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Year: 2018

On the wellposedness of the KdV equation on the space of pseudomeasures

Kappeler, Thomas ; Molnar, Jan

Abstract: In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudomeasures, also referred to as the Fourier Lebesgue space F฀∞(T,R), where F฀∞(T,R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space F฀s,∞(T,R) with −1/2Sobolev space H−1(T,R) to be in F฀∞(T,R) in terms of asymptotic behavior of spectral quantities of the Hill operator −฀2x+q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

DOI: https://doi.org/10.1007/s00029-017-0347-1

Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-149834 Journal Article Accepted Version

Originally published at: Kappeler, Thomas; Molnar, Jan (2018). On the wellposedness of the KdV equation on the space of pseudomeasures. Selecta Mathematica, 24(2):1479-1526. DOI: https://doi.org/10.1007/s00029-017-0347-1 arXiv:1610.00278v1 [math.AP] 2 Oct 2016 ouin vligi h ore eegespace Lebesgue Fourier the in evolving solutions nti ae ecnie h nta au rbe o h Ko the for problem value circle initial the the on consider equation we paper this In Introduction 1 u oli oipoetersl fBugi [ Bourgain of result the improve to is goal Our siiildt.Tespace The data. initial as ∗ † atal upre yteSisNtoa cec Foundati Science National Swiss the by supported Partially atal upre yteSisNtoa cec Foundati Science National Swiss the by supported Partially H ntesaeo eidcped-esrs lorfre oa to referred also space pseudo-measures, Lebesgue periodic of space the on h ro sacaatrzto o distribution a in for key A characterization a data. initial is as proof measures the Borel small for Bourgain of ∂ oooy culy thlso n egtdFuirLebesg Fourier weighted any on holds F it Actually, topology. (secondary) Classification. Subject AMS 2000 proved. Keywords. are algebra Wiener the on equation KdV the for uniiso h iloperator Hill the of quantities ntewlpsdeso h KdV the of wellposedness the On t ℓ − u s, nti ae epoeawlpsdesrsl fteKVequat KdV the of result wellposedness a prove we paper this In 1 ∞ = ( T ( T − , R , ∂ R ob in be to ) x 3 qaino h pc of space the on equation with ) u 6 + d qain elpsdes iko coordinates Birkhoff well-posedness, equation, KdV F hmsKappeler Thomas u∂ ℓ − T ∞ F pseudomeasures 1 x ( = ℓ / ,x u, T ∞ 2 F , R ( ℓ R s < T 0 ∞ / ,where ), , coe ,2016 4, October Z R ossso -eidcdistributions 1-periodic of consists , 6 ntrso smttcbhvo fspectral of behavior asymptotic of terms in ) − ∈ n mrvso eloens result wellposedness a on improves and 0 Abstract ∂ T x 2 F + t , 1 ℓ q ∞ nadto,wlpsdesresults wellposedness addition, In . ( a Molnar Jan , ∗ T ∈ , R R 71 piay 55,35D05 35Q53, (primary) 37K10 sedwdwt h weak* the with endowed is ) . F 2 ℓ ngoa eloens for wellposedness global on ] 0 ∞ q ihsalBrlmeasures Borel small with nteSblvspace Sobolev the in on on † h Fourier the s rdetof gredient twgd Vries rteweg-de espace ue q ∈ ion S ′ ( T , (1) R ) whose Fourier coefficients q = q, eikπx , k Z, satisfy (q ) Z ℓ∞(Z, R) and k h i ∈ k k∈ ∈ q0 = 0. Here and in the sequel we view for convenience 1-periodic distributions as 2-periodic ones and denote by f, g the L2-inner product 1 2 f(x)g(x) dx h i 2 0 extended by duality to S′(R/2Z, C) C∞(R/2Z, C). We point out that q = × R 2k+1 0 for any k Z since q is a 1-periodic distribution. We succeed in dropping ∈ the smallness condition on the initial data and can allow for arbitrary initial data q Fℓ∞. In fact, our wellposedness results hold true for any of the spaces ∈ 0 Fℓs,∞ with 1/2

s (qk)k∈Z s,∞ := sup k qk , α :=1+ α . k k k∈Zh i | | h i | | Informally stated, our result says that for any 1/2 1 converging −1 (m) to q in H , the corresponding sequence ( (t, q )) > of solutions of (1) 0 S m 1 with initial data q(m) converges to γ(t) in H−1 for any t (a, b). In[17] it was 0 ∈ proved that the KdV equation is globally in time C0-wellposed meaning that for any q H−1 (1) admits a solution γ : R H−1 with initial data in the ∈ 0 → 0 above sense and for any T > 0 the solution map : H−1 C([ T, T ], H−1) is S 0 → − 0 continuous. Note that for any 1/2 < s 6 0, Fℓs,∞ continuously embeds into − 0 −1 F s,∞ F s,∞ F −s,1 H0 . On ℓ0 we denote by τw∗ the weak* topology σ( ℓ0 , ℓ0 ). We refer to Appendix B for a discussion. F s,∞ Theorem 1.1 For any q ℓ0 with 1/2 < s 6 0, the solution curve s,∈∞ − t (t, q) evolves in Fℓ . It is bounded, sup R (t, q) < and 7→ S 0 t∈ kS ks,∞ ∞ continuous with respect to the weak* topology τw∗. ⋊

Remark 1.2. It is easy to see that for generic initial data, the solution curve 3 t Airy(t, q) of the Airy equation, ∂tu = ∂xu, is not continuous with respect 7→ S F s,∞ − to the norm topology of ℓ0 . A similar result holds true for the KdV equation at least for small initial data – see Section 4. ⊸

We say that a subset V Fℓs,∞ is KdV-invariant if for any q Fℓs,∞, ⊂ 0 ∈ 0 (t, q) V for any t R. S ∈ ∈ Theorem 1.3 Let V Fℓs,∞ with 1/2

October 4, 2016 2 Then for any T > 0, the restriction of the solution map to V is weak* S continuous,

: (V, τ ) C([ T, T ], (V, τ )). ⋊ S w∗ → − w∗ By the same methods we also prove that the KdV equation is globally Cω- F 0,1 wellposed on the Wiener algebra ℓ0 – see Section 5 where such a result is N,1 proved for the weighted Fourier Lebesgue space Fℓ , N Z> . 0 ∈ 0 Method of proof. Theorem 1.1 and Theorem 1.3 are proved by the method of normal forms. We show that the restriction of the Birkhoff map Φ: H−1 0 → −1/2,2 F s,∞ ℓ0 q z(q)=(zn(q))n∈Z, constructed in [10], to ℓ0 is a map with 7→s+1/2,∞ Z C values in ℓ0 ( , ), having the following properties:

Theorem 1.4 For any 1/2

Remark 1.5. Note that by [10] for any q H−1, Φ(Iso(q)) = where ∈ 0 TΦ(q) −1/2,2 = z˜ = (˜z ) Z ℓ : z˜ = Φ(q) k Z . (2) TΦ(q) { k k∈ ∈ 0 | k| | k| ∀ ∈ } Furthermore, since by Theorem 1.4 for any q Fℓs,∞, 1/2 1 – cf. k·kσ,p − ∞ − Lemma B.1 from Appendix B. ⊸

Key ingredient for studying the restriction of the Birkhoff map to the Ba- F s,∞ nach spaces ℓ0 are pertinent asymptotic estimates of spectral quantities of the Schrödinger operator ∂2 +q, which appear in the estimates of the Birkhoff − x coordinates in [10, 7] – see Section 2. The proofs of Theorem 1.1 and Theo- rem 1.3 then are obtained by studying the restriction of the solution map , S −1 F s,∞ defined in [17] on H0 , to ℓ0 . To this end, the KdV equation is expressed in Birkhoff coordinates z =(zn)n∈Z. It takes the form

∂ z = iω z , ∂ z =iω z , n > 1, t n − n n t −n n −n 1 where ωn, n > 1, are the KdV frequencies. For q H0 , these frequencies are ∈1 1 2 3 defined in terms of the KdV Hamiltonian H(q)= 0 2 (∂xq) + q dx. When R

October 4, 2016 3 viewed as a function of the Birkhoff coordinates, H is a real analytic function of the actions In = znz−n, n > 1, alone and ωn is given by

ωn = ∂In H.

For q H−1, the KdV frequencies are defined by analytic extension – see [11] ∈ 0 for novel formulas allowing to derive asymptotic estimates. Related results. The wellposedness of the KdV equation on T has been extensively studied - c.f. e.g. [20] for an account on the many results obtained so far. In particular, based on [1] and [18] it was proved in [3] that the KdV equation is globally uniformly C0-wellposed and Cω-wellposed on the Sobolev spaces Hs(T, R) for any s > 1/2. In[17] it was shown that the KdV equation 0 − is globally C0-wellposed in the Sobolev spaces Hs, 1 6 s < 1/2 and in 0 − [11] it was proved that for 1 0, the solution map − − Hs C([ T, T ], Hs) is nowhere locally uniformly continuous. In [20], it was 0 → − 0 shown that the KdV equation is illposed in Hs for s< 1. Most closely related 0 − to Theorem 1.1 and Theorem 1.3 are the wellposedness results of Bourgain [2] for initial data given by Borel measures which we have already discussed at the beginning of the introduction and the recent wellposedness results in [7] on the Fourier Lebesgue spaces Fℓs,p for 1/2 6 s 6 0 and 2 6 p< . 0 − ∞ Notation. We collect a few notations used throughout the paper. For any s,p s R and 1 6 p 6 , denote by ℓ C the C-Banach space of complex valued ∈ ∞ 0, sequences given by

s,p ℓ := z =(z ) Z C : z = 0; z < , 0,C { k k∈ ⊂ 0 k ks,p ∞} where 1/p sp p s z s,p := n zn , 1 6 p< , z s,∞ := sup n zn , k k Zh i | | ! ∞ k k k∈Zh i | | Xk∈ s,p and by ℓ0 the real subspace s,p s,p ℓ := z =(z ) Z ℓ : z = z k > 1 . 0 { k k∈ ∈ 0,C −k k ∀ } s,p R s,p By ℓR we denote the -subspace of ℓ0 consisting of real valued sequences R F s,p z = (zk)k∈Z in . Further, we denote by ℓ0,C the Fourier Lebesgue space, introduced by Hörmander,

s,p ′ s,p Fℓ := q SC(T):(q ) Z ℓ 0,C { ∈ k k∈ ⊂ 0,C} where qk, k Z, denote the Fourier coefficients of the 1-periodic distribution ∈ ikπx 2 q, q = q,ek , ek(x) := e , and , denotes the L -inner product, f, g = 1 2 h i h· ·i ′ h i f(x)g(x) dx, extended by duality to a sesquilinear form on C(R/2Z) 2 0 S × ∞ R Z F s,p F s,p CCR ( /2 ). Correspondingly, we denote by ℓ0 the real subspace of ℓ0,C, s,p ′ s,p Fℓ := q SC(T):(q ) Z ℓ . 0 { ∈ k k∈ ⊂ 0 }

October 4, 2016 4 s s F s,2 F s,2 In case p = 2, we also write H0 [H0,C] instead of ℓ0 [ ℓ0,C] and refer to it as s,2 s,2 Sobolev space. Similarly, for the sequences spaces ℓ0 and ℓ0,C we sometimes s s write h0 [h0,C]. Occasionally, we will need to consider the sequence spaces s,p s,p s,p s,p ℓC (N) ℓ (N, C) and ℓR (N) ℓ (N, R) defined in an obvious way. ≡ ≡ Note that for any z ℓs,p, I := z z > 0 for all k > 1. We denote by ∈ 0 k k −k Tz the torus given by

s,p := z˜ = (˜z ) Z ℓ :z ˜ z˜ = z z , k > 1 . Tz { k k∈ ∈ 0 k −k k −k } For 1 6 p< , is compact in ℓs,p for any z ℓs,p but for p = , it is not ∞ Tz 0 ∈ 0 ∞ s,∞ R s,p compact in ℓ for generic z. For any s and 1 6 p< , the dual of ℓ0 ′ −s,p ′ ∈ ∞ ′ is given by ℓ0 where p is the conjugate of p, given by 1/p + 1/p = 1. In ′ ′ case p =1 we set p = and in case p = we set p = 1. We denote by τw∗ ∞s,∞ ∞ the weak* topology on ℓ0 and refer to Appendix B for a discussion of the properties of τw∗.

