N -1 j=0 1 -q j z z = (-1) N N -1 j=0 q j + O( 1 z ) .
Alternatively, the asymptotics of Q(z)/z N can be computed using [3, (7.8)],

Q(z) z N = det(Id -zM N -1 ) z N = (-1) N det(M N -1 ) + O( 1 z ) and hence (69) (-1) N N -1 j=0 q j = (-1) N det(M N -1
) .

Here M N -1 is the N × N matrix (M np ) 0≤n,p≤N -1 with

M np = + √ µ n+1 γ n+1 f p |1 (λ p -λ n -1) f n+1 |1
where we used that u ∈ O N and hence f n+1 |1 = 0 for any 0

≤ n ≤ N -1. Clearly, det(M N -1 ) = N n=1 + √ µ n γ n f 0 |1 f N |1 det ( 1 λ p -λ n -1 ) 0≤n,p≤N -1 .
Since by the formula for the Cauchy determinant, For estimating g k -g k-1 we first need to study the normalizing constants µ n ≡ µ n (u). By [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] , ∀ p = n .

det ( 1 λ p -(λ n + 1) ) 0≤n,p≤N -1 = (-1) N (N -1)/2 0≤p<q≤N -1 (λ p -λ q ) 2 0≤n,p≤N -1 (λ p -(λ n + 1)
Lemma 22. Let 0 ≤ s < 1/2. For any u ∈ H -s r,0 , 0 < µ n ≤ 1 and To estimate 1 -µ n , write it as a sum I n + II n where

0 ≤ 1 -µ n ≤ γ n n + γ n p =n γ p (p -n) 2 (1 + γ n )(1 + γ p ) , ∀ n ≥ 1 . Proof. Since λ n = n -∞ k=n+1 γ k , n ≥ 0, one has (71) 0 ≤ γ n λ n -λ 0 = γ n n + n k=1 γ k ≤ γ n 1 + γ n < 1 .
I n := 1 -(1 - γ n λ n -λ 0 ) p =n (1 -b np ) ≥ 0 , II n := 1 - p =n (1 -b np ) ≥ 0 .
By (71) and (73) it then follows that

0 ≤ I n ≤ γ n λ n -λ 0 ≤ γ n n , ∀ n ≥ 1 .
To estimate II n note that 0 ≤ II n = 1 -exp - . By Young's inequality one then infers

C 1 u -s C 2 g n 1-τ 1-s 1-s ≤ 1 q C 1 u -s C 2 q + (1 - τ 1 -s ) g n 1-s
where 1 q = τ 1-s . Hence (81) yields

1 q g n 1-s ≤ 1 + ∞ k=1 γ k + 1 q C 1 u -s C 2 q or g n 1-s ≤ 1 -s τ + 1 -s τ ∞ k=1 γ k + C 1 u -s C 2 1-s τ ,
showing that (g n ) n≥0 is bounded in H 1-s c .

Lemma 23 and Lemma 24 lead to the following Corollary 10. For any u ∈ H 1-s r,0 with 0 ≤ s < 1/2, the sequence (g n ) n≥0 converges strongly in H We have now all the ingredients to prove Proposition 9. Proof of Proposition 9. Let 0 ≤ s < 1/2. First we prove item (i). By [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]Lemma 8] it follows that for any N ≥ 1, sup 0≤n≤N f n (•, u) 1-s is bounded on bounded subsets of potentials u ∈ H -s r,0 . Choose any sequence (u k ) k≥1 in H -s r,0 which converges weakly to an element u in H -s r,0 . Then sup k≥1,0≤n≤N

f n (•, u k ) 1-s < ∞ .
Hence there exists a subsequence (u k j ) j≥1 of (u k ) k≥1 so that for any 0 ≤ n ≤ N, f n (•, u k j ) h n in H 1-s + . By [4, Lemma 7], lim j→∞ λ n (u k j ) = λ n (u) for any 0 ≤ n ≤ N . Since f n (•, u k j ) → h n in H 1-s-τ + where τ ( 1 2 -s)/2, it follows that u k j f n (•, u k j ) uh n weakly in H -s c , implying that L u h n = λ n (u)h n , ∀ 0 ≤ n ≤ N . Since f n (•, u k j ) → h n strongly in L 2 + , it follows from the normalization conditions of f n (cf. (62)) that h n = 1 for any 0 ≤ n ≤ N and 1|h 0 > 0 , h n |e ix h n-1 > 0 , ∀ 1 ≤ n ≤ N .

