KdV & KAM

Version 4.6 February 2003 Not for Circulation

Thomas Kappeler Universit¨at Z¨urich, Institut f¨ur Mathematik Winterthurerstrasse 190, CH-8057 Z¨urich [email protected] J¨urgen P¨oschel Mathematisches Institut A, Universit¨at Stuttgart Pfaffenwaldring 57, D-70569 Stuttgart [email protected]

Thomas Kappeler

J¨ urgen P¨ oschel

KdV & KAM

c Kappeler & P¨oschel Finalst version

In memory of ¨ J URGEN M OSER teacher mentor friend

Preface

This book is concerned with two aspects of the theory of integrable partial differential equations. The first aspect is a normal form theory for such equations, which we exemplify by the periodic Korteweg de Vries equation – undoubtedly one of the most important nonlinear, integrable pdes. This makes for the ‘KdV’ part of the title of the book. The second aspect is a theory for Hamiltonian perturbations of such pdes. Its prototype is the so called KAM theory, developed for finite dimensional systems by Kolmogorov, Arnold and Moser. This makes for the ‘KAM’ part of the title of the book. To be more specific, our starting point is the periodic KdV equation considered as an infinite dimensional, integrable Hamiltonian system admitting a complete set of independent integrals in involution. We show that this leads to a single, global, real analytic system of Birkhoff coordinates – the cartesian version of action-angle coordinates –, such that the KdV Hamiltonian becomes a function of the actions alone. In fact, these coordinates work simultaneously for all Hamiltonians in the KdV hierarchy. While the existence of global Birkhoff coordinates is a special feature of KdV, local Birkhoff coordinates may be constructed via our approach for many integrable pdes anywhere in phase space. Specifically this holds true for the defocusing nonlinear Schr¨odinger equation, for which parallel results were developed in [51]. The global coordinates make it evident that all solutions of the periodic KdV equation are periodic, quasi-periodic, or almost-periodic in time. It also provides a convenient handle to study small Hamiltonian perturbations, by applying a suitable generalization of the KAM theory to partial differential operators. To check the pertinent nondegeneracy conditions, we construct Birkhoff normal forms up to order six to gain sufficient control over the KdV frequencies as functions of the actions. In fact, these Birkhoff normal forms are just the first terms in the power series expansion of the KdV Hamiltonian in Birkhoff coordinates. Finally, we describe the set up, assumptions and conclusions of a general infinite dimensional KAM theorem, that is applicable here and goes back to Kuksin. The situation differs from more conventional applications of KAM to pdes in that the per-

X

Preface

turbations are given by unbounded operators. This is only partially compensated by a smoothing effect of the small divisors. In addition, one has to modify the iteration scheme and use normal forms which also depend on angular variables. Only recently, monographs on KAM theory for integrable pdes appeared, by Bourgain [17], Craig [29], and Kuksin [75]. Of these, the first two choose a different approach, setting up a functional equation and applying a Lyapunov-Schmidt decomposition scheme pioneered by Craig & Wayne [28]. The latter employs a normal form theory for Lax-integrable pdes near finite dimensional tori, which is based on the Its-Matveev formula. In contrast, the normal form theory presented in this book with its global features goes much further. It allows us to obtain a perturbation theory for KdV from an abstract KAM theorem of a particularly simple form, and to use Birkhoff normal forms to check the relevant nondegeneracy conditions. Moreover, this normal form might turn out to be useful for other long time stability results for perturbed integrable pdes such as Nekhoroshev estimates. This book is not only intended for the handful of specialists working at the intersection of integrable partial differential equations and Hamiltonian perturbation theory, but also researchers farther away from these fields. In fact, it is our intention to reach out to graduate students as well. It is for this reason that first of all, we have included a chapter on the classical theory, describing the finite dimensional background of integrable Hamiltonian systems and their perturbation theory according to the theory initiated by Kolmogorov, Arnold and Moser. Secondly, we made the book self-contained, omitting only those proofs which can be found in well known textbooks. We therefore included numerous appendices – some of them, we hope, of independent interest – on topics from complex analysis on Hilbert spaces, spectral theory of Schr¨odinger operators, Riemann surface theory, representation of holomorphic differentials, and certain aspects of the KdV equation such as the KdV hierarchy and new formulas for the KdV frequencies. Thirdly, we wrote the book in a modular manner, where each of its five main chapters – chapters II to VI – as well as its appendices may be read independently of each other. Every chapter has its own introduction, and the notation is explained. As a result, there is some natural repetition and overlap among them. Moreover, the results of these chapters are summarized in the very first chapter, titled “The Beginning”, and here too we took the liberty to quote from the introductions to the later chapters. We consider these repetitions a benefit for the reader rather than a nuisance, since it allows him, or her, to peruse the material in a nonlinear manner. This book took many years to complete, and during this long time we benefitted from discussions and collaborations with many friends and colleagues. We would like to thank all of them, in particular Benoˆıt Gr´ebert, with whom we developped parallel results for the defocusing nonlinear Schr¨odinger equation in [51], and J¨urg Kramer, for his contribution to the nondegeneracy result for the first KdV Hamiltonian. Most of all we are indebted to J¨urgen Moser, who initiated this joint effort and never failed to encourage us as long as he was able to do so. We dedicate this book to him. The second author also gratefully acknowledges the hospitality of the Forschungsinstitut at the ETH Z¨urich and the Institute of Mathematics at the University of

Preface

XI

Z¨urich during many periods of our collaborative efforts, as well as the support of the Deutsche Forschungsgemeinschaft, while the first author gratefully acknowledges the support of the Swiss National Science Foundaton and of the European Research Training Network HPRN-CT-1999-00118. Finally we would like to thank Jules Hobbes for his never tiring TEXpertise from the very first lines through many, many revisions up to the final, press-ready output, and J¨urgen Jost and Springer Verlag for their pleasant cooperation to make this book happen. Last, but not least we thank our families for their patience and support during these many years. Z¨urich/Stuttgart February 14/16, 2003