2 Spectral theory

In this section we consider the Schrödinger operator

L(q)= ∂2 + q, (3) − x which appears in the Lax pair formulation of the KdV equation. Our aim is to relate the regularity of the potential q to the asymptotic behavior of certain spectral data. −1 −1 R Z C Let q be a complex potential in H0,C := H0 ( / , ). In order to treat periodic and antiperiodic boundary conditions at the same time, we consider the differential operator L(q) = ∂2 + q, on H−1(R/2Z, C) with domain of − x definition H1(R/2Z, C). See Appendix C for a more detailed discussion. The 2 spectral theory of L(q), while classical for q L0,C, has been only fairly recently −1 ∈ extended to the case q H C – see e.g. [5, 9, 16, 19, 22, 7] and the references ∈ 0, therein. The spectrum of L(q), called the periodic spectrum of q and denoted by spec L(q), is discrete and the eigenvalues, when counted with their multiplicities and ordered lexicographically – first by their real part and second by their imaginary part – satisfy

λ+(q) 4 λ−(q) 4 λ+(q) 4 , λ±(q)= n2π2 + nℓ2 . (4) 0 1 1 ··· n n

Furthermore, we define the gap lengths γn(q) and the mid points τn(q) by

λ+(q)+ λ−(q) γ (q) := λ+(q) λ−(q), τ (q) := n n , n > 1. (5) n n − n n 2 For q H−1 we also consider the operator L (q) defined as the operator ∂2+ ∈ 0,C dir − x −1 C 1 C q on Hdir ([0, 1], ) with domain of definition Hdir([0, 1], ). See Appendix C

October 4, 2016 5 as well as [5, 9, 16, 19, 22] for a more detailed discussion. The spectrum of

Ldir(q) is called the Dirichelt spectrum of q. It is also discrete and given by a sequence of eigenvalues (µn)n>1, counted with multiplicities, which when ordered lexicographically satisfies

µ 4 µ 4 µ 4 , µ = n2π2 + nℓ2 . (6) 1 2 2 ··· n n For our purposes we need to characterize the regularity of potentials q in weighted Fourier Lebesgue spaces in terms of the asymptotic behavior of certain spectral quantities. A normalized, symmetric, monotone, and submultiplicative weight is a function w: Z R, n w , satisfying → 7→ n

wn > 1, w−n = wn, w|n| 6 w|n|+1, wn+m 6 wnwm, for all n, m Z. The class of all such weights is denoted by M. For w M, ∈ ∈ R F w,s,p F s,p s , and 1 6 p 6 , denote by ℓ0,C the subspace of ℓ0,C of distributions ∈ ∞ w,s,p f whose Fourier coefficients (f ) Z are in the space ℓ = z = (z ) Z n n∈ 0,C { n n∈ ∈ ℓs,p : z < where for 1 6 p< 0,C k kw,s,p ∞} ∞ 1/p p sp p f w,s,p := (fn)n∈Z w,s,p = wn n fn , α :=1+ α , k k k k Z h i | | ! h i | | nX∈ and for p = , ∞ s f w,s,∞ := (fn)n∈Z w,s,∞ = sup wn n fn . k k k k n∈Z h i | |

To simplify notation, we denote the trivial weight wn 1 by o and write s,p o,s,p ≡ Fℓ C Fℓ C . 0, ≡ 0, As a consequence of (4)–(6) it follows that for any q H−1, the sequence ∈ 0,C of gap lengths (γ (q)) > and the sequence (τ (q) µ (q)) > are both in n n 1 n − n n 1 ℓ−1,2(N). For q Fℓw,s,∞, 1/2

Theorem 2.1 Let w M and 1/2 ℓC (N) and the map ∈ 0, n n 1 ∈ w,s,∞ w,s,∞ Fℓ ℓ (N), q (γ (q)) > , 0,C → C 7→ n n 1 is locally bounded.

w,s,∞ w,s,∞ (ii) For any q Fℓ , one has (τ µ (q)) > ℓ (N) and the map ∈ 0,C n − n n 1 ∈ C w,s,∞ w,s,∞ Fℓ ℓ (N), q (τ (q) µ (q)) > , 0,C → C 7→ n − n n 1 is locally bounded. ⋊

October 4, 2016 6 A key ingredient for studying the restriction of the Birkhoff map of the KdV −1 F s,∞ equation, defined on H0 , to ℓ0 is the following spectral characterization for a potential q H−1 to be in Fℓs,∞. ∈ 0 0 Theorem 2.2 Let q H−1 with gap lengths γ(q) ℓs,∞ for some 1/2

(i) q Fℓs,∞. ∈ 0 (ii) Iso(q) Fℓs,∞. ⊂ 0 (iii) Iso(q) is weak* compact. ⋊

Remark 2.3. For any 1/2 < s 6 0, there are potentials q Fℓs,∞ so that − ∈ Iso(q) is not compact in Fℓs,∞ – see item (iii) in Lemma 3.5. ⊸

In the remainder of this section we prove Theorem 2.1 and Theorem 2.2 by extending the methods, used in [8, 21, 4] for potentials q L2, for singular ∈ potentials. We point out that the spectral theory is only developed as far as needed.

2.1 Setup

2 1 2 2 We extend the L -inner product f, g = f(x)g(x) dx on LC(R/2Z) h i 2 0 ≡ 2 R Z C ′ R Z ∞ R Z iπnx Z L ( /2 , ) by duality to C( /2 ) CC ( R/2 ). Let en(x) = e , n , M R S × F w,s,p ∈ and for w , s , and 1 6 p 6 denote by ℓ⋆,C the space of 2- ∈ ∈ ∞ ′ periodic, complex valued distributions f C(R/2Z) so that the sequence of ∈ S w,s,p their Fourier coefficients f = f,e is in the space ℓ = z = (z ) Z n h ni C { n n∈ ⊂ C : z < . To simplify notation, we write Fℓs,p Fℓo,s,p. k kw,s,p ∞} ⋆,C ≡ ⋆,C In the sequel we will identify a potential q Fℓw,s,∞ with the corre- ∈ 0,C F w,s,∞ sponding element n∈Z qnen in ℓ⋆,C where qn is the nth Fourier coef- ficient of the potential obtained from q by viewing it as a distribution on P R/2Z instead of R/Z, i.e., q = q,e , whereas q = q,e = 0 2n h 2ni 2n+1 h 2n+1i and q0 = q, 1 = 0. We denote by V the operator of multiplication by q h i1 with domain HC(R/2Z). See Appendix C for a detailed discussion of this op- erator as well as the operator L(q) introduced in (3). When expressed in its 1 Fourier series, the image Vf of f = f e HC(R/2Z) is the distribu- n∈Z n n ∈ tion Vf = q f e H−1(R/2Z). To prove the asymptotic n∈Z m∈Z n−m m n ∈P C estimates of the gap lengths stated in Theorem 2.1 we need to study the eigen- P P
value equation L(q)f = λf for sufficiently large periodic eigenvalues λ. For −1 1 q H , the domain of L(q) is HC(R/2Z) and hence the eigenfunction f is ∈ 0,C an element of this space. It is shown in Appendix C that for q Fℓs,∞ with ∈ 0,C F s+2,p 2 F s,∞ 1/2 < s 6 0 and 2 6 p 6 , one has f ℓ⋆,C and ∂xf, Vf ℓ⋆,C . − ∞ + ∈− 2 2 ∈ Note that for q = 0 and any n > 1, λn (0) = λn (0) = n π , and the eigenspace

October 4, 2016 7 + − corresponding to the double eigenvalue λn (0) = λn (0) is spanned by en and e . Viewing L(q) λ±(q) for n large as a perturbation of L(0) λ±(0), we −n − n − n are led to decompose Fℓs,∞ into the direct sum Fℓs,∞ = , ⋆,C ⋆,C Pn ⊕Qn = span e , e , = span e : k = n . (7) Pn { n −n} Qn { k 6 ± } The L2-orthogonal projections onto and are denoted by P and Q , Pn Qn n n respectively. It is convenient to write the eigenvalue equation Lf = λf in 2 the form Aλf = Vf, where Aλf = ∂xf + λf and V denotes the operator of multiplication with q. Since Aλ is a Fourier multiplier, we write f = u + v, where u = Pnf and v = Qnf, and decompose the equation Aλf = Vf into the two equations

Aλu = PnV (u + v),Aλv = QnV (u + v), (8) referred to as P - and Q-equation. Since q H−1, it follows from [16] that ∈ 0,C λ±(q) = n2π2 + nℓ2 . Hence for n sufficiently large, λ±(q) S where S n n n ∈ n n denotes the closed vertical strip

S := λ C : Rλ n2π2 6 12n , n > 1. (9) n { ∈ | − | } Note that λ C : Rλ > 0 S . Given any n > 1, u , and { ∈ } ⊂ n>1 n ∈ Pn λ S , we derive in a first step from the Q-equation an equation for Vv which ∈ n S for n sufficiently large can be solved as a function of u and λ. In a second step, for λ a periodic eigenvalue in Sn, we solve the P equation for u after having substituted in it the expression of Vv. The solution of the Q-equation is then easily determined. Towards the first step note that for any λ S , ∈ n A : Fℓs+2,p is boundedly invertible as for any k = n, λ Qn ∩ ⋆,C →Qn 6 min λ k2π2 > min Rλ k2π2 > n2 k2 > 1. (10) λ∈Sn| − | λ∈Sn| − | | − | In order to derive from the Q-equation an equation for Vv, we apply to it the −1 operator VAλ to get −1 Vv = VAλ QnV (u + v)= TnV (u + v), where

T T (λ) := VA−1Q : Fℓw,s,∞ Fℓw,s,∞. n ≡ n λ n ⋆,C → ⋆,C It leads to the following equation forv ˇ := Vv

(Id T (λ))ˇv = T (λ)V u. (11) − n n

To show that Id Tn(λ) is invertible, we introduce for any s R, w M, and − w,s,∞ ∈ ∈ l Z the shifted norm of f Fℓ C , ∈ ∈ ⋆, s f w,s,∞;l := fel w,s,∞ = wk+l k + l fk Z , k k k k h i k∈ ℓp

October 4, 2016 8 and denote by T the operator norm of T viewed as an operator on k nkw,s,∞;l n Fℓw,s,∞ with norm . Furthermore, we denote by R f, N > 1, the ⋆,C k·kw,s,∞;l N tail of the Fourier series of f Fℓw,s,∞, ∈ ⋆,C

RN f = fkek. |kX|>N Lemma 2.4 Let 1/2 < s 6 0, w M, and n > 1 be given. For any − ∈ q Fℓw,s,∞ and λ S , ∈ 0,C ∈ n T (λ): Fℓw,s,∞ Fℓw,s,∞ n ⋆,C → ⋆,C is a bounded linear operator satisfying the estimate c T (λ) 6 s q , (12) k n kw,s,∞;±n n1/2−|s| k kw,s,∞ where cs > 1 is a constant depending only on and decreasing monotonically in s. In particular, cs does not depend on q nor on the weight w. ⋊

Proof. Let s and w be given as in the statement of the lemma. Note that −1 F s,∞ F s+2,∞ F s,∞ Aλ : ℓ⋆,C ℓ⋆,C is bounded for any λ Sn and hence for any f ℓ⋆,C , s,∞ → ∈ −1 ∈ q Fℓ C , and λ S , the multiplication of A Q f with q, defined by ∈ 0, ∈ n λ n

−1 qm−kfk VAλ Qnf = 2 2 em (13) Z λ k π  mX∈ |kX|6=n −   ′ R Z −1 is a distribution in SC( /2 ). Note that Tn(λ)f = VAλ Qnf and that its norm T (λ)f satisfies for any λ S , k n kw,s,∞;n ∈ n w k + n |s| m k |s| q f 6 m+n m−k k Tnf w,s,∞;n sup h i h − i|s| | ||s| | | |s| , k k m∈Z n + k n k m + n m k k + n |kX|6=n | || − |h i h − i h i where we have used (10). Since m k 6 m + n n + k , 1/2

1 w q w f 6 m−k m−k k+n k Tnf w,s,∞;n 2 sup 1−2|s| | |s| | | |s|| k k m∈Z n + k n k m k k + n |kX|6=n | | | − | h − i h i 1 6 2 q w,s,∞ f w,s,∞;n.  n + k (1−2|s|) n k k k k k |kX|6=n | | | − |  

October 4, 2016 9 One checks that 1 c 6 s , n + k (1−2|s|) n k n1/2−|s| |kX|6=n | | | − | Going through the arguments of the proof one sees that the same kind of

estimates also lead to the claimed bound for T f . Ú k n kw,s,∞;−n Lemma 2.4 can be used to solve, for n sufficiently large, the equation (11) as well as the Q-equation (8) in terms of any given u and λ S . ∈Pn ∈ n w,s,∞ Corollary 2.5 For any q Fℓ C with 1/2 < s 6 0 and w M, there ∈ 0, − ∈ exists ns = ns(q) > 1 so that,

2c q 6 n1/2−|s|, (14) sk kw,s,∞ s with c > 1 the constant in (12) implying that for any λ S , T (λ) is a 1/2 s ∈ n n contraction on Fℓw,s,∞ with respect to the norms shifted by n, T (λ) 6 ⋆,C ± k n kw,s,∞;±n 1/2. The threshold ns(q) can be chosen uniformly in q on bounded subsets F w,s,∞ of ℓ0,C . As a consequence, for n > ns(q), equation (11) and (8) can be uniquely solved for any given u , λ S , ∈Pn ∈ n vˇ = K (λ)T (λ)V u Fℓs,∞, K K (λ) := (Id T (λ))−1, u,λ n n ∈ ⋆,C n ≡ n − n (15) v = A−1Q V u + A−1Q vˇ = A−1Q K V u Fℓs+2,∞ . (16) u,λ λ n λ n u,λ λ n n ∈ ⋆,C ∩Qn

In particular, one has vˇu,λ = Vvu,λ. ⋊

Remark 2.6. By the same approach, one can study the inhomogeneous equation

(L λ)f = g, g Fℓs,∞, − ∈ ⋆,C for λ S and n > n . Writing f = u + v and g = P g + Q g, the Q-equation ∈ n s n n becomes

A v = Q V (u + v) Q g = Q Vv + Q (V u g) λ n − n n n − leading for any given u and λ S to the unique solutionv ˇ of the ∈ Pn ∈ n equation corresponding to (11)

vˇ = Vv = K T (V u g) Fℓs,∞, n n − ∈ ⋆,C and, in turn, to the unique solution v Fℓs+2,∞ of the Q-equation ∈ ⋆,C ∩Qn v = A−1Q (V u g)+ A−1Q K T (V u g)= A−1Q K (V u g). ⊸ λ n − λ n n n − λ n n −

October 4, 2016 10 2.2 Reduction

In a next step we study the P -equation Aλu = PnV (u+v)of (8). For n > ns(q), u , and λ S , substitute in it the solutionv ˇ of (11), given by (15), ∈Pn ∈ n u,λ

Aλu = PnV u + Pnvˇu,λ = Pn(Id + KnTn)V u.