Hence h n = f n (•, u) for any 0 ≤ n ≤ N . Since the subsequence (u k j ) j≥1 with the property that for any 0 ≤ n ≤ N, f n (•, u k j ) h n in H 1-s + was chosen arbitrarily, it follows that

f n (•, u k ) f n (•, u) , ∀ 0 ≤ n ≤ N ,
and in turn that

g n (•, u k ) g n (•, u) , ∀ 0 ≤ n ≤ N .
This establishes claim (i). Item (iii) follows from Lemma 23 and item (ii) from Corollary 82 except for the phase of a. It remains to prove that a is a real, positive number. By Lemma 21, a = 1 for any potential u ∈ N ≥1 O N . Since N ≥1 O N is dense in H -s r,0 (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]) it then suffices to show that a ≡ a(u) depends continuously on u ∈ H -s r,0 , or equivalently that the map H -s r,0 → L 2 c , u → g ∞ (•, u) is continuous. This can be seen as follows. Let (u k ) k≥1 be any sequence in H -s r,0 which converges in H -s r,0 . Denote its limit by u. Then by the triangle inequality, g ∞ (•, u k ) -g ∞ (•, u) is bounded by

g ∞ (•, u k ) -g n (•, u k ) + g n (•, u k ) -g n (•, u) + g n (•, u) -g ∞ (•, u) .
By (74) in Lemma 23, for any given ε > 0, there exists n 0 ≥ 1 so that

g ∞ (•, u k ) -g n (•, u k ) ≤ ε , ∀ n ≥ n 0 , ∀ k ≥ 1 and g n (•, u) -g ∞ (•, u) ≤ ε , ∀ n ≥ n 0 .
Since by item (i), lim k→∞ g n 0 (•, u k ) -g n 0 (•, u) = 0 there exists k ε ≥ 1 so that g n 0 (•, u k ) -g n 0 (•, u) ≤ ε , ∀ k ≥ k ε . Altogether we have proved that

g ∞ (•, u k ) -g ∞ (•, u) ≤ 3ε , ∀ k ≥ k ε .
Since ε > 0 can be chosen arbitrarily, one has lim k→∞ g ∞ (•, u k ) = g(•, u) in L 2 c .

Appendix B. The notion of normally analytic maps

In this appendix, we briefly review the notion of normally analytic maps. We restrict our attention to the setup encountered in Section 3. Without any further reference, we use the notation established in the paper. Let 0 ≤ s < 1/2, r > 0 and assume that f : B -s c,0 (r) → C is an analytic function given by its Taylor series f (u) = ∞ k=0 f k (u) where

f k (u) = |α|=k f k α u α , f k α ∈ C, α ∈ N Z 0 .
Here we used the multi-index notation u α = n∈Z u(n) αn with α = (α n ) n∈Z ∈ N Z 0 and |α| = n∈Z α n . Definition 1. The map f : B -s c,0 (r) → C is said to be normally analytic if f is analytic where

f (u) := ∞ k=0 f k (u) , f k (u) := |α|=k |f k α | u α ,
locally uniformly converges on B -s c,0 (r). Similarly, we say that an analytic map F = (F k ) k≥1 : B -s c,0 (r) → 1,2-2s (N, C) is normally analytic if each of its component F k is normally analytic.

1

  ,σ (N, C) := {z = (z n ) n≥1 : z n ∈ C ∀ n ∈ N, z 1,σ < ∞} , with the norm z 1,σ := n≥1 n σ |z n | .

  Vert n (ρ) := {λ ∈ C : |λ -n| ≥ ρ; | λ -n| ≤ 1/2}, whereas for n = 0, we define Vert 0 (ρ) := {λ ∈ C : |λ| ≥ ρ; λ ≤ 1/2}.

Remark 2 .