TK/JP

Contents

Chapter I The Beginning 1

Overview 1

Chapter II Classical Background 2 3 4 5

Hamiltonian Formalism 19 Liouville Integrable Systems 27 Birkhoff Integrable Systems 34 KAM Theory 39

Chapter III Birkhoff Coordinates 6 7 8 9 10 11 12

Background and Results 51 Actions 63 Angles 69 Cartesian Coordinates 74 Orthogonality Relations 85 The Diffeomorphism Property 91 The Symplectomorphism Property 102

Chapter IV Perturbed KdV Equations 13 14 15 16 17

The Main Theorems 111 Birkhoff Normal Form 118 Global Coordinates and Frequencies The KAM Theorem 133 Proof of the Main Theorems 139

127

XIV

Contents

Chapter V The KAM Proof 18 19 20 21 22

Set Up and Summary of Main Results The Linearized Equation 152 The KAM Step 160 Iteration and Convergence 165 The Excluded Set of Parameters 171

145

Chapter VI Kuksin’s Lemma 23

Kuksin’s Lemma 177

Chapter VII Background Material A B C

Analyticity 187 Spectra 194 KdV Hierarchy 207

Chapter VIII Psi-Functions and Frequencies D E F

Construction of the Psi-Functions A Trace Formula 223 Frequencies 227

211

Chapter IX Birkhoff Normal Forms G H I J

Two Results on Birkhoff Normal Forms 233 Birkhoff Normal Form of Order 6 240 Kramer’s Lemma 248 Nondegeneracy of the Second KdV Hamiltonian

Chapter X Some Technicalities K L M

Symplectic Formalism 257 Infinite Products 260 Auxiliary Results 262

References 267 Index 275 Notations

252

List of Figures

1 2 3 4 5 6 7 8 9 10

a-cycles 57 √ s Signs of 1 − λ2 62 p Signs of c 12 (λ) − 4 for real q 63 Labeling of periodic eigenvalues as q varies 64 Isolating neighbourhoods 65 A generic 1-function 198 The set [[a, b]] 202 a- and b-cycles for N = 2 224 a 0 - and b-cycles with basepoint λ0 for N = 2 224 p Signs of c 12 (λ) − 4 for real q 282

I The Beginning

1 Overview In this book we consider the Korteweg-de Vries (KdV) equation u t = −u x x x + 6uu x . The KdV equation is an evolution equation in one space dimension which is named after the two Dutch mathematicians Korteweg and de Vries [66] – see also Boussinesq [18] and Rayleigh [113]. It was proposed as a model equation for long surface waves of water in a narrow and shallow channel. Their aim was to obtain as solutions solitary waves of the type discovered in nature by Russell [114] in 1834. Later it became clear that this equation also models waves in other homogeneous, weakly nonlinear and weakly dispersive media. Since the mid-sixties the KdV equation received a lot of attention in the aftermath of the computational experiments of Kruskal and Zabusky [69], which lead to the discovery of the interaction properties of the solitary wave solutions and in turn to the understanding of KdV as an infinite dimensional integrable Hamiltonian system. Our purpose here is to study small Hamiltonian perturbations of the KdV equation with periodic boundary conditions. In the unperturbed system all solutions are periodic, quasi-periodic, or almost-periodic in time. The aim is to show that large families of periodic and quasi-periodic solutions persist under such perturbations. This is true not only for the KdV equation itself, but in principle for all equations in the KdV hierarchy. As an example, the second KdV equation will also be considered. The KdV Equation Let us recall those features of the KdV equation that are essential for our purposes. It was observed by Gardner [46], see also Faddeev & Zakharov [40], that the KdV equation can be written in the Hamiltonian form ∂u d ∂H = ∂t dx ∂u

2

I The Beginning

with the Hamiltonian H (u) =

Z S1

1 2 2 ux


+ u 3 dx,

where ∂ H/∂u denotes the L 2 -gradient of H , representing the Fr´echet derivative of H with respect to the standard scalar product on L 2 . Since we are interested in spatially periodic solutions, we take as the underlying phase space the Sobolev space H N = H N (S 1 ; R),

S 1 = R/Z,

of real valued functions with period 1, where N ≥ 1 is an integer, and endow it with the Poisson bracket proposed by Gardner, Z ∂ F d ∂G {F ,G} = dx. 1 S ∂u(x) dx ∂u(x) Here, F and G are differentiable functions on H N with L 2 -gradients in H 1 . This makes H N a Poisson manifold, on which the KdV equation may also be represented in the form u t = {u, H } familiar from classical mechanics. We note that the initial value problem for the KdV equation on the circle S 1 is well posed on every Sobolev space H N with N ≥ 1: for initial data u o ∈ H N it has been shown by Temam for N = 1, 2 [128] and by Saut & Temam for any real N ≥ 2 [121] that there exists a unique solution evolving in H N and defined globally in time. For further results on the initial value problem see for instance [78, 88, 126] as well as the more recent results [14, 15, 64]. The KdV equation admits infinitely many conserved quantities, or integrals, in involution, and there are many ways to construct such integrals [46, 94, 95]. Lax [77] obtained a set of Poisson commuting integrals in a particularly elegant way by considering the spectrum of an associated Schr¨odinger operator. For u ∈ H 0 = L 2 = L 2 (S 1 , R) consider the differential operator L=−

d2 +u dx 2

on the interval [0, 2] of twice the length of the period of u with periodic boundary conditions. It is well known [80, 82, 84] that its spectrum, denoted spec(u), is pure point and consists of an unbounded sequence of periodic eigenvalues λ0 (u) < λ1 (u) ≤ λ2 (u) < λ3 (u) ≤ λ4 (u) < . . . . Equality or inequality may occur in every place with a ‘≤’-sign, and one speaks of the gaps (λ2n−1 (u), λ2n (u)) of the potential u and its gap length γn (u) = λ2n (u) − λ2n−1 (u),

n ≥ 1.

If some gap length is zero, one speaks of a collapsed gap, otherwise of an open gap.