Using that Id + KnTn = Kn one then obtains Aλu = PnKnV u or Bnu = 0, where

B B (λ): , u (A P K (λ)V )u. (17) n ≡ n Pn →Pn 7→ λ − n n Lemma 2.7 Assume that q Fℓs,∞ with 1/2 n ∈ 0,C − s with n given by Corollary 2.5, λ S is an eigenvalue of L(q) if and only if s ∈ n det(Bn(λ)) = 0. ⋊

Proof. Assume that λ S is an eigenvalue of L = L(q). By Lemma C.2 there ∈ n exists 0 = f Fℓs+2,∞ so that Lf = λf. Decomposing f = u + v 6 ∈ ⋆,C ∈ Pn ⊕Qn it follows by the considerations above and the assumption n > n that u = 0 s 6 and B (λ)u = 0. Conversely, assume that det(B (λ)) = 0 for some λ S . n n ∈ n Then there exists 0 = u so that B (λ)u = 0. Since n > n , there exist 6 ∈ Pn n s vˇ and v as in (15) and (16), respectively. Then v v Fℓs+2,∞ u,λ u,λ ≡ u,λ ∈ ⋆,C ∩Qn solves the Q-equation by Corollary 2.5. To see that u solves the P -equation, note that Bn(λ)u = 0 implies that

Aλu = PnKnV u = Pn(Id + KnTn)V u.

As by (15),v ˇu,λ = KnTnV u, and asv ˇu,λ = Vv, one sees that indeed

Aλu = PnV u + PnVv. Ú

Remark 2.8. Solutions of the inhomogeneous equation (L λ)f = g for g − ∈ Fℓs,∞, λ S , and n > n can be obtained by substituting into the P -equation ⋆,C ∈ n s A u = P V u + P Vv P g λ n n − n the expression for Vv obtained in Remark 2.6, Vv = K T (V u g), to get n n − A u = P V u + P K T V u P g P K T g λ n n n n − n − n n n = P (Id + K T )V u P (Id + K T )g. n n n − n n n

Using that Id + KnTn = Kn one concludes that

B (λ)u = P K (λ)g. (18) n − n n Conversely, for any solution u of (18), f = u + v, with v being the element in s+2,∞ s+2,∞ Fℓ C given in Remark 2.6, satisfies (L λ)f = g and f Fℓ C . ⊸ ⋆, − ∈ ⋆,

October 4, 2016 11 We denote the matrix representation of a linear operator F : with Pn →Pn respect to the orthonormal basis e , e of also by F , n −n Pn Fe ,e Fe ,e F = h n ni h −n ni . Fen,e−n Fe−n,e−n h i h i In particular,

2 2 λ n π 0 an bn Aλ = − 2 2 ,PnKnV = , 0 λ n π b−n a−n  −    where for any λ S and n > n the coefficients of P K V are given by ∈ n s n n a a (λ) := K Ve ,e , a a (λ) := K Ve ,e , n ≡ n h n n ni −n ≡ −n h n −n −ni b b (λ) := K Ve ,e , b b (λ) := K Ve ,e . n ≡ n h n −n ni −n ≡ −n h n n −ni Note that for any λ S , the functions a (λ) and b (λ) have the following ∈ n ±n ±n series expansion

a (λ)= T (λ)lVe ,e , b (λ)= T (λ)lVe ,e . (19) ±n h n ±n ±ni ±n h n ∓n ±ni > > Xl 0 Xl 0 Furthermore, by a straightforward verification it follows from the expression of k an in terms of the representation of Kn = k>0 Tn(λ) and V in Fourier space that for any n > n s P a = K Ve ,e = a . (20) n h n −n −ni −n Hence,

2 2 λ n π an(λ) bn(λ), Bn(λ)= − − −2 2 . (21) b−n(λ) λ n π an(λ)  − − −  In addition, if q is real valued, then

a (λ)= a (λ), b (λ)= b (λ), λ S . (22) n n −n n ∈ n Lemma 2.9 Suppose q Fℓw,s,∞ with 1/2 < s 6 0 and w M. Then for ∈ 0,C − ∈ any n > ns, with ns as in Corollary 2.5, the coefficients an(λ) and b±n(λ) are analytic functions on the strip S and for any λ S n ∈ n (i) a (λ) 6 2 T (λ) q , | n | k n kw,s,∞;±nk ks,∞ (ii) w 2n s b (λ) q 6 2 T (λ) q . ⋊ 2nh i | ±n − ±2n| k n kw,s,∞;±nk kw,s,∞ Proof. Let us first prove the claimed estimate for b (λ) q . Since T (λ) 6 | n − 2n| k n kw,s,∞;n 1/2 for n > n and λ S , the series expansion (19) of b converges uniformly s ∈ n n on Sn to an analytic function in λ. Moreover, we obtain from the identity

Kn =Id+ TnKn

b = Ve ,e + T K Ve ,e = q + T K Ve ,e . n h −n ni h n n −n ni 2n h n n −n ni

October 4, 2016 12 Furthermore, for any f Fℓw,s,∞ we compute ∈ ⋆,C w 2n s f,e = w 2n s fe ,e 6 fe = f . 2nh i |h ni| 2nh i |h n 2ni| k nkw,s,p k kw,s,p;n Consequently, using that T 6 1/2 and hence K 6 2, one k nkw,s,p;n k nkw,s,p;n gets

w 2n s b q 6 T K Ve 6 2 T Ve 2nh i | n − 2n| k n n −nkw,s,p;n k nkw,s,p;nk −nkw,s,p;n = 2 T q . k nkw,s,p;nk kw,s,p

The estimates for b q and a are obtained in a similar fashion. Ú | −n − −2n| | n| The following refined estimate will be needed in the proof of Lemma 2.13 in Subsection 2.4.

w,s,∞ Lemma 2.10 Let q Fℓ C with w M and 1/2 < s 6 0. Then for any ∈ 0, ∈ − f Fℓs,∞ and λ S with n > n , ∈ ⋆,C ∈ n s w 2n s T f,e 6 c′ ε (n) q f 2nh i |h n ±ni| s s k kw,s,pk kw,s,p;±n ′ where cs > cs > 1 is independent of q, n, and λ, and

loghni n , s = 0, εs(n)=  1 , 1/2

c˜ loghni 1 s , s = 0, 6 n 2 2 1−|s|  c˜s n m 1 2 , 1/2

Altogether we thus have proved the claim. Ú

October 4, 2016 13 The preceding lemma together with (14) implies that for any n > ns, the function det B (λ)=(λ n2π2 a )2 b b is analytic in λ S and can be n − − n − n −n ∈ n considered a small perturbation of (λ n2π2)2 provided n > n is sufficiently − s large.

Lemma 2.11 Suppose q Fℓs,∞ with 1/2 < s 6 0. Choose n = n (q) > 1 ∈ 0,C − s s as in Corollary 2.5. Then for any n > ns, det(Bn(λ)) has exactly two roots

ξn,1 and ξn,2 in Sn counted with multiplicity. They are contained in

D := λ : λ n2π2 6 4n1/2 S n { | − | }⊂ n and satisfy

1/2 ξn,1 ξn,2 6 √6 sup bn(λ)b−n(λ) . ⋊ (23) | − | λ∈Sn| |

Proof. Since for any n > n and λ S , T (λ) 6 1/2, one concludes s ∈ n k n ks,∞;±n from the preceding lemma that a (λ) 6 q and, with b (λ) 6 q + | n | k ks,∞ | ±n | | ±2n| b (λ) q , that | ±n − ±2n| 2n s b (λ) 6 2 q . h i | ±n | k ks,∞ Furthermore, by (14),

2 q 6 n1/2−|s|. k ks,∞ s Therefore, for any λ, µ S , ∈ n 1/2 |s| an(µ) + bn(λ)b−n(λ) 6 1 + 2 2n q s,∞ | | | | h i k k (24)  |s|  1/2 2 2 < 6n q s,∞ 6 4n = inf λ n π . k k λ∈∂Dn| − | It then follows that det B (λ) has no root in S D . Indeed, assume that n n \ n ξ S is a root, then ξ n2π2 a (ξ) = b (ξ)b (ξ) 1/2 and hence ∈ n | − − n | | n −n | ξ n2π2 6 a (ξ) + b (ξ)b (ξ) 1/2 < 4n1/2, | − | | n | | n −n | implying that ξ D . In addition, (24) implies that by Rouché’s theorem the ∈ n two analytic functions λ n2π2 and λ n2π2 a (λ), defined on the strip S − − − n n have the same number of roots in Dn when counted with multiplicities. As a consequence (λ n2π2 a (λ))2 has a double root in D . Finally, (24) also − − n n implies that

1/2 2 2 sup bn(λ)b−n(λ) < inf λ n π sup an(λ) λ∈Sn| | λ∈∂Dn| − |− λ∈Sn| | 2 2 6 inf λ n π an(λ) λ∈∂Dn| − − | and hence again by Rouché’s theorem, the analytic functions (λ n2π2 a (λ))2 − − n and (λ n2π2 a (λ))2 b (λ)b (λ) have the same number of roots in D . − − n − n −n n

October 4, 2016 14 Altogether we thus have established that det(B (λ))=(λ n2π2 a (λ))2 n − − n − bn(λ)b−n(λ) has precisely two roots ξn,1, ξn,2 in Dn.

To estimate the distance of the roots, write det Bn(λ) as a product g+(λ)g−(λ) where g (λ)= λ n2π2 a (λ) ϕ (λ) and ϕ (λ)= b (λ)b (λ) with an ± − − n ∓ n n n −n arbitrary choice of the sign of the root for any λ. Eachp root ξ of det(Bn) is either a root of g+ or g− and thus satisfies

ξ n2π2 + a (ξ) ϕ (ξ) . ∈ { n ± n } As a consequence,

ξn,1 ξn,2 6 an(ξn,1) an(ξn,2) + max ϕn(ξn,1) ϕn(ξn,2) | − | | − | ± | ± | (25) 6 sup ∂λan(λ) ξn,1 ξn,2 +2 sup ϕn(λ) . λ∈Dn| || − | λ∈Dn| | Since

dist(D , ∂S ) > 12n 4n1/2 > 8n, n n − one concludes from Cauchy’s estimate and the estimate 2 q 6 n1/2−|s| k ks,∞ following from (14) that q supλ∈Sn an(λ) s,∞ 1 sup ∂λan(λ) 6 | | 6 k k 6 . λ∈Dn| | dist(Dn, ∂Sn) 8n 16 Therefore, by (25),

2 ξn,1 ξn,2 6 6 sup bn(λ)b−n(λ) | − | λ∈Dn| |

as claimed. Ú

2.3 Proof of Theorem 2.1 (i)

Let q Fℓw,s,∞ with 1/2

λ+ 4 λ− 4 λ+ 4 , and λ± = n2π2 + nℓ2 . 0 1 1 ··· n n

It follows from a standard counting argument that for n > ns with ns as in Corollary 2.5 that λ± S and λ± / S for any k = n. It then follows n ∈ n n ∈ k 6 from Lemma 2.7 and Lemma 2.11 that ξ ,ξ = λ−, λ+ and hence { n,1, n,2} { n n } γ = λ+ λ− satisfies n n − n γ = ξ ξ , n > n . | n| | n,1 − n,2| ∀ s Lemma 2.12 If q Fℓw,s,∞ with w M and 1/2 < s 6 0, then for any ∈ 0,C ∈ − N > ns, 16c T γ(q) 6 4 T q + s q 2 . ⋊ k N kw,s,∞ k N kw,s,∞ N 1/2−|s| k kw,s,∞

October 4, 2016 15 Proof of Theorem 2.1 (i). By Lemma 2.11,

γn = ξn,1 ξn,2 6 √3 sup bn(λ) + sup b−n(λ) | | | − | λ∈Sn| | λ∈Sn| |   6 √3 q + q + sup b (λ) q + sup b (λ) q . | 2n| | −2n| | n − 2n| | −n − −2n|  λ∈Sn λ∈Sn 

It then follows from Lemma 2.4 and Lemma 2.9 that for n > N with N := ns 4c w 2n s γ 6 √3 w 2n s q + w 2n s q + s q 2 . 2nh i | n| 2nh i | 2n| 2nh i | −2n| n1/2−|s| k kw,s,∞   w,s,∞ Thus, (γn(q))n>1 ℓC (N). As ns can be chosen locally uniformly in q w,s,∞ ∈ w,s,∞ w,s,∞ ∈

Fℓ , the map Fℓ ℓ (N), q (γ (q)) > is locally bounded. Ú 0,C 0,C → C 7→ n n 1

2.4 Jordan blocks of L(q)

To treat the Dirichlet problem, we develop the methods of [7], where the case F s,p q ℓ0,C with 1/2 6 s 6 0 and 2 6 p < was considered, to the case ∈ − s,∞ with 1/2 < s 6 0 and p = . If q Fℓ C is not real valued, then the − ∞ ∈ 0, operator L(q) might have complex eigenvalues and the geometric multiplicity of an eigenvalue could be less than its algebraic multiplicity.

We choosen ˇs > ns, where ns as in Corollary 2.5, so that in addition

λ± 6 (ˇn 1)2π2 +n ˇ /2, n< nˇ , | n | s − s ∀ s λ± n2π2 6 n/2, n > nˇ , (26) | n − | ∀ s λ± are 1-periodic [1-antiperiodic] if n even [odd] n > nˇ . n ∀ s F w,s,∞ Note thatn ˇs can be chosen uniformly on bounded subsets of ℓ0,C since F w,s,∞ −1 ℓ0,C embeds compactly into H0,C. For n > nˇs we further let

+ − + − Null(L λn ) Null(L λn ), λn = λn , En = − ⊕ − 6 Null(L λ+)2, λ+ = λ−.  − n n n We need to estimate the coefficients of L(q) when represented with re- En + − spect to an appropriate orthonormal basis of E n. In the case where λn = λn the matrix representation will be in Jordan normal form. By Lemma C.3, F s+2,∞ 2 2 C + 2 En ℓ⋆,C ֒ L := L ([0, 2], ). Denote by fn En an L -normalized ⊂ → + 2 ∈ eigenfunction corresponding to λn and by ϕn an L -normalized element in En so that f +, ϕ forms an L2-orthonormal basis of E . Then the following { n n} n lemma holds.