 2 (i) Note that for any n ≥ 0 and 0 < ρ ≤ 1/4, D n+1 (ρ) = D n (ρ)+1. Since by Theorem 2 one has λ n (u) ∈ D n (ρ) for any u ∈ H -s c,0

  first prove the claimed upper bound for |η n (λ, u)|. Let n ≥ 1. Since for any k ≥ 0, λ k (u) ∈ D k (1/4) and λ k (u) + 1 ∈ D k+1 (1/4) one has for any µ ∈ ∂D n (1/3) and λ ∈ D n (1/4)

  analytic. In addition, Γ(B -s c,0 (r s, * )) is bounded in 1,1-2s + . Proof. Since B -s c,0 (r s, * ) → 1,2-2s + , u → (F n (u)) n≥1 (Proposition 2) and B -s c,0 (r s, * ) → ∞ + , u → ( 1 nκn(u) ) n≥1 (Proposition 3) are analytic, it follows from (58) that Γ : B -s c,0 (r s, * ) → 1,1-2s + is analytic. Furthermore, from Proposition 2 and Proposition 3 one infers that Γ(B -s c,0 (r s, * )) is bounded in 1,1-2s + .

1 0e 3 . 3 .

 133 tx dt| ≤ |x|e |x| and that for any y ∈ C with |y| ≤ 1/4, | log(1 -y)| = | Combining the estimates derived yields the desired bound, |nκ n (u) -The claimed analyticity of the map √ nκ n n≥1 then follows from [8, Theorem A.3].

4 5 4 p

 4 =n |γ p | ≤ e 1/16 .

4 |γ n |σ exp( 5 4 4 p

 44 |γ n |σ) , where σ := p =n |γp| |λp-λn| can be estimated as σ ≤ 5 =n |γ p | ≤ 1 4 , implying that

Furthermore, recall that

  for any given τ ∈ C and r > 0, D τ (r) denotes the open disc {λ ∈ C : |λ -τ | < r}.

Figure 5 . 1 .

 51 Figure 5.1. L u has one eigenvalue in each white disc (schematic).

  ) and since 0≤n,p≤N -1(λ p -(λ n + 1)) = (-1) N (N +1)/2 0≤n,p≤N -1 |λ p -(λ n + 1)| it follows together with f N |1 = 1|f N that (-1) N det M N -1 = c N 1|f N where (70) c N = N n=1 + √ µ n γ n f 0 |1 | f N |1 | 2 0≤p<q≤N -1 (λ p -λ q ) 2 0≤n,p≤N -1 |λ p -(λ n + 1)| > 0 .Combining (68) -(70) one concludes a N = c N > 0. Since |a N | = 1 this shows that a N = 1.

1 .

 1 (0 < s < 1/2), µ n , n ≥ 1, is given byµ n = (1 -γ n λ n -λ 0 )Using that λ n+1 = λ n + 1 + γ n+1 , one obtains by algebraic transformations,µ n = (1 -γn λn-λ 0 ) p =n (1 -b np ) where b np := γ n γ p (λ p -λ n )(λ p-1 -λ n-1 )

Furthermore, for any p ≥ n + 1 , 1 .( 1 -

 111 (λ p -λ n )(λ p-1 -λ n-1 ) = (p -n) + p k=n+1 γ k (p -n) + p-1 k=n γ k ≥ ((p -n) + γ p )((p -n) + γ n ) .Similarly, for 1 ≤ p ≤ n -1, exchanging the role of n and p,(λ p -λ n )(λ p-1 -λ n-1 ) ≥ ((n -p) + γ n )((n -p) + γ p ) .Hence for anyp = n (72) b np ≤ γ n γ p (|n -p| + γ p )(|n -p| + γ n ) ≤ γ n γ p (1 + γ p )(1 + γ n ) < b np ) ≤ 1 .