1 Overview

3

For u = u(t, · ) depending also on t define the corresponding operator L(t) = −

d2 + u(t, · ). dx 2

Lax observed that u is a solution of the KdV equation if and only if d L = [B,L], dt where [B,L] = B L − L B denotes the commutator of L with the anti-symmetric operator d3 d d B = −4 3 + 3u + 3 u. dx dx dx It follows by an elementary calculation that the solution of d U = BU, dt

U (0) = I,

defines a family of unitary operators U (t) such that U ∗ (t)L(t)U (t) = L(0). Consequently, the spectrum of L(t) is independent of t, and so the periodic eigenvalues λn = λn (u) are conserved quantities under the evolution of the KdV equation, a fact first observed by Gardner, Greene, Kruskal & Miura [47]. Thus, the flow of the KdV equation defines an isospectral deformation on the space of all potentials in H N . From an analytical point of view, however, the periodic eigenvalues are not satisfactory as integrals, as λn is not a smooth function of u whenever the corresponding gap collapses. But McKean & Trubowitz [89] showed that the squared gap lengths γn2 (u),

n ≥ 1,

together with the average Z u(x) dx

[u] = S1

form another set of integrals, which are real analytic on all of L 2 and Poisson commute with each other. Moreover, the squared gap lengths together with the average determine uniquely the periodic spectrum of a potential [48]. The space L 2 thus decomposes into the isospectral sets  Iso(u) = v ∈ L 2 : spec(v) = spec(u) , which are invariant under the KdV flow and may also be characterized as  Iso(u) = v ∈ L 2 : gap lengths(v) = gap lengths(u), [v] = [u] . As shown by McKean & Trubowitz [89] these are compact connected tori, whose dimension equals the number of positive gap lengths and is infinite generically. Moreover, as the asymptotic behavior of the gap lengths characterizes the regularity of a

4

I The Beginning

potential in exactly the same way as its Fourier coefficients do [84], we have u ∈ HN

Iso(u) ⊂ H N

⇔

for each N ≥ 1. Hence also the phase space H N decomposes into a collection of tori of varying dimension which are invariant under the KdV flow. Angle-Action and Birkhoff Coordinates In classical mechanics the existence of a foliation of the phase space into Lagrangian invariant tori is tantamount, at least locally, to the existence of angle-action coordinates. This is the content of the Liouville-Arnold-Jost theorem. In the infinite dimensional setting of the KdV equation, however, the existence of such coordinates is far less clear as the dimension of the foliation is nowhere locally constant. Invariant tori of infinite and finite dimension each form dense subsets of the foliation. Nevertheless, angle-action coordinates can be introduced globally in the form of Birkhoff coordinates as we describe now. They will form the basis of our study of perturbations of the KdV equation. To formulate the statement we define the phase spaces more precisely. For any integer N ≥ 0, let  H N = u ∈ L 2 (S 1 , R) : kuk N < ∞ , where 2 2 X 2N |k| u(k) kuk2N = u(0) ˆ ˆ + k∈Z

is defined in terms of the discrete Fourier transform uˆ of u. The Poisson structure { · , · } is degenerate on H N and admits the average [ · ] as a Casimir function. The leaves of the corresponding symplectic foliation are given by [u] = const. Instead of restricting the KdV Hamiltonian to each leaf, it is more convenient to fix one such leaf, namely  H0N = u ∈ H N : [u] = 0 , which is symplectomorphic to each other leaf by a simple translation, and consider the mean value as a parameter. On H0N the Poisson structure is nondegenerate and induces a symplectic structure. Writing u = v + c with [v] = 0 and c = [u], the Hamiltonian then takes the form H (u) = Hc (v) + c3 with Hc (v) =

Z S1

1 2 2 vx

+v

3




Z dx + 6c S1

1 2 2 v dx.

We consider Hc as a 1-parameter family of Hamiltonians on H0N . We remark that Z 1 H0 = v 2 dx 2 S1

1 Overview

5

corresponds to translation and is the zero-th Hamiltonian of the KdV hierarchy, as described in appendix C. To describe the angle-action variables on H0N we introduce the model space hr = `r2 × `r2 with elements (x, y), where n o X `r2 = x ∈ `2 (N, R) : kxkr2 = n 2r |xn |2 < ∞ . n≥1

We endow hr with the standard Poisson structure, for which {xn ,ym } = δnm , while all other brackets vanish. The following theorem was first proven in [5] and [6]. A quite different approach for this result – and the one we expand on here – was first presented in [60]. For a related result for the nonlinear Schr¨odinger equation see [51]. Theorem 1.1. There exists a diffeomorphism 9 : h1/2 → H00 with the following properties. (i) 9 is one-to-one, onto, bi-analytic, and preserves the Poisson bracket. (ii) For each N ≥ 0, the restriction of 9 to h N +1/2 , denoted by the same symbol, is a map 9 : h N +1/2 → H0N , which is one-to-one, onto, and bi-analytic as well. (iii) The coordinates (x, y) in h3/2 are global Birkhoff coordinates for KdV. That is, for any c ∈ R, the transformed Hamiltonian Hc B 9 depends only on xn2 + yn2 , n ≥ 1, with (x, y) being canonical coordinates. Thus, in the coordinates (x, y) the KdV Hamiltonian is a real analytic function of the actions alone: Hc = Hc (I1 , I2 , . . . ),

In =

1 2 (x + yn2 ), 2 n

with equations of motion x˙n = ωn (I )yn , where ωn = ωc,n =

y˙n = −ωn (I )xn ,

∂ Hc (I ), ∂ In

I = (In )n≥1 .

The whole system appears now as an infinite chain of anharmonic oscillators, whose frequencies depend on their amplitudes in a nonlinear and real analytic fashion.