F s,∞ ′ Lemma 2.13 Let q ℓ0,C with 1/2 nˇs – ∈ − ′ with nˇs given by (26) – so that for any n > ns,

(L λ+)ϕ = γ ϕ + η f +, − n n − n n n n

October 4, 2016 16 where η C satisfies the estimate n ∈ η 6 16( γ + b (λ+) + b (λ+) ). | n| | n| | n n | | −n n | The threshold n′ can be chosen locally uniformly in q Fℓs,∞. ⋊ s ∈ 0,C Proof. We begin by verifying the claimed formula for (L λ+)ϕ in the case − n n where λ+ = λ−. Let f − be an L2-normalized eigenfunction corresponding to n 6 n n λ−. As f − E there exist a, b C with a 2 + b 2 = 1 and b = 0 so that n n ∈ n ∈ | | | | 6 1 a f − = af + + bϕ or ϕ = f − f +. n n n n b n − b n Hence 1 a Lϕ = λ−f − λ+f +. n b n n − b n n − Substituting the expression for fn into the latter identity then leads to a (L λ+)ϕ =(λ− λ+)ϕ + (λ− λ+)f + = γ ϕ + η f + − n n n − n n b n − n n − n n n n where η = γ a/b. In the case λ+ is a double eigenvalue of geometric mul- n − n n tiplicity two, ϕn is an eigenfunction of L and one has ηn = 0. Finally, in the case λ+ is a double eigenvalue of geometric multiplicity one, (L λ+)ϕ is in n − n n the eigenspace E+ E as claimed. n ⊂ n To prove the claimed estimate for η , we view (L λ+)ϕ = γ ϕ + n − n n − n n η f + as a linear equation with inhomogeneous term g = γ ϕ + η f +. By n n − n n n n identity (18) one has

B P ϕ = γ P K ϕ η P K f +, n n n n n n n − n n n n + + 2 where Kn Kn(λn ) and Bn Bn(λn ). To estimate ηn, take the L -inner ≡ ≡ + product of the latter identity with Pnfn to get

η P K f +,P f + = γ I II, (27) nh n n n n n i n − where

I = P K ϕ ,P f + , II = B P ϕ ,P f + . h n n n n n i h n n n n n i We begin by estimating P K f +,P f + . Using that K = Id+ T K one h n n n n n i n n n gets

+ + + 2 + + P K f ,P f = P f 2 + T K f ,P f , h n n n n n i k n n kL h n n n n n i and by Cauchy-Schwarz

1/2 + + + 2 + T K f ,P f 6 T K f ,e P f 2 . |h n n n n n i| |h n n n mi| k n n kL m∈{±Xn} 

October 4, 2016 17 + + Note that P f 2 6 f 2 = 1. Moreover, by Lemma 2.10 one has k n n kL k n kL log n + + n Cs q s,∞ Knfn s,∞;±n, s = 0, TnKnfn ,e±n 6 k k k k |h i|  1 C q K f + , 1/2 nˇ so that k n k0,2;±n s s 1 T K f +,P f + 6 . |h n n n n n i| 8 ′ + By increasing n if necessary, Lemma 2.14 below assures that P f 2 > 1/2. s k n n kL Thus the left hand side of (27) can be estimated as follows

1 1 1 η P K f +,P f + > η = η , n > n′ . (28) nh n n n n n i | n| 4 − 8 8| n| ∀ s  

Next let us estimate the term I = P K ϕ ,P f + in (27). Using again h n n n n n i Kn =Id+ TnKn one sees that

I = P ϕ ,P f + + T K ϕ ,P f + . h n n n n i h n n n n n i + + Clearly, P ϕ ,P f 6 ϕ 2 f 2 6 1 and arguing as above for the |h n n n n i| k nkL k n kL second term, one then concludes that

I 6 1 + 1/8, n > n′ . (29) | | ∀ s + Finally it remains to estimate II = BnPnϕn,Pnfn . Using again ϕn L2 = + h i k k f 2 = 1, we conclude from the matrix representation (21) of B that k n kL n + + + 2 2 B P ϕ ,P f 6 B ϕ 2 f 2 6 λ n π a + b + b . |h n n n n n i| k nkk nkL k n kL | n − − n| | n| | −n| + Since det Bn(λn ) = 0, one has 1 λ+ n2π2 a = b b 1/2 6 ( b + b ), | n − − n| | n −n| 2 | n| | −n| ′ and hence it follows that for all n > ns that

II 6 2( b + b ). (30) | | | n| | −n|

Combining (28)-(30) leads to the claimed estimate for ηn. Ú

It remains to prove the estimate of Pn used in the proof of Lemma 2.13. To this end, we introduce for n > n the Riesz projector P : L2 E given by s n,q → n (see also Appendix C)

1 −1 Pn,q = (λ L(q)) dλ. 2πi 2 2 − Z|λ−n π |=n

October 4, 2016 18 Lemma 2.14 Let q Fℓs,∞ with 1/2 < s 6 0. Then there exists n˜ > nˇ ∈ 0,C − s s – with nˇ given by (26) – so that for any eigenfunction f Fℓs+2,p of L(q) s ∈ ⋆,C corresponding to an eigenvalue λ S with n > n˜ , ∈ n s 1 P f 2 > f 2 . k n kL 2k kL

The threshold n˜s can be chosen locally uniformly for q. ⋊

Proof. In Lemma C.4 we show that as n , → ∞

P P 2 ∞ = o(1), k n,q − nkL →L locally uniformly in q Fℓs,∞. Clearly, P f = f, hence ∈ 0,C n,q

P f 2 > P f 2 (P P )f 2 > 1+ o(1) f 2 . Ú k n kL k n,q kL − k n,q − n kL k kL
2.5 Proof of Theorem 2.1 (ii)

s,∞ We begin with a brief outline of the proof of Theorem 2.1 (ii). Let q Fℓ C ∈ 0, with 1/2 < s 6 0. Since according to [16] for any q H−1 the Dirichlet − ∈ 0,C eigenvalues, when listed in lexicographical ordering and with their algebraic 2 2 2 multiplicities, µ1 4 µ2 4 , satisfy the asymptotics µn = n π +nℓn, they are ··· −1 simple for n > ndir, where ndir > 1 can be chosen locally uniformly for q H0,C. 2 ∈ For any n > ndir let gn be an L -normalized eigenfunction corresponding to

µn. Then

1 1 g H C := g H ([0, 1], C) : g(0) = g(1) = 0 . n ∈ dir, { ∈ } F s,∞ ′ Now let q ℓ0,C with 1/2 ndir and denote by En the two dimensional subspace introduced 2 in Section 2.4. We will choose an L -normalized function G˜n in En so that its 1 restriction G to the interval = [0, 1] is in H C and close to g . We then n I dir, n show that µ λ+ can be estimated in terms of (L λ+)G , G , where n − n h dir − n n niI f, g denotes the L2-inner product on , f, g = 1 f(x)g(x) dx. As by h iI I h iI 0 Lemma 2.13 R (L λ+)G˜ = O( γ + b (λ+) + b (λ+) ), − n n | n| | n n | | −n n | the claimed estimates for µ τ = µ λ+ + γ /2 then follow from the n − n n − n n estimates of γ of Theorem 2.1 (i) and the ones of b q , b q of n n − 2n −n − −2n Lemma 2.9 (ii). ˜ + 2 The function Gn is defined as follows. Let fn , ϕn be the L -orthonormal 1 basis of En chosen in Section 2.4. As En HC(R/2Z), its elements are con- ⊂ + tinuous functions by the Sobolev embedding theorem. If fn (0) = 0, then + + fn (1) = 0 as fn is an eigenfunction of the 1-periodic/antiperiodic eigenvalue λ+ of L(q) and we set G˜ = f +. If f +(0) = 0, then we define G˜ (x) = n n n n 6 n

October 4, 2016 19 r ϕ (0)f +(x) f +(0)ϕ (x) , where r > 0 is chosen in such a way that n n n − n n n 1 G˜ (x) 2 dx = 1. Then G˜ (0) = G˜ (1) = 0 and since G˜ is an element of 0 | n | n
n n E its restriction G := G˜ is in H1 . R n n n I dir,C Denote by Πn,q the Riesz projection, introduced in Appendix C,

1 −1 Πn,q := (λ Ldir(q)) dλ. 2πi 2 2 − Z|λ−n π |=n It has span(g ) as its range, hence there exists ν C so that n n ∈

Πn,qGn = νngn.

F s,∞ ′′ ′ Lemma 2.15 Let q ℓ0,C with 1/2 < s 6 0. Then there exists ns > ns ′ ∈ − ′′ with ns as in Lemma 2.13 so that for any n > ns

ν (µ λ+)g = β η Π (f + ) γ Π (ϕ ) , (31) n n − n n n n n,q n I − n n,q n I
where βn C with βn 6 1 and ηn is the off-diagonal coefficient in the matrix ∈ | | + + representation of (L λn ) with respect to the basis fn , ϕn , introduced in − En { } Lemma 2.13, and

1/2 6 ν 6 3/2. (32) | n| n′′ can be chosen locally uniformly for q Fℓs,∞. ⋊ s ∈ 0,C Proof. Write G = ν g + h , where h = (Id Π )G . Then n n n n n − n,q n (L λ+)G = ν (µ λ+)g +(L λ+)h . dir − n n n n − n n dir − n n On the other hand, G = G˜ , where G˜ E is given by G˜ = α f + +β ϕ n n I n ∈ n n n n n n C 2 2 1 with αn, βn satisfying αn + βn = 1 and Gn Hdir,C. Hence by ∈ | | | ′ | ∈ Lemma C.1 and Lemma 2.13, for n > ns,

(L λ+)G =(L λ+)G˜ = β (η f + γ ϕ ) . dir − n n − n n I n n n − n n I

Combining the two identities and using that Πn,qhn = 0 and that Πn,q com- mutes with (L λ+), one obtains, after projecting onto span(g ), iden- dir − n n tity (31).

It remains to prove (32). Taking the inner product of Πn,qGn = νngn with gn one gets

ν = ν g , g = Π G , g . n nh n niI h n,q n niI

Let sn(x) = √2 sin(nπx) and denote by Πn = Πn,0 the orthogonal projection onto span s . Recall that P : L2 E is the Riesz projection onto E . In { n} n,q → n n Lemma C.4 we show that as n , → ∞

Π Π 2 ∞ , P P 2 ∞ = o(1), (33) k n,q − nkL (I)→L (I) k n,q − nkL →L

October 4, 2016 20 F s,∞ ˜ locally uniformly in q ℓ0,C . Thus using ΠnGn = Πn(PnGn) I and recalling 2 ∈2 that Gn L2(I) = gn L2(I) = 1 we obtain k k k k v = Π G , g + (Π Π )G , g = P G˜ , Π g + o(1). n h n n niI h n,q − n n niI h n n n niI Moreover, it follows from (33) that uniformly in 0 6 x 6 1

Π g (x) = eiφn s (x)+ o(1), n , n n n → ∞ with some real φn. Similarly, again by (33), uniformly in 0 6 x 6 2 P G˜ (x)= a e (x)+ b e (x)+ o(1), n , n n n n n −n → ∞ where, since G 2 = 1 and G (0) = 0, the coefficients a and b can be k nkL (I) n n n chosen so that a 2 + b 2 = 1, a + b = 0. | n| | n| n n iψn That is PnG˜n(x) = e sn(x)+ o(1) with some real ψn and hence

P G˜ , Π g = eiψn−iφn s ,s + o(1) = eiψn−iφn + o(1), n . h n n n niI h n niI → ∞ From this we conclude ν =1+ o(1), n . | n| → ∞ Therefore, 1/2 6 ν 6 3/2 for all n > n′′ provided n′′ > n′ is sufficiently | n| s s s large. ′′ Going through the arguments of the proof one verifies that ns can be chosen

locally uniformly in q. Ú

Lemma 2.15 allows to complete the proof of Theorem 2.1 (ii).

Proof of Theorem 2.1 (ii). Take the inner product of (31) with gn and use that ν > 1/2 by Lemma 2.15 to conclude that | n| 1 µ λ+ 6 β η Π (f + ), g + γ Π (ϕ ), g . (34) 2| n − n | | n| | n|h n,q n I niI | n||h n,q n I niI| 2
Recall that β 6 1 and note that for any f, g LC( ) | n| ∈ I

Π f, g 6 Π f, g + (Π Π )f, g 6 (1 + o(1)) f 2 g 2 , |h n,q iI| |h n iI| |h n,q − n iI| k kL (I)k kL (I) where for the latter inequality we used that by Lemma C.4 (ii), Πn,q + k − Π 2 2 = o(1) as n . Since f 2 = ϕ 2 = 1 and nkL (I)→L (I) → ∞ k n kL (I) k nkL (I) g 2 =1, (34) implies that k nkL (I) µ λ+ 6 (2 + o(1))( η + γ ) | n − n | | n| | n| yielding with Lemma 2.13 the estimate µ τ 6 (3 + o(1)) γ +(32+ o(1))( γ + b (λ+) + b (λ+) ). | n − n| | n| | n| | n n | | −n n | By Theorem 2.1 (i) and Lemma 2.9 (ii) it then follows that (τn µn)n>1 w,s,∞ − ∈ ℓC (N). Going through the arguments of the proof one verifies that the map w,s,∞ w,s,∞

Fℓ ℓ (N), q (τ µ ) > is locally bounded. Ú 0,C → C 7→ n − n n 1

October 4, 2016 21 2.6 Adapted Fourier Coefficients

The bounds of the operator norm T and the coefficients a and b k nkw,s,∞;n n ±n of P K (λ)V , λ S , obtained in Lemma 2.4 and Lemma 2.9, respectively, n n ∈ n are uniform in λ S and in q on bounded subsets of Fℓw,s,∞. In addition, ∈ n 0,C they are also uniform with respect to certain ranges of p and the weight w. To give a precise statement we introduce the balls

w,s,∞ w,s,∞ s,∞ s,∞ B := q Fℓ C : q 6 m , B := q Fℓ C : q 6 m . m { ∈ 0, k kw,s,∞ } m { ∈ 0, k ks,∞ } Then according to Lemma 2.4, given m> 0 and 1/2 N , (35) n1/2−|s| m,s ′ where cs > cs > 1 is chosen as in Lemma 2.10. This estimate implies that T (λ) 6 1/2, λ S , w M, q Bw,s,∞. k n kw,s,∞;n ∀ ∈ n ∈ ∈ 2m Lemma 2.16 Let 1/2 < s 6 0 and m > 1. For n > N with N given − m,s m,s as in (35), the coefficients a and b are analytic functions on S Bs,∞. n ±n n × m Moreover, their restrictions to S Bw,s,∞ for any w M satisfy n × 2m ∈ 2 8csm (i) an S ×Bw,s,∞ 6 6 m/4. | | n 2m n1/2−|s| s log n ′ 2 log n ⋊ (ii) w2n 2n b±n q±2n S ×Bw,s,∞ 6 h i8csm 6 h im/4. h i | − | n 2m n1−|s| n1/2