- 1 1

 1 log(1 -b np ) . Since by the mean value theorem, for any x ≥ 0, 1-e -x = x 0 e -t dt ≤ x one infers 0 ≤ II n ≤ p =n -log(1 -b np ) .Similarly, for 0≤ y ≤ 1, one has -log(1 -y) = y 0 -t dt ≤ y/(1 -y), implying together with (72) that 0 ≤ II n ≤ γ n )(1 + γ p ) 1 + γ n + γ p ≤ (1 + γ n )(1 + γ p ) . Therefore, 0 ≤ II n ≤ γ n p =n γ p (p -n) 2 (1 + γ n )(1 + γ p ) .Lemma 22 is used to prove the following andT u (e inx g n ) = Π(ue inx g n ) = ue inx g n -(Id -Π)(ue inx g n )where ue inx g n denotes the element in H -s c , given by the bounded linear functional H s c → C , v → u , ve inx g n and •, • denotes the dual pairing between H -s c and H s c . Here we used thatH s c × H 1-s c → H s c , (v, f ) → vf is a bounded bilinear map.Multiplying both sides of the identity L u f n = λ n f n by e -inx , the above computations lead to the following identity in H -s c , (79)Dg n = (λ n -n)g n + ug n -e -inx (Id -Π)(ue inx g n ) .For any given m ∈ Z, denote by Π ≥m the L 2 -orthogonal projection onH s c , s ∈ R, defined by Π ≥m : k∈Z f (k)e ikx → k≥m f (k)e ikx ,and let Π ≤m := Id -Π ≥m+1 . One then hasΠ ≥-n g n = g n , Π ≥-n Dg n = Dg n ,and Π ≥-n e -inx (Id -Π)(ue inx g n ) = 0 .Hence (79) yields (80)Dg n = (λ n -n)g n + Π ≥-n (ug n ) ,implying the estimateDg n -s ≤ |λ n -n| g n -s + Π ≥-n (ug n ) -s . Using that g n -s ≤ g n = 1, n -λ n = ∞ k=n+1 γ k , and the definition of the norm • -s , g n 1-s ≤ Dg n -s + | g n |1 | ≤ Dg n -s + 1 , one then obtains (81) g n 1-s ≤ 1 + ∞ k=1 γ k + Π ≥-n (ug n ) -s . Since H -s r,0 → 1,1-2s + , u → (γ n (u)) n≥1 is a continuous map and Π ≥-n (ug n ) -s ≤ ug n -s ,it remains to estimate ug n -s . Recall that τ := ( 1 2 -s)/2. Then there exists a constant C 1 ≥ 1 so that ug n -s ≤ C 1 u -s g n 1-s-τ , ∀n ≥ 0 .Using that g n = 1 it follows by interpolation between L 2 c and H 1-s c that g n 1-s-τ ≤ C 2 g n

  1-s c to g ∞ . In particular, g ∞ is in H 1-s c . It is given by g ∞ = ae D -1 u , a ∈ C, |a| = 1 .Proof. By Lemma 24, (g n ) n≥0 is bounded in H 1-s c . Since by Lemma 23,g n → g ∞ in L 2 c , it follows that g ∞ is in H 1-s c and that g n g ∞ weakly in H 1-s c . Therefore g n → g ∞ strongly in H 1-s-τ c and hence ug n → ug ∞ strongly in H -s c . Since Π ≤-n-1 (ug n ) -s ≤ Π ≤-n-1 (ug n -ug ∞ ) -s + Π ≤-n-1 (ug ∞ ) -s ≤ ug n -ug ∞ -s + Π ≤-n-1 (ug ∞ ) -s and Π ≤-n-1 (ug ∞ ) -s → 0 as n → ∞, one then concludes that lim n→∞ Π ≥-n (ug n ) = ug ∞ strongly in H -s c . Since lim n→∞ (λ n -n) = 0 it then follows from (80) that (Dg n ) n≥0 converges strongly to Dg ∞ in H -s c and that (82) Dg ∞ = ug ∞ .It implies that g ∞ = ae D -1 u where a ∈ C. Since g ∞ = 1 by Lemma 23 one has |a| = 1.

  Denote by | u| the sequence (| u(n)|) n∈Z . It then follows in a straightforward way from the definition of f being normally analytic that for any u ∈ B -s c,0 (r) f (| u|) = ∞ k=0 |α|=k |f k α || u| α .

T.K. partially supported by the Swiss National Science Foundation. P.T. partially supported by the Simons Foundation, Award #526907.

is analytic as well. Since by Theorem 1(i), for any n ≥ 1, γ n is analytic on U -s w it then follows that Γ : U -s w → 1,1-2s + is analytic. By Proposition 6 and Proposition 7, Γ(U -s w ) is bounded in 1,1-2s + .

Proof of Corollary 1. Using the arguments of the proof of Corollary 6(i) -(iii), the claimed identities follow.

Arguing as in the proof of Corollary 6(iv) and using Theorem 5, it follows that for any u ∈ U -s w , κ n (u), n ≥ n 2 , admit product representations,

where the infinite products are absolutely convergent. Furthermore, Theorem 5 allows to define the scaling factors κ n also for the integers 0 ≤ n < n 2 . For any 1 ≤ n < n 2 , κ n : U -s w → C is defined by the right hand side of (59), whereas for n = 0,

.