6

I The Beginning

These results are not restricted to the KdV Hamiltonian. They simultaneously apply to every real analytic Hamiltonian in the Poisson algebra of all Hamiltonians which Poisson commute with all actions I1 , I2 , . . . . In particular, one obtains Birkhoff coordinates for every Hamiltonian in the KdV hierarchy defined in appendix C. As an example, we will later also consider the second KdV Hamiltonian. The existence of Birkhoff coordinates makes it evident that every solution of the KdV equation is almost-periodic in time. In the coordinates of the model space every solution is given by p xn (t) = 2Ino sin(θno + ωn (I o )t), p yn (t) = 2Ino cos(θno + ωn (I o )t), where (θ o , I o ) corresponds to the initial data u o . Hence, it winds around the underlying invariant torus  TI o = (x, y) : xn2 + yn2 = 2Ino , n ≥ 1 , The solution in the original space H0N is thus winding around the embedded torus 9(TI o ), and expanding 9 into its Taylor series, it is of the form u(t) = 9(x(t), y(t)) X o = 9k (I o , θ o ) eihk,ω(I )it . k∈Z∞ ,|k|<∞

P N o o Here, hk,ωi = n kn ωn , and each 9k (I , θ ) is an element of H0 . Thus, every solution is almost-periodic in time. We remark that the solution above can also be represented in terms of the Riemann theta function. The corresponding formula is due to Its & Matveev [33]. Among all almost-periodic solutions there is a dense subset of quasi-periodic solutions, which are characterized by a finite number of frequencies and correspond to finite gap potentials. To describe them more precisely, let A ⊂ N be a finite index set, and consider the set of A-gap potentials  G A = u ∈ H00 : γn (u) > 0 ⇔ n ∈ A . That is, u ∈ G A if and only if precisely the gaps (λ2n−1 (u), λ2n (u)) with n ∈ A are open. Clearly, u ∈ G A ⇔ Iso(u) ⊂ G A , and all finite gap potentials are smooth, in fact real analytic, as almost all gap lengths are zero. As might be expected there is a close connection between the set G A and the subspace  h A = (x, y) ∈ h0 : xn2 + yn2 > 0 ⇔ n ∈ A .

1 Overview

7

Addendum to Theorem 1.1. The canonical transformation 9 also has the following property. (iv) For every finite index set A ⊂ N, the restriction 9 A of 9 to h A is a map 9A : hA → GA, which is one-to-one, onto, and bi-analytic. It follows that G A is a real analytic, invariant submanifold of H00 , which is completely foliated into invariant tori Iso(u) of the same dimension |A|. The KdV flow consists of a quasi-periodic winding around each such torus which is characterized by the frequencies ωn with n ∈ A. The above representation reduces to a convergent, real analytic representation X o u(t) = 9k (I Ao , θ Ao ) eihk,ω A (I A )it , k∈Z A

with I A = (In )n∈A , and similarly defined θ A and ω A . Outline of the Proof of Theorem 1.1 The proof of the theorem splits into four parts. First we define actions In and angles θn for a potential q following a procedure introduced for finite dimensional integrable systems. The formula for the actions In , due to Flaschka & McLaughlin [43], is given entirely in terms of the periodic spectrum of the potential. The angles θn , which linearize the KdV equation, were introduced even earlier by a number of authors, namely Dubrovin, Its, Krichever, Matveev, Novikov [32, 33, 35, 36, 57] (see also [34]), McKean & van Moerbeke [88], and McKean & Trubowitz [89, 90]. They are defined in terms of the Riemann surface 6(q) associated with the periodic spectrum of a potential as explained in section 6. We show that each In is real analytic on L 20 , while each θn , taken modulo 2π, is real analytic on the dense open domain L 20 X Dn , where Dn denotes the subvariety of potentials with collapsed n-th gap. Next, we define the cartesian coordinates xn and yn canonically associated to In and θn . Although defined originally only on L 20 X Dn , we show that they extend real analytically to a complex neighbourhood W of L 20 . Surely, the angle θn blows up when γn collapses, but this blow up is compensated by the rate at which In vanishes in the process. In particular, for real q the resulting limit will vanish. Then we show that the thus defined map  : q 7→ (x, y) is a diffeomorphism between L 20 and h1/2 . The main problem here is to verify that dq  is a linear isomorphism at every point q. This is done with the help of orthogonality relations among the coordinates, which are in fact their Poisson brackets. For the nonlinear Schr¨odinger equation the corresponding orthogonality relations have first been established by McKean & Vaninsky [91, 92]. It turned out that many of their ideas can also be used in the case of KdV. It is immediate that each Hamiltonian in the KdV hierarchy becomes a function of the actions alone, using their characterization in terms of the asymptotic expansion of one of the Floquet multipliers w(λ) and hence as spectral invariants.

8

I The Beginning

Finally, we verify that  preserves the Poisson bracket. As it happens, it is more convenient to look at the associated symplectic structures. This way, we only need to establish the regularity of the gradient of θn at special points, not everywhere. Thus, we equivalently show that  is a symplectomorphism. This will complete the proof of the main results of Theorem 1.1. Perturbations of the KdV Equation Our aim is to investigate whether sufficiently small Hamiltonian perturbations of the KdV equation, namely   ∂u d ∂ Hc ∂K = +ε , ∂t dx ∂u ∂u admit almost-periodic solutions as well, winding around invariant tori in phase space. In the classical setting of integrable Hamiltonian systems of finitely many degrees of freedom this question is answered by the theory of Kolmogorov, Arnold and Moser, known as KAM theory. It states that with respect to Lebesgue measure the majority of the invariant tori of a real analytic, nondegenerate integrable system persist under sufficiently small, real analytic Hamiltonian perturbations. They are only slightly deformed and still completely filled with quasi-periodic motions. The base of this partial foliation of the phase space into invariant tori, however, is no longer open, but has the structure of a Cantor set: it is a nowhere dense, closed set with no isolated points. Till now, there is no general infinite dimensional KAM theory to establish the persistence of infinite dimensional tori with almost-periodic solutions for Hamiltonian systems arising from partial differential equations, such as the KdV equation. There does exist a KAM theory to this effect, but it is restricted to systems, in which the coupling between the different modes of oscillations is of short range type. See [108] and the references therein. Such a theory does not apply here, and we can not make any statement about the persistence of almost-periodic solutions. But as noted above, there are also families of finite dimensional tori on real analytic submanifolds, corresponding to finite gap solutions and filling the space densely. In the classical setting a KAM theorem about the persistence of such lower dimensional tori was first formulated by Melnikov [93], and proven later by Eliasson [38]. It was independently extended to the infinite dimensional setting of partial differential equations by Kuksin [70, 72]. In the following we describe this kind of perturbation theory for finite gap solutions, following [107, 109]. A be a compact set of positive Again, let A ⊂ N be a finite index, and let 0 ⊂ R+ Lebesgue measure. We then set T0 =