Proof. The claimed analyticity follows from the representations (19) of an and

b±n and the bounds from Lemma 2.4, Lemma 2.9, and Lemma 2.10. Ú

Lemma 2.17 Let 1/2 < s 6 0 and m > 1. For each n > N with N − m,s m,s given as in (35), there exists a unique real analytic function

2 s,∞ C 2 2 8csm αn : B2m , αn n π Bs,∞ 6 6 m/4, → | − | 2m n1/2−|s| such that λ n2π2 a (λ, ) 0 identically on Bs,∞. ⋊ − − n · |λ=αn ≡ 2m Proof. We follow the proof of [21, Lemma 5]. Let E denote the space of analytic 2 s,∞ C 2 2 8csm functions α: B2m with α n π Bs,∞ 6 1/2−|s| equipped with the usual → | − | 2m n metric induced by the topology of uniform convergence. This space is complete

– cf. [6, Theorem A.4]. Fix any n > Nm,s and consider on E the fixed point problem for the operator Λn,

Λ α := n2π2 + a (α, ). n n · By (35), each such function satisfies

2 2 1/2 α n π Bs,∞ 6 2m< 4n , | − | 2m

October 4, 2016 22 and hence maps the ball Bs,∞ into the disc D = λ n2π2 6 4n1/2 S . 2m n {| − | }⊂ n Therefore, by Lemma 2.16

2 2 2 8csm Λnα n π Bs,∞ 6 an S ×Bs,∞ 6 , | − | 2m | | n 2m n1/2−|s| meaning that Λn maps E into E. Moreover, Λn contracts by a factor 1/4 by Cauchy’s estimate,

an s,∞ 2 Sn×B2m 1 8csm 1 ∂ a s,∞ 6 | | 6 6 . λ n Dn×B 1/2 1/2−|s| | | 2m dist(Dn, ∂Sn) 12n 4n n 4 −

Hence, we find a unique fixed point αn = Λnαn with the properties as claimed. Ú

To simplify notation define α−n := αn for n > 1. For any given m > 1, (m) s,∞ define the map Ω on B2m by

(m) Ω (q)= q2ne2n + bn(αn(q), q)e2n, 06=|nX|XMm,s where Mm,s > Nm,s is chosen such that

8c′ 1 s 6 sup 1/2−|s| . (36) n>Mm,s n 16m

Thus, for n > Mm,s the Fourier coefficients of the 1-periodic function r = (m) Ω (q) are r2n = bn(αn(q)), r−2n = b−n(α−n(q)), and

0 r2n Bn(αn(q), q)= − . r−2n 0 −  These new Fourier coefficients are adapted to the lengths of the corresponding spectral gaps, whence we call Ω(m) the adapted Fourier coefficient map on s,∞ Bm .

Proposition 2.18 For 1/2 < s 6 0 and m > 1, Ω(m) maps Bs,∞ into − m Fℓs,∞. Further, for every w M, its restriction to Bw,s,∞ is a real analytic 0,C ∈ m diffeomorphism

(m) w,s,∞ (m) w,s,∞ F w,s,∞ Ω w,s,∞ : Bm Ω (Bm ) ℓ0,C Bm → ⊂ such that

(m) sup dqΩ Id 6 1/16, (37) w,s,∞ w,s,∞ q∈Bm k − k and Bw,s,p Ω(m)(Bw,s,p). Moreover, m/2 ⊂ m 1 q 6 Ω(m)(q) 6 2 q , q Bw,s,∞. ⋊ (38) 2k kw,s,∞ k kw,s,∞ k kw,s,∞ ∈ m

October 4, 2016 23 s,∞ Proof. Since αn maps B2m into Sn for n > Nm,s, each coefficient bn(αn(q), q) is well defined for q Bs,∞, and by Lemma 2.16 ∈ 2m 2 s s 8csm w2n 2n bn(αn) q2n Bs,∞ 6 w2n 2n bn q2n S ×Bs,∞ 6 . h i | − | 2m h i | − | n 2m n1/2−|s| (m) s,∞ Hence the map Ω is defined on B2m and

(m) s sup Ω (q) q w,s,∞ = sup sup w2n 2n bn(αn) q2n Bw,s,∞ w,s,∞k − k w,s,∞ > h i | − | 2m q∈B2m q∈B2m |n| Mm,s 1 6 2 8csm sup 1/2−|s| n>Mm,s n m2 m 6 6 , 16m 16 by our choice (36)of Mm,s. Consequently, the inverse function theorem (Lemma A.4) applies proving that Ω(m) is a diffeomorphism onto its image which covers w,s,p Bm/2 . If q is real valued, then αn(q) is real and hence by (22) b−n(αn(q)) = b (α (q)) implying that Ω(m)(q) is real valued as well. Altogether we thus − n n have proved that Ω(m) is real analytic. Finally, we note that by Cauchy’s estimate (m) sup w,s,∞ Ω (q) q (m) q∈B2m w,s,∞ 1 sup dqΩ Id 6 k − k 6 , w,s,∞ w,s,∞ q∈Bm k − k m 16 hence in view of the mean value theorem for any q , q Bw,s,p 0 1 ∈ m 1 (m) (m) (m) Ω (q1) Ω (q0)= 1+ d(1−t)q0+tq1 Ω Id dt (q1 q0), − 0 − −  Z    we find that

1 (m) (m)

q q 6 Ω (q ) Ω (q ) 6 2 q q . Ú 2k 1 − 0kw,s,∞ k 1 − 0 kw,s,∞ k 1 − 0kw,s,∞

2.7 Proof of Theorem 2.2

Proposition 2.19 Let 1/2 1, and w M. If q Bs,∞ and − ∈ ∈ m Ω(m)(q) Bw,s,∞, ∈ m/2 w,s,∞ w,s,∞ then q B Fℓ C . ⋊ ∈ m ⊂ 0, (m) s,∞ Proof. By Proposition 2.18, the map Ω is defined on Bm and a real analytic diffeomorphism onto its image; for w M, the restriction of Ω(m) to Bw,s,∞ ∈ m ⊂ Bs,∞ Fℓw,s,∞ is again a real analytic diffemorphism onto its image and by m ∩ 0,C Lemma 2.18 this image contains Bw,s,∞. Thus, if Ω(m) maps q Bs,∞ to m/2 ∈ m r = Ω(m)(q) Bw,s,∞, ∈ m/2 then we must have

(m) −1 w,s,∞ w,s,∞

F Ú q = (Ω ) Bw,s,∞ (r) Bm ℓ0,C . m/2 ∈ ⊂

October 4, 2016 24 To proceed, we want to bound the Fourier coefficients of r = Ω(m)(q) in terms of the gap lengths of q.

s,∞ Lemma 2.20 Let 1/2 < s 6 0, m > 1, and suppose that q Bm , r = (m) − ∈ Ω (q), and n > Mm,s with Mm,s given as in (36). If 1 r r = 0, and 6 2n 6 9, −2n 6 9 r −2n then

r r 6 γ (q) 2 6 9 r r . ⋊ | 2n −2n| | n | | 2n −2n|

Proof. We follow the proof of [21, Lemma 10]. To begin, we write det Bn(λ)= g+(λ)g−(λ) with

g (λ) := λ n2π2 a (λ) ϕ (λ), ϕ (λ)= b (λ)b (λ). ± − − n ∓ n n n −n p The assumption on r±2n implies that g± are continuous, even analytic, func- tions of λ. Indeed, recall that r±2n = b±n(αn), thus

ϕ (α )= b (α )b (α ) = 0, ρ := ϕ (α ) > 0, n n n n −n n 6 n | n n | p so we may choose ϕn(λ) as a fixed branch of the square root locally around

λ = αn. To obtain an estimate of the domain of analyticity, we consider the disc Do := λ : λ α 6 2ρ . Since by assumption n > M it follows n { | − n| n} m,s from (36) together with cs > 1 that m n1/2−|s| 6 , n > M . 4 128 ∀ m,s Lemma 2.16 then yields

m n1/2−|s| an 6 6 . (39) | |Sn 4 128

To estimate bn note that | |Sn |s| s s bn 6 2n ( 2n bn q2n + 2n q2n ). | |Sn h i h i | − |Sn h i | | Since q Bs,∞, 2n s q 6 m and hence again by Lemma 2.16 one has ∈ m h i | 2n| 1/2 |s| |s| n bn 6 2n (m/2+ m) 6 4n m 6 . (40) | |Sn 8

Cauchy’s estimate and definition (9) of Sn then gives b 1/2 ±n Sn n /8 1 ∂λb±n Do 6 | | 6 6 , (41) | | n dist(Do , ∂S ) 12n n2π2 α 2ρ 88 n n −| − n|− n 2 2 1/2−|s| where we used that by Lemma 2.17, αn n π 6 m/4 6 n /128 and 1/2 | − | by (40), ρn 6 n /8. Note that by (39), the same estimate holds for ∂λan,

∂λan o 6 1/88 (42) | |Dn

October 4, 2016 25 Thus by the mean value theorem, for any λ Do , ∈ n 1 b±n(λ) b±n(αn) o 6 ∂λb±n o 2ρn 6 ρn, (43) | − |Dn | |Dn 44 implying that ϕ (λ)2 is bounded away from zero for λ Do . Hence ϕ (λ) is n ∈ n n analytic for λ Do . ∈ n By Lemma 2.11, det Bn(λ) = g+(λ)g−(λ) has precisely two roots in Sn which both are contained in D S . To estimate the location of these roots, n ⊂ n we approximate g±(λ) by h±(λ) defined by

h (λ)= λ n2π2 a (α ) ϕ (α ), + − − n n − n n h (λ)= λ n2π2 a (α )+ ϕ (α ). − − − n n n n Since α n2π2 a (α ) =0, one has n − − n n h (λ)= λ α ϕ (α ), h (λ)= λ α + ϕ (α ). + − n − n n − − n n n

Clearly, h+(λ) and h−(λ) each have precisely one zero λ+ = αn + ϕn(αn) and λ = α ϕ (α ), respectively. − n − n n We want to compare h+ and g+ on the disc

D+ := λ : λ (α + ϕ (α )) < ρ /2 Do . n { | − n n n | n }⊂ n

Since h+(αn + ϕn(αn)) = 0, we have

ρn h+ + = h+(λ) h+(αn + ϕn(αn)) + = . | |∂Dn | − |∂Dn 2 In the sequel we show that 4 ∂λϕn 6 , (44) | |Dn+ 88 yielding together with (42)

h+ g+ + 6 an(αn) an(λ) + + ϕn(αn) ϕn(λ) + | − |Dn | − |Dn | − |Dn ρn 6 ∂λan + + ∂λϕn + 2ρn < = h+ + . | |Dn | |Dn 2 | |∂Dn   Thus, it follows from Rouche’s theorem that g+ has a single root contained + in Dn . In a similar fashion, we find that g− has a single root contained in − Dn := λ : λ (αn ϕn(αn)) < ρn/2 . Since the roots of g±(λ) are roots of { | − − | }± det Bn(λ), they have to coincide with λn and hence ρ 6 λ+ λ− 6 3ρ , n | n − n | n which is the claim. + + o It remains to show the estimate (44) for ∂ ϕ on D . Note that Dn D λ n n ⊂ n and write b (λ) 1/2 b (λ) = b (λ)b (λ) 1/2 n . | n | | n −n | b (λ) −n

October 4, 2016 26 By the assumption of this lemma, 1 6 |bn(αn)| 6 9 and by the estimate (43), 9 |b−n(αn)|

1 ρn ρn 6 bn(αn) 6 3ρn, bn(λ) bn(αn) o 6 . 3 | | | − |Dn 44 Thus, by the triangle inequality we obtain

41 1 1 + 1 133 ρn = ρn 6 bn Do 6 3+ ρn = ρn. 132 3 − 44 | | n 44 44     Treating b−n in an analogous way, we arrive at + − bn bn − , + 6 10, bn o bn o Dn Dn

which in view of ( 41) finally yields the desired estimate (44),

+ 1/2 − 1/2 ∂λb o − ∂λb o + n Dn bn n Dn bn 4 ∂λϕn o 6 | | + | | 6 . Ú Dn + − | | 2 bn o 2 bn o 88 Dn Dn

Lemma 2.21 If q H−1 with gap lengths γ (q ) ℓs,∞, 1/2 < s 6 0, 0 ∈ 0 0 ∈ − then Iso(q ) is a -norm bounded subset of Fℓs,∞. In particular, q is in 0 k·ks,∞ 0 0 F s,∞ ⋊ ℓ0 .