All these infinite products are bounded on U -s w and hence analytic on U -s w . We point out, that in the case u is real valued, these definitions coincide with the ones in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF](s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]

Finally we consider the spectral invariants µ n ≡ µ n (u), n ≥ 1, discussed at the end of Section 4. For u ∈ L 2 r,0 , they admit the product representations (28),

It follows from Theorem 5 that µ n , n ≥ 1, analytically extend to U -s w and are bounded on U -s w . Theorem 1(ii) leads to estimates of the maps κ n , n ≥ 1, and µ n , n ≥ 1, which will be used in subsequent work to show that for any 0 ≤ s < 1/2, the Birkhoff map analytically extends to a neighborhood of H -s r,0 in H -s c,0 .

Proposition 8. Let 0 ≤ s < 1/2, w ∈ H -s r,0 , and let U -s w be the neighborhood of w in H -s c,0 of Proposition 6. Then there exists n 3 ≥ n 2 with n 2 as in Lemma 18 so that the following holds. (i) For any u ∈ U -s w and n ≥ n 3 ,

One then concludes as in the proof of Proposition 5 that

Combining the estimates derived yields

(ii) Note that for any n ≥ 1, κ n : U -s w → C is continuous and κ n (w) > 0 since w is assumed to be real (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]). Hence there exists a neighborhood Ũ -s w of w so that

By shrinking Ũ -s w if needed, the same arguments as in the case of κ n show that

(iii) By item (i) and (ii), for any n ≥ 1, the principal branch of the square roots nκ n (u) and µ n (u) are well defined on Ũ -s w . They are analytic and uniformly in n bounded on Ũ -s w . The claimed analyticity of the sequences √ nκ n n≥1 √ µ n n≥1 then follows from [8, Theorem A.3].

Appendix A. Asymptotics of eigenfunctions of L u

Let u ∈ H -s r,0 with 0 ≤ s < 1/2. Denote by f n ≡ f n (•, u), n ≥ 0, the eigenfunctions of the Lax operator L u , corresponding to the eigenvalues λ n ≡ λ n (u), introduced in [3] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF] (0 < s < 1/2). Recall that for any n ≥ 0, f n is in H 1-s + and satisfies f n = 1, whereas the phase of f n is determined recursively by

The aim of this appendix is to prove asymptotic estimates of (63)

Proposition 9. For any 0 ≤ s < 1/2 the following holds: (i) For any n ≥ 0, the map

is weakly sequentially continuous.

(ii) For any u ∈ H -s r,0 , the sequence (g n (•, u)) n≥0 converges in H 1-s c and the limit g ∞ ≡ g ∞ (•, u) is given by

Lemma 23. Let 0 ≤ s < 1/2. For any u ∈ H -s r,0 , (g k ) k≥0 is a Cauchy sequence in L 2 c . More precisely, one has for any n ≥ 1,

where τ = ( 1 2 -s)/2 and where the constant C > 0 can be chosen locally uniformly with respect to u ∈ H -s r,0 . As a consequence, the limit g ∞ of (g k ) k≥0 in L 2 c is given by the telescoping sum, (75)

which is absolutely convergent in L 2 c . Furthermore, g ∞ = 1. Proof. We write g n as a telescoping sum, g n = g 0 + n k=1 (g k -g k-1 ) and note that

By [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [START_REF] Gérard | Sharp wellposedness of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solutions[END_REF]

Similarly, since

Combining (76) -(78) yields the claimed estimate (74). Since g k = 1 for any k ≥ 0, the limit g ∞ also satisfies g ∞ = 1.

We further investigate the sequence (g n ) n≥0 .

Lemma 24. For any u ∈ H 1-s r,0 with 0 ≤ s < 1/2, (g n ) n≥0 is a bounded sequence in H 1-s c . Proof. Let u be an element in H 1-s r,0 with 0 ≤ s < 1/2. Recall that L u = D -T u is a bounded operator H 1-s + → H -s + (cf. [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] (s = 0) and [4] (0 < s < 1/2)). One has for any n ≥ 0, De inx g n = ne inx g n + e inx Dg n