[

TI ⊂ G A,

T I = 9 A (TI ),

I ∈0

where  TI = (x, y) ∈ h A : xn2 + yn2 = 2In , n ∈ A ∼ = T A × { I },

1 Overview

9

T with T = R/2πZ the circle of length 2π. Notice that T0 ⊂ N ≥0 H0N in view of the Addendum to Theorem 1.1. We show that under sufficiently small real analytic Hamiltonian perturbations of the KdV equation the majority of these tori persists together with their translational flows, the tori being only slightly deformed. The following theorem was first proven by Kuksin for the case c = 0. A a compact subset of positive Theorem 1.2. Let A ⊂ N be a finite index set, 0 ⊂ R+ Lebesgue measure, and N ≥ 1. Assume that the Hamiltonian K is real analytic in a complex neighbourhood U of T0 in H0,NC and satisfies the regularity condition

∂K : U → H0,NC , ∂u



∂ K sup

∂K



≤ 1. = sup

∂u

u∈U ∂u N N ;U

Then, for any real c, there exists an ε0 > 0 depending only on A, N , c and the size of U such that for |ε| < ε0 the following holds. There exist (i) a nonempty Cantor set 0ε ⊂ 0 with meas(0 − 0ε ) → 0 as ε → 0, (ii) a Lipschitz family of real analytic torus embeddings 4 : Tn × 0ε → U ∩ H0N , (iii) a Lipschitz map χ : 0ε → Rn , such that for each (θ, I ) ∈ Tn × 0ε , the curve u(t) = 4(θ + χ(I )t, I ) is a quasiperiodic solution of   d ∂ Hc ∂K ∂u = +ε ∂t dx ∂u ∂u winding around the invariant torus 4(Tn ×{I }). Moreover, each such torus is linearly stable. Remark 1. Note that the L 2 -gradient of a function on H0N has mean value zero by the definition of the gradient. On the other hand, the L 2 -gradient of a function on the larger space H N usually has mean value different from zero, and the gradient of its restriction to H0N is the projection of the former onto H0N :
∇ K | H N = ProjH N ∇ K = ∇ K − [∇ K ], 0

0

with ∇ = ∂/∂u. This, however, does not affect their Hamiltonian equations, since the derivative of the constant function [∇ K ] vanishes. Therefore, we will not explicitly distinguish between these two gradients. Remark 2. We already mentioned that \ GA ⊂ H0N . N ≥0

Thus, the quasi-periodic solutions remain in H0N if the gradient of K is in H0N .

10

I The Beginning

Remark 3. The regularity assumption on K entails that K depends only on u, but not on its derivatives. So the perturbation effected by K is of lower order than the unperturbed KdV equation. In view of the derivation of the KdV equation as a model equation for surface waves of water in a certain regime – see for example [136] – it would be interesting to obtain perturbation results which also include terms of higher order, at least in the region where the KdV approximation is valid. However, results of this type are still out of reach, if true at all. Remark 4. We point out that the perturbing term ε∂ K /∂u need not be a differential operator. For example, K (u) =

Z

u 3 dx

2

S1

has an L 2 -gradient ∂K = 6u 2 ∂u

Z

u 3 dx S1

to which the theorem applies as well. Remark 5. The invariant embedded tori are linearly stable in the sense that the variational, or linearized, equations of motion along such a torus are reducible to constant coefficient form whose spectrum is located on the imaginary axis. Hence, all Lyapunov exponents of such a torus vanish. Similar results hold for any equation in the KdV hierarchy. Consequently, for any given finite index set A, the manifold of A-gap potentials is foliated into the same family of invariant tori. The difference is only in the frequencies of the quasi-periodic motions on each of these tori. Therefore, similar results should also hold for the higher order KdV equations, once we can establish the corresponding nonresonance conditions. As an example we consider the second KdV equation, which reads ∂t u = ∂x5 u − 10u∂x3 u − 20∂x u∂x2 u + 30u 2 ∂x u. Its Hamiltonian is H (u) = 2

Z S1

1 2 2 uxx


+ 5uu 2x + 52 u 4 dx,

which is defined on H 2 . Again, with u = v + c, where [v] = 0, we get 5 H 2 (u) = Hc2 (v) + c4 2 with Hc2 (v) = H 2 (v) + 10cH 1 (v) + 30c2 H 0 (v).

1 Overview

11

Here, H (v) = 1

Z S1


1 2 v + v 3 dx, 2 x

1 H (v) = 2 0

Z S1

v 2 dx

are the familiar KdV Hamiltonian and the zero-th Hamiltonian of translation, respectively. We study this Hamiltonian on the space H0N with N ≥ 2, considering c as a real parameter. A a compact subset of positive Theorem 1.3. Let A ⊂ N be a finite index set, 0 ⊂ R+ Lebesgue measure, and N ≥ 3. Assume that the Hamiltonian K is real analytic in a complex neighbourhood U of T0 in H0,NC and satisfies the regularity condition

∂K : U → H0,NC−2 , ∂u



∂ K sup

≤ 1.