Proof. Suppose q is a real valued potential in H−1 with gap lengths γ(q ) 0 0 0 ∈ ℓs,∞ for some 1/2 < s 6 0. We can choose 1/2 <σ 1 and hence ℓs,∞ ֒ ℓσ,p. Consequently, by [7, Corollary 3 − σ,p → we have q0 Fℓ . Moreover, by [7, Corollary 4] the isospectral set Iso(q0) is ∈ σ,p compact in Fℓ , hence there exists R > 0 so that Iso(q0) is contained in the σ,p F s,∞ ball BR . To prove that Iso(q0) is a bounded subset of ℓ , we choose

m = 4(R + γ(q ) ). (45) k 0 ks,∞ Further, let w = n −σ+s, n Z. Then w M, ℓw,σ,∞ = ℓs,∞, and γ(q ) n h i ∈ ∈ 0 ∈ ℓw,σ,∞, while for any q Iso(q ) we have ∈ 0 q Bσ,∞. ∈ m (m) σ,∞ The map Ω is well defined on Bm and

r r(q) = Ω(m)(q) Fℓσ,∞. ≡ ∈ 0 Since r is real valued, we have r = r for all n Z. Suppose n > M , −n n ∈ | | m,s then it follows from Lemma 2.20 that for any q Iso(q ) with r = 0 that ∈ 0 | n| 6 r = r r 1/2 6 γ (q) = γ (q ) . | n| | n −n| | n | | n 0 |

The same estimate holds true when rn = 0. In particular, it follows that r F w,σ,∞ | | ∈ ℓ0 . To satisfy the smallness assumption of Proposition 2.19 for r w,σ,∞, ε ε k k ε|n| we modify the weight w: let w be the weight defined by wn = min(wn, e ),

October 4, 2016 27 Z ε ε ε Z n . Note that w−n = wn, wn > 1, and w|n| 6 w|n|+1 for any n . ∈ ε ep ε∈ For ε > 0 sufficiently small, one verifies that log wn+m 6 log wn + log wm for any n, m Z. Thus for ε > 0 sufficiently small, wε is submultiplicative and ∈ therefore wε M – see [21, Lemma 9] for details. Moreover, ∈ ε2n σ σ r(q) wε,σ,∞ 6 sup e 2n q2n + sup w2n 2n rn k k |n|Mm,σ h i | | 6 e2εMm,σ q + γ(q ) . k kσ,∞ k 0 kw,σ,∞ Choosing ε> 0 sufficiently small, we conclude from (45) that

r(q) ε 6 2 q + γ(q ) 6 m/2. k kw ,σ,∞ k kσ,∞ k 0 ks,∞ ε Thus Proposition 2.19 applies yielding q Bw ,σ,∞. By the definition of wε, ∈ m w = wε holds for at most finitely many n, hence n 6 n

q ε 6 C q , k kw ,σ,∞ εk kw,σ,∞ where the constant Cε > 1 depends only on ε and Mm,σ, but is independent of q. Since q = q , it thus follows that q 6 C m for all k kw,σ,∞ k ks,∞ k ks,∞ ε q Iso(q ). Ú ∈ 0 −1 Proof of Theorem 2.2. Suppose q is a real valued potential in H0 with gap lengths γ(q) ℓs,∞ for some 1/2 < s 6 0. By the preceding lemma, Iso(q) ∈ − F s,∞ F σ,p is bounded in ℓ0 . Moreover, by [7], Iso(q) is compact in ℓ0 for any 2 6 p< and 1/2 6 σ 6 0 with (s σ)p> 1. Consequently, Iso(q) is weak* ∞ s,−∞ −

F Ú compact in ℓ0 by Lemma B.1.

ℓs,∞ 3 Birkhoff coordinates on F 0

The aim of this section is to prove Theorem 1.4. First let us recall the results −1 on Birkhoff coordinates on H0 obtained in [10]. W −1 Theorem 3.1 ([10, 15]) There exists a complex neighborhood of H0 within −1 −1/2 H C and an analytic map Φ: W h C , q (z (q)) Z with the following 0, → 0, 7→ n n∈ properties: (i) Φ is canonical in the sense that z , z = 1 ∂ z ∂ ∂ z dx = i for { n −n} 0 u n x u −n > all n 1, whereas all other brackets between coordinateR functions vanish. s s+1/2 (ii) For any s > 1, the restriction Φ Hs is a map Φ Hs : H h which − | 0 | 0 0 → 0 is a bianalytic diffeomorphism.

(iii) The KdV Hamiltonian H Φ−1, expressed in the new variables, is defined 3/2 ◦ on h0 and depends on the action variables alone. In fact, it is a real 3,1 N analytic function of the actions on the positive quadrant ℓ+ ( ),

3,1 3 ℓ (N) := (I ) > : I > 0 n > 1, n I < . ⋊ + { n n 1 n ∀ n ∞} > nX1

October 4, 2016 28 We will also need the following result (cf. [10, § 3]).

Theorem 3.2 ([10]) After shrinking, if necessary, the complex neighborhood W −1 −1 of H0 in H0,C of Theorem 3.1 the following holds: (i) Let Z = q H−1 : γ2 (q) = 0 for n > 1. The quotient I /γ2, defined n { ∈ 0 n 6 } n n on H−1 Z , extends analytically to W for any n > 1. Moreover, for any 0 \ n ε > 0 and any q W there exists n > 1 and an open neighborhood W ∈ 0 q of q in W so that I 8nπ n 1 6 ε, n > n p W . (46) γ2 − ∀ 0 ∀ ∈ q n

(ii) The Birkhoff coordinates (zn)n∈Z are analytic as maps from W into C and fulfill locally uniformly in W and uniformly for n > 1, the estimate

γ + µ τ z = O | n| | n − n| . | ±n| √n   (iii) For any q W and n > 1 one has I (q) = 0 if and only if γ (q) = 0. In ∈ n n particular, Φ(0) = 0. ⋊

3.1 Birkhoff coordinates

In [7] based on the results of [15], the restrictions of the Birkhoff map

−1 −1/2 Φ: H h , q (z (q)) Z, z (q) = 0, 0 → 0 7→ n n∈ 0 to the Fourier Lebesgue spaces Fℓs,p, 1/2 6 s 6 0, 2 6 p < , are studied. 0 − ∞ It turns out that the arguments developed in the papers [7, 13] can be adapted F s,p to prove Theorem 1.4. As a first step we extend the results in [7] for ℓ0 , 1/2 6 s 6 0, 2 6 p< , to the case p = . More precisely, we prove − ∞ ∞ Lemma 3.3 For any 1/2

W F s,∞ is real analytic and extends analytically to an open neighborhood s,∞ of ℓ0 F s,∞ in ℓ0,C . Its Jacobian d0Φs,∞ at q = 0 is the weighted Fourier transform

F s,∞ s+1/2,∞ 1 d0Φs,∞ : ℓ0 ℓ0 , f f,e2n → 7→ 2π max( n , 1)h i! | | n∈Z with inverse given by p

−1 s+1/2,∞ s,∞ (d Φ ) : ℓ Fℓ , (z ) Z 2π n z e . 0 s,∞ 0 → 0 n n∈ 7→ | | n 2n n∈Z X p In particular, Φs,∞ is a local diffeomorphism at q = 0. ⋊

October 4, 2016 29 Proof. The coordinate functions zn(q) are analytic functions on the complex neighborhood W H−1 of H−1 of Theorem 3.2. Since for any 1/2 1. By the asymptotics of the periodic and Dirichlet eigenvalues of Theorems 2.1, Φs,∞ maps the complex neighborhood W := W Fℓs,∞ of Fℓs,∞ into the space ℓs+1/2,∞ and is s,∞ ∩ 0,C 0 0,C locally bounded. Using [12, Theorem A.3], one sees that Φs,∞ is analytic.

The formulas for d0Φs,∞ and its inverse follow from [12, Theorem 9.7] by

continuity. Ú

In a second step, following arguments used in [13], we prove that Φs,∞ is onto.

Lemma 3.4 For any 1/2 < s 6 0, the map Φ : Fℓs,∞ ℓs+1/2,∞ is − s,∞ 0 → 0 onto. ⋊

Proof. Given any z ℓs+1/2,∞ h−1/2, there exists q H−1 so that Φ(q)= z. ∈ 0 ⊂ 0 ∈ 0 Moreover, by Theorem 3.2 (i) we have for all n sufficiently large 8nπI 1 n > . γ2 2 n

s+1/2,∞ s,∞ Since In = zn z−n and z ℓ , this implies γ(q) ℓ (N). Using Theo- ∈ 0 ∈ rem 2.2, we conclude that q Fℓs,∞. Since by definition Φ is the restriction ∈ 0 s,∞ F s,∞ of the Birkhoff map Φ to ℓ0 , we conclude that

Φs,∞(q)= z.

This completes the proof. Ú

3.2 Isospectral sets

Recall that for any z h−1/2, the torus h−1/2 was introduced in (2). ∈ 0 Tz ⊂ 0 Lemma 3.5 Suppose q Fℓs,∞ with 1/2

(ii) Φ (Iso(q)) = . s,∞ TΦ(q) (iii) If Φ(q) / cs = z ℓs,∞ : n sz 0 , then Φ(Iso(q)) is not compact ∈ 0 { ∈ 0 h i n → } F s,∞ ⋊ in ℓ0 .

October 4, 2016 30 Proof. (i) follows from Theorem 2.2. According to [10] the identity Φ(Iso(q)) = −1 F s,∞ Φ(q) holds for any q H0 and thus implies (ii). Suppose q ℓ0 is such T s ∈ ∈ that z = Φ(q) / c . Then there exists ε > 0 and a subsequence (ν ) > N ∈ 0 n n 1 ⊂ with ν so that n → ∞ ν s z > ε, n > 1. h ni | νn | ∀ N (m) (m) For every m define z z by setting z0 = 0 and for any k > 1, (m) ∈ ∈ T z−k = zk and

(m) zk, k = νm, zk = − z , otherwise.  k It follows that z(m1) z(m2) > 2ε for all m = m , hence is not k − ks,∞ 1 6 2 Tz compact. Ú

3.3 Weak* topology

In this subsection we establish various properties of Φs,∞ related to the weak* topology.

Lemma 3.6 For any 1/2 < s 6 0, the map Φ : Fℓs,∞ ℓs+1/2,∞ is − s,∞ 0 → 0 -norm bounded. ⋊ k·ks,∞ s,∞ F s,∞ Proof. It suffices to consider the case of the ball Bm ℓ0 of radius m > 1. s,∞ −1 ⊂ Since Bm embeds compactly into H0 , (46) implies that one can choose N > N such that for all q Bs,∞, m,s ∈ m 8nπI n 6 > 2 2, n N. γn Since z (q) 2 = I , we conclude with Lemma 2.12 that | n | n s+1/2 TN Φ(q) s+1/2,∞ = sup n zn k k |n|>Nh i | | s 6 sup n γn |n|>Nh i | | 16c 6 4 T q + s q 2 , q Bs,∞. k N ks,∞ N 1/2−|s| k ks,∞ ∈ m

Moreover, each of the finitely many remaining coordinate functions zn(q), n < −1 s,∞ | | N, is real analytic on H0 and hence bounded on the compact set Bm , which

proves the claim. Ú

Lemma 3.7 For any 1/2 < s 6 0, the map Φ : Fℓs,∞ ℓs+1/2,∞ maps − s,∞ 0 → 0 weak* convergent sequences to weak* convergent sequences. ⋊

October 4, 2016 31 (k) ∗ s,∞ (k) s,∞ Proof. Given q ⇁ q in Fℓ , there exists m > 1 so that (q ) > B . 0 k 1 ⊂ m Since q(k) q in H−1 and Φ: H−1 h−1/2 is continuous, it follows that → 0 0 → 0 (k) (k) −1/2 (k) Z z := Φ(q ) Φ(q) =: z in h0 . In particular, zn zn for all n . → (k) → s+1/2,∞ ∈ By the previous lemma it follows that (z )k>1 is bounded in ℓ0 and (k) ∗ s+1/2,∞ hence z ⇁ z in ℓ0 . Ú

Corollary 3.8 For any 1/2 1, the map − Φ : (Bs,∞, σ(Fℓs,∞, Fℓ−s,1)) (ℓs+1/2,∞, σ(ℓs+1/2,∞,ℓ−(s+1/2),1)) s,∞ m 0 0 → 0 0 0 is a homeomorphism onto its image. ⋊

s,∞ F s,∞ F −s,1 Proof. By Lemma B.1,(Bm , σ( ℓ0 , ℓ0 )) is metrizable. Hence by Lemma 3.6 and Lemma 3.7,themapΦ: (Bs,∞, σ(Fℓs,∞, Fℓ−s,1)) (ℓs+1/2,∞, σ(ℓs+1/2,∞,ℓ−(s+1/2),1)) m 0 0 → 0 0 0 is continuous. Since Φ : Fℓs,∞ ℓs+1/2,∞ is bijective and Bs,∞ is compact s,∞ 0 → 0 m with respect to the weak* topology, the claim follows. Ú

3.4 Proof of Theorem 1.4 and asymptotics of the KdV frequencies

Proof of Theorem 1.4. The claim follows from Lemma 3.3, Lemma 3.4, Lemma 3.5,

Lemma 3.6, and Corollary 3.8. Ú

Recall that in [17] the KdV frequencies ωn = ∂In H have been proved to −1 extend real analytically to H0 – see also [11] for more recent results.

Lemma 3.9 Uniformly on -norm bounded subsets of Fℓs,∞, 1/2

ω = (2nπ)3 6I + o(1). ⋊ n − n Proof. The claim follows immediately from [11, Theorem 3.6] and the fact that s,∞ −1/2,p

F F Ú ℓ0 embeds compactly into ℓ if (s + 1/2)p> 1.