∂u N −2;U

If c ∈ / E A2 , where the exceptional set E A2 is an at most countable subset of the real line not containing 0 and with at most |A| accumulation points, then the same conclusions as in Theorem 1.2 hold for the system with Hamiltonian Hc2 + εK . Remark 1. The gradient ∂ K /∂u is only required to be in H0N −2 . Still, the regularity assumption ensures that the perturbation is of lower order than the unperturbed equation. Remark 2. A more detailed description of the set E A2 is given in appendix J. It is likely that the theorem is true for all c ∈ R. We now give two simple examples of perturbations to which the preceding theorems apply. As a first example let Z K (u) = F(x, u) dx, S1

where F defines a real analytic map { λ ∈ R : |λ| < R } → H0N ,

λ 7 → F( · , λ),

for some R > 0 and N ≥ 1. Then, with f = ∂ F/∂λ, ∂K = f (x, u) − [ f (x, u)] ∂u belongs to H0N , and the perturbed KdV equation is u t = −u x x x + 6uu x + ε

d f (x, u). dx

Theorem 1.2 applies after fixing 0 and c for all sufficiently small ε. Of course, F may also depend on ε, if the dependence is, say, continuous.

12

I The Beginning

We remark that perturbations of the KdV equation with a Hamiltonian K as above can be characterized equivalently as local perturbations given by dd x f (x, u(x)), where f admits a power series expansion in the second argument, X f (x, λ) = f k (x)λk , k≥0

convergent in H0N . In this case, the perturbed equationRis again a partial differential equation, and its Hamiltonian K is given by K (u) = S 1 F(x, u) dx, where F is a primitive of f with respect to λ, F(x, λ) =

X f k (x) λk+1 . k+1 k≥0

As a second example let K (u) =

Z S1

F(x, u x ) dx,

where F is as above with N ≥ 3. More generally, F could also depend on u, but this adds nothing new. Then d ∂K =− f (x, u x ), ∂u dx with f = ∂ F/∂λ, belongs to H0N −2 , and the perturbed second KdV equation is ut = · · · − ε

d2 f (x, u x ). dx 2

To this second example, Theorem 1.3 applies. Outline of the Proof of Theorems 1.2 and 1.3 A prerequisite for developing a perturbation theory of KAM type is the existence of coordinates with respect to which the variational equations along the unperturbed motions on the invariant tori reduce to constant coefficient form. Often, such coordinates are difficult to construct even locally. Here, they are provided globally by Theorem 1.1. According to Theorem 1.1 the Hamiltonian of the KdV equation on the model space h3/2 is of the form Hc = Hc (I1 , I2 , . . . ),

In =

1 2 (x + yn2 ). 2 n

The equations of motion are thus x˙n = ωn (I )yn ,

y˙n = −ωn (I )xn ,

1 Overview

13

with frequencies ωn = ωc,n =

∂ Hc (I ), ∂ In

I = (In )n≥1 ,

that are constant along each orbit. So each orbit is winding around some invariant torus  TI = (x, y) : xn2 + yn2 = 2In , n ≥ 1 , where the parameters I = (In )n≥1 are the actions of its initial data. We are interested in a perturbation theory for families of finite-dimensional tori TI . So we fix an index set A ⊂ N of finite cardinality |A|, and consider tori with In > 0 ⇔ n ∈ A. The linearized equations of motion along any such torus have now constant coefficients and are determined by |A| internal frequencies ω = (ωn )n∈A and infinitely many external frequencies  = (ωn )n ∈A / . Both depend on the |A|-dimensional parameter ξ = (Ino )n∈A , since all other components of I vanish in this family. The KAM theorem for such families of finite dimensional tori requires a number of assumptions, among which the most notorious and unpleasant ones are the so called nondegeneracy and nonresonance conditions. In this case, they essentially amount to the following. First, the map ξ 7→ ω(ξ ) from the parameters to the internal frequencies has to be a local homeomorphism, which is Lipschitz in both directions. This is known as Kolmogorov’s condition in the classical theory. Second, for each k ∈ Z A and l ∈ ZNXA with 1 ≤ |l| ≤ 2, the zero set of any of the frequency combinations hk,ω(ξ )i + hl,(ξ )i has to be a set of measure zero. This is sometimes called Melnikov’s condition. The verification of these conditions for the KdV Hamiltonian requires some knowledge of its frequencies. One way to obtain this knowledge is to use Riemann surface theory: Krichever proved that the frequency map ξ 7 → ω(ξ ) is a local diffeomorphism everywhere on the space of A-gap potentials, see [68, 9], and Bobenko & Kuksin showed that the second condition is satisfied in the case c = 0 using Schottky uniformization [11]. Here, however, we follow a different and more elementary route to verify these conditions by computing the first coefficients of the Birkhoff normal form of the KdV Hamiltonian, which we explain now. In classical mechanics the Birkhoff normal form allows to view a Hamiltonian system near an elliptic equilibrium as a small perturbation of an integrable system.

14

I The Beginning

This tool is also applicable in an infinite dimensional setting. Writing X u= γn qn e2πinx n6=0

√ √ with weights γn = 2π |n| and complex coefficients q±n = (xn ∓iyn )/ 2, the KdV Hamiltonian becomes X X Hc = λn |qn |2 + γk γl γm qk ql qm n≥1

k+l+m=0

on h3/2 with λn = (2πn)3 + 6c · 2πn. Thus, at the origin we have an elliptic equilibrium with characteristic frequencies λn . To transform this Hamiltonian into its Birkhoff normal form up to order four two coordinate transformations are required: one to eliminate the cubic term, and one to normalize the resulting fourth order term. Both calculations are elementary. Expressed in real coordinates (x, y) the result is the following. Theorem 1.4. There exists a real analytic, symplectic coordinate transformation 8 in a neighbourhood of the origin in h3/2 , which transforms the KdV Hamiltonian on h3/2 into Hc B 8 =

1X 3X 2 λn (xn2 + yn2 ) − (xn + yn2 )2 + . . . , 2 4 n≥1

n≥1

where the dots stand for terms of higher order in x and y. The important fact about the non-resonant Birkhoff normal form is that its coefficients are uniquely determined independently of the normalizing transformation, as long as it is of the form “identity + higher order terms”. For this reason, these coefficients are also called Birkhoff invariants. Comparing Theorem 1.4 with Theorem 1.1 and viewing 9 as a global transformation into a complete Birkhoff normal form we thus conclude that the two resulting Hamiltonians on h3/2 must agree up to terms of order four. In other words, the local result provides us with the first terms of the Taylor series expansion of the globally integrable KdV Hamiltonian. Corollary 1.5. The canonical transformation 9 of Theorem 1.1 transforms the KdV Hamiltonian into the Hamiltonian X X Hc (I ) = λn In − 3 In2 + . . . , n≥1

n≥1

where In = 12 (xn2 + yn2 ), and the dots stand for higher order terms in (x, y). Thus, ωn (I ) =

∂ Hc (I ) = λn − 6In + . . . . ∂ In

Here, λn and hence ωn also depend on c.