4 Proofs of Theorems 1.1 and 1.3

-Proof of Theorem 1.1. According to [17], for any q Fℓs,∞ ֒ H−1, the solu ∈ 0 → 0 tion curve t S(q)(t) H−1 exists globally in time and is contained in Iso(q). 7→ ∈ 0 Since the latter is -norm bounded by Lemma 2.21, the solution curve is k·ks,∞ uniformly -norm bounded in time, k·ks,∞

sup S(q)(t) s,∞ 6 sup q˜ < . t∈Rk k q˜∈Iso(q)k k ∞

By [17], any coordinate function t (S(q))n(t), n Z, is continuous and hence s,∞ 7→ ∈

R (Fℓ , τ ), t S(q)(t) is a continuous map. Ú 7→ 0 w∗ 7→

October 4, 2016 32 Proof of Theorem 1.3. Suppose V Fℓs,∞ is a -norm bounded subset. ⊂ 0 k·ks,∞ Then there exists m > 1 so that V Bs,∞ and the weak* topology induced on ⊂ m Bs,∞ coincides with the norm topology induced from Fℓσ,p provided (s σ)p> m 0 − 1 – see Lemma B.1. Since by [7], for any 1/2 6 σ 6 0, 2 6 p< , the map − ∞ : (V, ) C([ T, T ], (V, )) S k·kσ,p → − k·kσ,p is continuous, it follows that

: (V, τ ) C([ T, T ], (V, τ )) S w∗ → − w∗

is continuous as well. Ú

Proof of Remark 1.2. Since by Lemma 3.3, the Birkhoff map Φ is a local dif- F s,∞ feomorphism near 0, it suffices to show for generic small initial data q in ℓ0 that the solution curve t (q)(t), expressed in Birkhoff coordinates, is not 7→ S continuous. But this latter claim follows in a straightforward way from the

asymptotics of the KdV frequencies of Lemma 3.9. Ú

5 Wiener Algebra

It turns out that by our methods we can also prove that the KdV equation is ω F 0,1 globally in time C -wellposed on ℓ0 , referred to as Wiener algebra. Actually, N,1 we prove such a result for any Fourier Lebesgue space Fℓ with N Z> . 0 ∈ 0

5.1 Birkhoff coordinates F N,1 In a first step we prove that ℓ0 admits global Birkhoff coordinates. More precisely, we show

Theorem 5.1 For any N Z> , the restriction Φ of the Birkhoff map ∈ 0 N,1 Φ to FℓN,1 takes values in ℓN+1/2,1 and Φ : FℓN,1 ℓN+1/2,1 is a real 0 0 N,1 0 → 0 analytic diffeomorphism, and therefore provides global Birkhoff coordinates on F N,1 ⋊ ℓ0 .

Before we prove Theorem 5.1, we need to review the spectral theory of the Schrödinger operator ∂2 + q for q FℓN,1. − x ∈ 0

5.2 Spectral Theory

The spectral theory of the operator L(q)= ∂2+q for q FℓN,1 was considered − x ∈ 0 in [21] where the following results were shown.

Theorem 5.2 ([21, Theorem 1 & 4]) Let N Z> . ∈ 0

October 4, 2016 33 F N,1 (i) For any q ℓ0,C , the sequence of gap lengths (γn(q))n>1, defined in (5) N,1 ∈ is in ℓC (N) and the map

N,1 N,1 Fℓ C ℓC (N), q (γ (q)) > , 0, → 7→ n n 1 is uniformly bounded on bounded subsets. F N,1 (ii) For any q ℓ0,C , the sequence (τn µn(q))n>1 – cf. (5)-(6) – is in N,1 ∈ − ℓC (N) and the map

N,1 N,1 Fℓ C ℓC (N), q (τ (q) µ (q)) > , 0, → 7→ n − n n 1 is uniformly bounded on bounded subsets. ⋊

In addition, the following spectral characterization for a potential q L2 ∈ F N,1 to be in ℓ0 holds.

2 Theorem 5.3 ([21, Theorem 3]) Let q L0 and assume that (γn(q))n>1 N,1 N,1 ∈ N,1 ∈ ℓ for some N Z> . Then q Fℓ and Iso(q) Fℓ . ⋊ R ∈ 0 ∈ 0 ⊂ 0

5.3 Proof of Theorem 5.1

Theorem 5.1 will follow from the following lemmas.

Lemma 5.4 For any N Z> ∈ 0 F N,1 N+1/2,1 ΦN,1 Φ : ℓ0 ℓ0 , q (zn(q))n∈Z, ≡ FℓN,1 → 7→ 0

W F N,1 is real analytic and extends analytically to an open neighborhood N,1 of ℓ0 F N,1 ⋊ in ℓ0,C .

Proof. The coordinate functions zn(q) = (Φ(q))n, n Z, are analytic functions −1 −1 ∈ on the complex neighborhood W H C of H of Theorem 3.2. Furthermore, ⊂ 0, 0 γ (q) + µ (q) τ (q) z (q)= O | n | | n − n | ±n √n   locally uniformly on W and uniformly in n > 1. By the asymptotics of the periodic and Dirichlet eigenvalues of Theorem 5.2, ΦN,1 maps the complex W W F N,1 F N,1 N+1/2,1 neighborhood N,1 := ℓ0,C of ℓ0 into the space ℓ0,C and is locally ∩ −(N+1/2),∞ bounded. It then follows from [6, Theorem A.4] that for any ξ ℓ0,C , W N,1 ∈ W the map q ξ, Φ(q) is analytic on ℓ0,C implying that Φ: N,1 N+1/2,1 7→ h i ∩ → ℓ0,C is weakly analytic. Hence by [6, Theorem A.3], ΦN,1 is analytic. Ú

Next, following arguments used in [13], we prove that ΦN,1 is onto.

N,1 N+1/2,1 Lemma 5.5 For any N Z> , the map Φ : Fℓ ℓ is onto. ⋊ ∈ 0 N,1 0 → 0

October 4, 2016 34 Proof. For any z ℓN+1/2,1 h1/2, there exists q L2 so that Φ(q) = z. ∈ 0 ⊂ 0 ∈ 0 Moreover, by Theorem 3.2 (i) we have for all n sufficiently large 8nπI 1 n > . γ2 2 n

Since I = z z and z ℓN+1/2,1, this implies γ(q) ℓN,1(N). Using Theo- n n −n ∈ 0 ∈ rem 5.3, we conclude that q FℓN,1. Since by definition Φ is the restriction ∈ 0 N,1 F N,1 of the Birkhoff map Φ to ℓ0 , we conclude

ΦN,1(q)= z.

This completes the proof. Ú

N,1 Lemma 5.6 For any q Fℓ with N Z> , ∈ 0 ∈ 0 d Φ : FℓN,1 ℓN+1/2,1 q N,1 0 → 0 is a linear isomorphism. ⋊

Proof. By Theorem 3.1, d Φ: H−1 h−1/2 is a linear isomorphism for any q 0 → 0 −1 F N,1 q H0 . Since dqΦN,1 = dqΦ FℓN,1 for any q ℓ0 , it follows from ∈ 0 ∈ N,1 N+1/2,1 Lemma 5.4 that dqΦN,1 : Fℓ ℓ is one-to one. To show that dqΦN,1 0 → 0 is onto, note that by Theorem 3.1, d Φ : FℓN,1 ℓN+1/2,1 is a weighted 0 N,1 0 → 0 Fourier transform and hence a linear isomorphism. It therefore suffices to show that d Φ d Φ : FℓN,1 ℓN+1/2,1 is a compact operator implying that q N,1 − 0 N,1 0 → 0 dqΦN,1 is a Fredholm operator of index zero and thus a linear isomorphism. To show that d Φ d Φ : FℓN,1 ℓN+1/2,1 is compact we use that by [14, q N,1 − 0 N,1 0 → 0 Theorem 1.4], for any q HN , the restriction of d Φ to HN has the property ∈ 0 q 0 that d Φ d Φ: HN hN+3/2 is a bounded linear operator. In view of the q − 0 0 → 0 fact that FℓN,1 ֒ HN is bounded and hN+3/2 ֒ ℓN+1/2,1 is compact, it 0 → 0 0 →c 0 follows that d Φ d Φ : FℓN,1 ℓN+1/2,1 q N,1 − 0 N,1 0 → 0

is a compact operator. Ú

5.4 Frequencies

Finally, we need to consider the KdV frequencies introduced in Subsection 3.4. F N,1 They are viewed either as functions on ℓ0 or as functions of the Birkhoff N+1/2,1 coordinates on ℓ0 .

Lemma 5.7 The KdV frequencies ωn, n > 1, admit a real analytic extension N,1 N,1 to a common complex neighborhood W of Fℓ , N Z> , and for any 0 ∈ 0 r > 1 have the asymptotic behavior ω 8n3π3 = ℓr , n − n locally uniformly on WN,1. ⋊

October 4, 2016 35 N,1 2 -Proof. Since Fℓ ֒ L C this is an immediate consequence of [11, Theo 0,C → 0, rem 3.6]. Ú

5.5 Wellposedness

We are now in position to prove that the KdV equation is globally in time Cω- N,1 wellposed on Fℓ for any N Z> . First we consider the KdV equation in 0 ∈ 0 Birkhoff coordinates. Let : (t, z) (ϕt (z)) Z denote the flow in Birkhoff SΦ 7→ n n∈ coordinates with coordinate functions

ϕt (z) = eiωn(z)tz , n Z. n n ∈

Lemma 5.8 For any N Z> and T > 0, the map ∈ 0 : ℓN+1/2,1 C([ T, T ],ℓN+1/2,1), z (t (t, z)), SΦ 0 → − 0 7→ 7→ SΦ is real-analytic. ⋊

Proof. Since ω 8n3π3 = o(1) locally uniformly, this is an immediate conse- n − quence of [11, Theorem E.1]. Ú

ω Theorem 5.9 For any N Z>0, the KdV equation is globally in time C - F N,1 ∈ wellposed on ℓ0 . More precisely, for any T > 0, the map : FℓN,1 C([ T, T ], FℓN,1), q (t (t, q)), S 0 → − 0 7→ 7→ S is real analytic. ⋊

Proof. The claim follows immediately from the Lemma 5.8 and the fact estab- lished in Theorem 5.1 that the Birkhoff map is a real analytic diffeomorphism N,1 N+1/2,1

Φ: Fℓ ℓ . Ú 0 → 0

A Auxiliaries

Lemma A.1 For any 1/2 <σ< there exists a constant C > 0 so that for ∞ σ > 1 2σ−1 loghni any n 1, |m|6=n |m2−n2|σ is bounded by Cσ/n if 1/2 <σ< 1, Cσ n if σ = 1, and C /nσ if σ > 1. ⋊ P σ

Proof. [7, Lemma A.1]. Ú

For any s R and 1 6 p 6 denote by ℓs,p ℓs,p(Z, C) the sequence ∈ ∞ C ≡ space

s,p ℓC = z =(z ) Z C : z < . { k k∈ ⊂ k ks,p ∞} Lemma A.2 Suppose 1/2 < s 6 0. For any 1 6 σ 1 one has ℓs,∞ ֒ ℓσ,p and the embedding is compact. In − C → C ⋉ .particular, for any ε> 0, ℓs,∞ ֒ h−1/2+s−ε C → C

October 4, 2016 36 Proof. By Hölder’s inequality

1/p 1/2 σp p s −(s−σ)p m am 6 sup m am m , Z Zh i | | ! m∈ h i | | Zh i ! mX∈   mX∈ s,∞ σ,p provided (s σ)p> 1. Hence ℓC ֒ ℓC . The compactness follows from the − → p well known characterization of compact subsets in ℓ . Ú

The following result is well known – cf. [7, Lemma 20].

t Lemma A.3 (i) Let 1 6 t < 1/2. For a = (am)m∈Z hC and b = 1 − − ∈ (b ) Z hC, the convolution a b =( Z a b ) Z is well defined m m∈ ∈ ∗ m∈ n−m m n∈ and P a b 6 C a b . k ∗ kt,2 tk kt,2k k1,2 s,∞ (ii) Let 1/2 6 s 6 0 and s 3/2

Lemma A.4 Let E be a complex Banach space and denote for r > 0, Br = x E : x 6 r . If f : B E is analytic for some m > 1, and { ∈ k k } m → sup f(x) x 6 m/8, x∈Bm| − | then f is an analytic diffeomorphism onto its image, and this image covers

Bm/2. ⋊

B Facts on the weak * topology

In this subsection we collect various properties of the weak* topology τw∗ = s,∞ −s,1 s,∞ σ(ℓ0 ,ℓ0 ) on ℓ0 needed in the course of the paper. Lemma B.1 Let s R. ∈ s,∞ (i) The closed unit ball of ℓ0 is weak* compact, weak* sequentially compact, and the topology induced by the weak* topology on this ball is metrizable.

(m) s,∞ s,∞ (ii) For any sequence (x ) > ℓ and x ℓ the following statements m 1 ⊂ 0 ∈ 0 are equivalent:

(a) (x(m)) is weak* convergent to x. (b) (x(m)) is -norm bounded and componentwise convergent, i.e. k·ks,∞ (m) (m) Z sup x s,∞ < , lim xn = xn, n . m>1k k ∞ m→∞ ∀ ∈

October 4, 2016 37 (c) (x(m)) is -norm bounded and x(m) x in ℓσ,p for some σ 1. ∞ − (iii) For any subset A ℓs,∞ the following statements are equivalent: ⊂ 0 (a) A is weak* compact. (b) A is weak* sequentially compact. (c) A is -norm bounded and weak* closed. k·ks,∞ (d) A is -norm bounded and A is a compact subset of ℓσ,p for some k·ks,∞ 0 σ 1. ∞ − (iv) On any -norm bounded subset A ℓs,∞, the topology induced by k·ks,∞ ⊂ 0 the σ,p-norm, provided (s σ)p> 1, coincides with the topology induced k·k s,∞− ⋊ by the weak* topology of ℓ0 .

C Schrödinger Operators

In this appendix we review definitions and properties of Schrödinger operators 2 ∂x + q with a singular potential q used in Section 2 – see e.g. [9] and [23]. − 1 1 Boundary conditions. Denote by HC[0, 1] = H ([0, 1], C) the Sobolev space of functions f : [0, 1] C which together with their distributional derivative 2 → 1 ∂xf are in LC[0, 1]. On HC[0, 1] we define the following three boundary condi- tions (bc),

(per +) f(1) = f(0); (per ) f(1) = f(0); (dir) f(1) = f(0) = 0. − − 1 The corresponding subspaces of HC[0, 1] are defined by

1 1 H = f HC[0, 1] : f satisfies (bc) , bc { ∈ } −1 1 ′ 1 and their duals are denoted by Hbc :=(Hbc) . Note that Hper + can be canon- ically identified with the Sobolev space H1(R/Z, C) of 1-periodic functions f : R C which together with their distributional derivative are in L2 (R, C). → loc 1 1 R Z C Analogously, Hper − can be identified with the subspace of H ( /2 , ) consist- ing of functions f : R C with f,∂ f L2 (R, C) satisfying f(x+1) = f(x) → x ∈ loc − for all x R. In the sequel we will not distinguish these pairs of spaces. Fur- ∈ 1 1 1 thermore, note that Hdir is a subspace of Hper + as well as of Hper −. Denote 2 1 by , bc the extension of the L -inner product f, g I = 0 f(x)g(x) dx to a h· ·i −1 1 h i sesquilinear pairing of Hbc and Hbc. Finally, we record thatR the multiplication H1 H1 H1 , (f, g) fg, (47) bc × bc → per + 7→ and the complex conjugation H1 H1 , f f are bounded operators. bc → bc 7→

October 4, 2016 38 −1 Multiplication operators. For q Hper + define the operator Vbc of multipli- 1 −1 ∈ 1 cation by q, Vbc : Hbc Hbc as follows: for any f Hbc, Vbcf is the element −1 → ∈ in Hbc given by

V f, g := q, fg , g H1 . h bc ibc h iper + ∈ bc

In view of (47), Vbc is a well defined bounded linear operator.