1 Overview

15

By further computing some additional terms of order six in the expansion above, we gain sufficient control over the frequencies ω to verify all nondegeneracy and nonresonance conditions for any c. Incidentally, the normal form of Theorem 1.4 already suffices to prove the persistence of quasi-periodic solutions of the KdV equation of sufficiently small amplitude under small Hamiltonian perturbations. In addition, if the perturbing term ∂ K /∂u is of degree three or more in u, then no small parameter ε is needed to make the perturbing terms small, as it suffices to work in a sufficiently small neighbourhood of the equilibrium solution u ≡ 0. We will not expand on this point in this book. A Remark on the KAM Proof Previous versions of the KAM theorem for partial differential equations such as [72, 109] were concerned with perturbations that were given by bounded nonlinear operators. This was sufficient to handle, among others, nonlinear Schr¨odinger and wave equations on a bounded interval, see for example [12, 76, 110]. This is not sufficient, however, to deal with perturbations of the KdV equation, as here the term d ∂K dx ∂u is an unbounded operator. This entails some subtle difficulties in the proof of the KAM theorem, as we outline now. Write the perturbed Hamiltonian as H = N + P, where N denotes some integrable normal form and P a general perturbation. The KAM proof employs a rapidly converging iteration scheme of Newton type to handle small divisor problems, and involves an infinite sequence of coordinate transforma tions. At each step a transformation 8 is constructed as the time-1-map X tF t=1 of a Hamiltonian vector field X F that brings the perturbed Hamiltonian H = N + P closer to some new normal form N+ . Its generating Hamiltonian F as well as the correction Nˆ to the given normal form N are a solution of the linearized equation {F ,N } + Nˆ = R, where R is some suitable truncation of the Taylor and Fourier expansion of P. Then 8 takes the truncated Hamiltonian H 0 = N + R into H 0 B 8 = N+ + R+ , where N+ = N + Nˆ is the new normal form and Z 1 {(1 − t) Nˆ + t R,F} B X tF dt R+ = 0

the new error term arising from R. Accordingly, the full Hamiltonian H = N + P is transformed into H B 8 = N+ + R+ + (P − R) B 8. See section 19 for details. What makes this scheme more complicated than previous ones is the fact that the vector field X R generated by R represents an unbounded operator, whereas the vector

16

I The Beginning

field X F generated by the solution F of the linearized equation has to represent a bounded operator to define a bona fide coordinate transformation. For most terms in F this presents no problem, because they are obtained from the corresponding terms in R by dividing with a large divisor. There is no such smoothing effect, however, for that part of R of the form 1X Rn (θ; ξ )(xn2 + yn2 ), 2 n ∈A /

where θ = (θn )n∈A are the coordinates on the torus T A , and ξ the parameters mentioned above. We therefore include these terms in Nˆ and hence in the new normal form N+ . However, subsequently we have to deal with a generalized, θ-dependent normal form X 1X N= ωn (ξ )In + n (θ; ξ )(xn2 + yn2 ). 2 n∈A

n ∈A /

This, in turn, makes it difficult to obtain solutions of the linearized equation with useful estimates. In [74] Kuksin obtained such estimates and thus rendered the iterative construction convergent. It requires a delicate discussion of a linear small divisor equation with large variable coefficients, which we reproduce in section 23. Existing Literature Theorem 1.1 was first given in [5] and [6]. A quite different approach to this result, and the one detailed here, was first presented in [60] and extended to the nonlinear Schr¨odinger equation in [51]. At the heart of the argument are orthogonality relations which first have been established in the case of the nonlinear Schr¨odinger equation by McKean & Vaninsky [91, 92]. A version of Theorem 1.2 in the case c = 0 is due to Kuksin [71, 74]. In the second paper, he proves a KAM theorem of the type discussed above which is needed to deal with perturbations given by unbounded operators, and combines it with earlier results [9, 11] concerning nonresonance properties of the KdV frequencies and the construction of local coordinates so that the linearized equations of motions along a given torus of finite gap potentials reduce to constant coefficients [71]. The proof of Theorem 1.2 presented in this book is different from the approach in [71, 74]. Instead of the local coordinates constructed in [71] we use the global, real analytic angle-action coordinates given by Theorem 1.1 to obtain quasi-periodic solutions of arbitrary size for sufficiently small and sufficiently regular perturbations of the KdV equation. To verify the relevant nonresonance conditions we follow the line of arguments used in [76] where small quasi-periodic solutions for nonlinear Schr¨odinger equations were obtained, and explicitly compute the Birkhoff normal form of the KdV Hamiltonian up to order 4 and a few terms of order 6. We stress again that our results are concerned exclusively with the existence of quasi-periodic solutions. Nothing is known about the persistence of almost-periodic solutions. The KAM theory of [108] for such solutions is not applicable here, since

1 Overview

17

the nonlinearities effect a strong, long range coupling among all “modes” in the KdV equation. There are, however, existence results for certain simplified problems. Bourgain [16, 17] considered the Schr¨odinger equation iu t = u x x − V (x)u − |u|2 u on [0, π] with Dirichlet boundary conditions, depending on some analytic potential V . Given an almost-periodic solution of the linear equation with very rapidly decreasing amplitudes and nonresonant frequencies, he showed that the potential V may be modified so that this solution persists for the nonlinear equation. The potential serves as an infinite dimensional parameter, which has to be chosen properly for each initial choice of amplitudes. This result is obtained by iterating the LyapunovSchmidt reduction scheme introduced by Craig & Wayne [28]. A similar result was obtained independently in [111] by iterating the KAM theorem about the existence of quasi-periodic solutions. As a result, one obtains for – in a suitable sense – almost all potentials V a set of almost-periodic solutions, which – again in a suitable sense – has density one at the origin. See [111] for more details.