−1 1 1 Lemma C.1 Let q Hper +. For any g Hdir, the restriction (Vper ±g) H1 : Hdir ∈ ∈ dir → C 1 C ⋊ coincides with Vdirg : Hdir . → Proof. Since any h H1 is also in H1 , the definitions of V and V ∈ dir per + per + dir imply

V g, h = q, gh = V g, h , h per + iper + h iper + h dir idir which gives (Vper +g) 1 = Vdirg. Similarly, one sees that Vper −g 1 = Hdir Hdir

Vdirg. Ú

It is convenient to introduce also the space H1 H1 and define the per + ⊕ per − multiplication operator V of multiplication by q

V : H1 H1 H−1 H−1 , (f, g) (V f, V g). per + ⊕ per − → per + ⊕ per − 7→ per + per − We note that H1 H1 can be canonically identified with H1(R/2Z, C), per + ⊕ per − H1(R/2Z, C) H1 H1 , f (f +,f −), → per + ⊕ per − 7→ where f +(x) = 1 (f(x)+ f(x + 1)) and f −(x) = 1 (f(x) f(x + 1)). Its dual 2 2 − is denoted by H−1(R/2Z, C). 1 1 R Z C 1 Fourier basis. The spaces Hper ±, H ( /2 , ) and Hdir and their duals admit the following standard Fourier basis. Recall from Appendix A that for any s R and 1 6 p 6 , we denote by ℓs,p ℓs,p(Z, C) the sequence space ∈ ∞ C ≡ s,p ℓ = z =(z ) Z C : z < . C { k k∈ ⊂ k ks,p ∞} Basis for H1 , H−1 . Any element f H1 [H−1 ] can be repre- per + per + ∈ per + per + imπx 1 −1 sented as f = f e , e (x) := e , where (f ) Z hC [h ] and m∈Z m m m m m∈ ∈ C f = f,eP , f = 0, m Z. 2m h 2miper + 2m+1 ∀ ∈ Furthermore, for any q = q e H−1 , m∈Z m m ∈ per + P −1 Vper +f = qn−mfm en Hper +. Z Z ! ∈ nX∈ mX∈ −1 Note that by Lemma A.3, ( m∈Z qn−mfm)n∈Z is in hC . P

October 4, 2016 39 Basis for H1 , H−1 . Any element f H1 [H−1 ] can be repre- per − per − ∈ per − per − 1 −1 sented as f = f e where (f ) Z hC [hC ] and m∈Z m m m m∈ ∈ f =Pf,e , f = 0, m Z. 2m+1 h 2m+1iper − 2m ∀ ∈ Similarly, for any q = q e H−1 , m∈Z m m ∈ per + P −1 Vper −f = qn−mfm en Hper −. Z Z ! ∈ nX∈ mX∈ Basis for H1(R/2Z, C), H−1(R/2Z, C). Any element f H1(R/2Z, C)[H−1(R/2Z, C)] ∈ can be represented as f = f e where f = f,e . Here f, g := m∈Z m m m h mi h i 1 2 f(x)g(x) dx denotes the normalized L2-inner product on [0, 2] extended 2 0 P 1 R R toR a sesquilinear pairing between H ( /2Z, ) and its dual. In particular, for f H1(R/2Z, C), and m Z, ∈ ∈ 1 2 f,e = f(x)e−imπx dx. h mi 2 Z0 (For any q = q e H−1 ֒ H1(R/2Z, C m∈Z m m ∈ per + → P −1 Vf = qn−mfm en H (R/2Z, C). Z Z ! ∈ nX∈ mX∈ 1 −1 2 Basis for Hdir, Hdir : Note that (√2 sin(mπx))m>1 is an L -orthonormal ba- sis of L2([0, 1], C). Hence any element f H1 can be represented as 0 ∈ dir 1 f(x)= f,s s (x)= f sins (x), s (x)= √2 sin(mπx), h miI m 2 m m m > Z mX1 mX∈ where f sin = 1 f(x)s (x) dx, m Z. For any element g H−1 one gets by m 0 m ∈ ∈ dir duality R

1 sin sin g = gm sm, gm = g,sm dir. 2 Z h i mX∈ sin sin Z −2 sin 2 One verifies that g−m = gm for all m and m∈Z m gm < . For −1 − ∈ h i | | ∞ any q H C with q < and 1

October 4, 2016 40 it follows that for any m Z ∈ 1 sin sin fn q,snsm per + = fn q, cos((m n)πx) per +. 2 m−n even h i m−n even h − i Xn∈Z Xn∈Z Note that q, cos((m n)πx) is well defined as cos((m n)πx) H1 h − iper + − ∈ per + if m n is even. If m n is odd, we decompose the difference of the cosines in 1 − − Hper + as follows

cos((m n)πx) cos((m + n)πx) − − = (cos((m n)πx) cos(πx)) (cos((m + n)πx) cos(πx)) − − − − and then obtain, using again that f sin = f sin for all n Z, −n − n ∈ 1 sin sin fn q,snsm per + = fn q, cos((m n)πx) cos(πx) per +. 2 m−n odd h i m−n odd h − − i Xn∈Z Xn∈Z Altogether we have shown that

1 cos sin Vdirf = qm−nfn sm, 2 Z Z ! mX∈ nX∈ where Z cos q, cos(kπx) per +, if k even, qk = h i ∈ (48)  q, cos(kπx) cos(πx) , if k Z odd. h − iper + ∈ Since by assumption q < with 1

1/2 2t cos 2 m qm 6 Ct q t,2. (49) Zh i | | ! k k mX∈ Schrödinger operators with singular potentials. For any q H−1 denote by ∈ 0,C L(q) the unbounded operator ∂2 + V acting on H−1(R/2Z, C) with domain − x H1(R/2Z, C). As H1(R/2Z, C)= H1 H1 and V = V V the per + ⊕ per − per + ⊕ per − operator L(q) leaves the spaces H1 invariant and L(q)= L (q) L (q) per ± per + ⊕ per − with L (q) = ∂2 + V . Hence the spectrum spec(L(q)) of L(q), also per ± − x per ± referred to as spectrum of q, is the union spec(L (q)) spec(L (q)) of per + ∪ per − the spectra spec(Lper ±(q)) of Lper ±(q). The spectrum spec(L(q)) is known to be discrete and to consist of complex eigenvalues which, when counted with multiplicities and ordered lexicographically, satisfy

λ+ 4 λ− 4 λ+ 4 , λ± = n2π2 + nℓ2 , 0 1 1 ··· n n – see e.g. [16]. For any q H−1 there exists N > 1 so that ∈ 0,C λ± n2π2 6 n/2, n > N, λ± 6 (N 1)2π2 + N/2, n

October 4, 2016 41 −1 where N can be chosen locally uniformly in q on H0,C. Since for q = 0 and + + + n > 0, ∆(λ2n(0), 0) = 2and ∆(λ2n+1(0), 0) = 2, all λ2n(0) are 1-periodic and + − all λ2n+1(0) are 1-antiperiodic eigenvalues of q = 0. By considering the compact −1 interval [0, q] = tq : 0 6 t 6 1 H C it then follows after increasing N, if { } ⊂ 0, necessary, that for any n > N

λ+(q), λ−(q) spec(L (q)), [spec(L (q))] if n even [odd]. (51) n n ∈ per + per − For any q H−1 and n > N the following Riesz projectors are thus well ∈ 0,C defined on H−1(R/2Z, C)

1 −1 Pn,q := (λ L(q)) dλ. 2πi 2 2 − Z|λ−n π |=n 1 1 For n > N even [odd], the range of Pn,q is contained in Hper + [Hper −]. For q = 0, Pn,0 coincides with the projector Pn introduced in (7). −1 2 Similarly, for any q H0,C denote by Ldir(q) the unbounded operator ∂x + −1 ∈ 1 − Vdir acting on Hdir with domain Hdir. Its spectrum spec(Ldir(q)) is known to be discrete and to consist of complex eigenvalues which, when counted with multiplicities and ordered lexicographically, satisfy

µ 4 µ 4 , µ = n2π2 + nℓ2 , 1 2 ··· n n – see e.g. [16]. By increasing the number N chosen above, if necessary, we can thus assume that

µ n2π2 < n/2, n > N, µ 6 (N 1)2π2 + N/2, n 6 N. (52) | n − | | n| −

In particular, for any n > N, µn is simple and the corresponding Riesz projector

1 −1 Πn,q := (λ Ldir(q)) dλ 2πi 2 2 − Z|λ−n π |=n −1 is well defined on Hdir . If q = 0, we write Πn for Πn,0. Regularity of solutions. In Section 2 we consider solutions f of the equation s,∞ (L(q) λ)f = g in Fℓ C and need to know their regularity. − ⋆, Lemma C.2 For any q Fℓs,∞ with 1/2 < s 6 0, the following holds: For ∈ 0,C − any g Fℓs,∞ and any λ C, a solution f H1(R/2Z, C) of the inhomoge- ∈ ⋆,C ∈ ∈ neous equation (L(q) λ)f = g is an element in Fℓs+2,∞. ⋊ − ⋆,C s,∞ 1 Proof. Let g Fℓ C and assume that f H (R/2Z, C) solves (L(q) λ)f = g. ∈ ⋆, ∈ − 2 F s,∞ Write (L λ)f = g as Aλf = Vf g where Aλ = ∂x + λ. Since q ℓ0,C , − − r,2 ∈ Lemma A.2 implies that q and g are in Fℓ C with r = s 1/2 ε where ⋆, − − ε > 0 is chosen such that r > 1. By Lemma A.3 (i), Vf Fℓr,2 and hence − ∈ ⋆,C F r,2 F r+2,2 Aλf = Vf g ℓ⋆,C implying that f ℓ⋆,C . Since s 3/2 6 1 < − ∈ ∈ F s,∞ − − − r 6 0, Lemma A.3 (ii) applies. Therefore, Vf ℓ⋆,C and using the equation s+2,∞∈

A f = Vf g once more one gets f Fℓ as claimed. Ú λ − ∈ ⋆,C

October 4, 2016 42 For any q Fℓs,∞ with 1/2 < s 6 0, and n > n as in Corollary 2.5, ∈ 0,C − s introduce

+ − + − Null(L(q) λn ) Null(L(q) λn ), λn = λn , En En(q) := − ⊕ − 6 ≡ Null(L(q) λ+)2, λ+ = λ−.  − n n n 1 1 Then En is a two-dimensional subspace of Hper + [Hper −] if n is even [odd]. The following result shows that elements in En are more regular.

Lemma C.3 For any q Fℓs,∞ with 1/2 < s 6 0 and for any n > n ∈ 0,C − s E (q) Fℓs+2,∞ H1 [Fℓs+2,∞ H1 ] if n is even [odd]. ⋊ n ⊂ ⋆,C ∩ per + ⋆,C ∩ per − Proof. By Lemma C.2 with g = 0, any eigenfunction f of an eigenvalue λ s+2,∞ + − + − of L(q) is in Fℓ C . Hence if λ = λ or if λ = λ and has geometric ⋆, n 6 n n n F s+2,∞ + − multiplicity two, then En ℓ⋆,C . Finally, if λn = λn is a double eigenvalue ⊂ + of geometric multiplicity 1 and g is an eigenfunction corresponding to λn , there exists an element f H1(R/2Z, C) so that (L λ+)f = g. Since g is an ∈ − n F s+2,∞ eigenfunction it is in ℓ⋆,C by Lemma C.2 and by applying this lemma once more, it follows that f Fℓs+2,∞. Clearly, E = span(g,f) and hence E ∈ ⋆,C n n ⊂ F s+2,∞ ± ℓ⋆,C also in this case. By (51), λn are 1-periodic [1-antiperiodic] eigenvalues of q if n is even [odd]. Hence E Fℓs+2,∞ H1 [Fℓs+2,∞ H1 ] if n is n ⊂ ⋆,C ∩ per + ⋆,C ∩ per − even [odd] as claimed. Ú

Estimates for projectors. The projectors Pn,q [Πn,q] with n > N and N −1 R Z C −1 given by (50)-(52) are defined on H ( /2 , ) [Hdir ] and have range in 1 R Z C 1 H ( /2 , )[Hdir]. The following result concerns estimates for the restriction of P [Π ] to L2 = L2([0, 2], C) [L2( ) = L2([0, 1], C)] needed in Section 2 n,q n,q I for getting the asymptotics of µ τ stated in Theorem 2.1 (ii). n − n −1 Lemma C.4 Assume that q H0,C with q t,2 < and 1 0 (only depending on t) and N > N (with N as above) so that for any n > N the following holds (log n)2 (i) P P 2 ∞ 6 C q k n,q − nkL →L t n1−|t| k kt,2 (log n)2 (ii) Π Π 2 ∞ 6 C q k n,q − nkL (I)→L (I) t n1−|t| k kt,2 The constant N ′ can be chosen locally uniformly in q. ⋊

Proof. [7, Lemma 25]. Ú

s,∞ Remark C.5. We will apply Lemma C.4 for potentials q Fℓ C with 1/2 < ∈ 0, − s 6 0 using the fact that by the Sobolev embedding theorem there exists s,∞ t ⊸ .t< 1/2 so that Fℓ C ֒ H C> 1 − − 0, → 0,

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