II Classical Background

In this book we consider the periodic KdV equation as an infinite dimensional integrable Hamiltonian system, and subject it to small Hamiltonian perturbations. To this end, we extend many concepts, ideas and notions from the classical finite dimensional theory, such as angle-action coordinates, Birkhoff normal forms, and in particular KAM theory. To set the stage we give a concise review of these notions in the classical setting. To keep things short we omit lengthy proofs and give references instead.

2 Hamiltonian Formalism The abstract Hamiltonian formalism can be described either in terms of symplectic manifolds, or in terms of Poisson manifolds. The former is more familiar in mathematics, while the latter, used often in physics, is more general. We will give descriptions of both set ups and show that they are equivalent in the case of nondegenerate Poisson structures. In the following, M denotes a smooth manifold of finite dimension without boundary, where ‘smooth’ means ‘infinitely often differentiable’. We assume M to be connected, but not necessarily compact. Symplectic Manifolds Definition 2.1. A symplectic form on a smooth manifold M is a closed, nondegenerate 2-form υ on M. The pair (M, υ) is called a symplectic manifold. Here, ‘closed’ means that dυ = 0. ‘Nondegenerate’ means that υ is nondegenerate on every tangent space of M, making it a symplectic vector space. Necessarily, the dimension of a symplectic manifold M is even, since otherwise υ had a nontrivial kernel. Moreover, M is orientable, since the n-fold wedge product υ ∧ · · · ∧ υ defines a volume form on M.

20

II Classical Background

Being nondegenerate, the symplectic form υ induces an isomorphism S : T M → T ∗M X 7→ υ B X between the tangent and cotangent bundle of M at each point, referred to as the symplectic structure, where υ B X = υ(X, · ) denotes the inner product or contraction of a form υ and a vector field X . Let J = S −1 : T ∗ M → T M be the inverse of this map. Every smooth function H : M → R then defines a vector field XH = J d H on M, which is the unique vector field satisfying υ B X H = d H. X H is called the Hamiltonian vector field associated with the Hamiltonian H , and M its underlying phase space. In turn, this vector field defines a flow 8t on M called the Hamiltonian flow of the Hamiltonian H . We usually denote this flow by X Ht to indicate its connection with the Hamiltonian H . It is an immediate consequence of the definitions and the skew symmetry of υ that the Hamiltonian H is constant along the flow lines of its Hamiltonian vector field X H : d H B X Ht = d H (X H ) = υ(X H , X H ) = 0. dt In classical mechanics this is known as the law of conservation of energy. Given a symplectic form, one defines the Poisson bracket of two smooth functions G and H as {G, H } = υ(X G , X H ). It is a skew form on the linear space C ∞ (M) of all smooth functions on M. In view of the definition of Hamiltonian vector fields, we have {G, H } = υ(X G , X H ) = dG(X H ). Consequently, the flow X Ht has the property that F˙ = {F , H } for any smooth function F on M, where d F˙ = F B X Ht = d F(X H ) t=0 dt denotes the derivation of F with respect to the vector field X H .

2 Hamiltonian Formalism

21

The Poisson bracket satisfies Leibniz rule, {F G, H } = F {G, H } + G {F , H }, and the Jacobi identity, {F , {G, H }} + {G, {H ,F}} + {H , {F ,G}} = 0. Being a local property the Jacobi identity is most easily verified in the Darboux coordinates introduced below. Alternatively, one may first show that for any nondegenerate 2-form υ one has {F , {G, H }} + {G, {H ,F}} + {H , {F ,G}} = dυ(X F , X G , X H ),

(2.1)

see for example [86, p. 85]. Note that to define Hamiltonian vector fields the closedness of the 2-form υ is not required. Rather, the above identity shows that the latter is equivalent to the Jacobi identity for its Poisson bracket. Poisson Manifolds Instead of describing the evolution of a system directly by a Hamiltonian vector field on the phase space, one may also describe it indirectly by the evolution of so called observables, such as smooth functions on the phase space. The starting point here is a Poisson bracket. Let F = C ∞ (M). Definition 2.2. A Poisson bracket on a smooth manifold M is a skew-symmetric bilinear map { · , · } : F × F → F, which satisfies Leibniz rule and the Jacobi identity. A smooth manifold with a Poisson bracket is called a Poisson manifold. A flow 8t on a Poisson manifold is called a Poisson system, if there exists a function H on M, called the Hamiltonian of the system, such that d F˙ = F B 8t = {F , H } dt t=0 for any F ∈ F. By skew symmetry of the Poisson bracket, {F ,F} = 0 for all F. Hence, again H˙ = 0 for a Poisson system with Hamiltonian H . The latter, however, is not necessarily uniquely determined by the above characterization. This description of a Poisson system is implicit. But it can be made explicit in terms of its structure map as follows. By linearity and the Leibniz rule, the map F 7 → {F , H } of F into itself is a derivation. Therefore, in a finite dimensional setting,

22

II Classical Background

there exists a unique vector field X H , the Hamiltonian vector field associated with H , such that {G, H } = X H G = hdG, X H i, where h · , · i denotes the dual pairing between T ∗ M and T M. By skew-symmetry of the Poisson bracket, hdG, X H i = − hd H , X G i. Moreover, the association between d H and X H is linear. Therefore, there exists a unique map K : T ∗ M → T M, called the Poisson structure, mapping each fiber T p∗ M linearly into T p M, such that X H = K d H. We then obtain {F , H } = hd F , X H i = hd F ,K d H i. K is skew-symmetric, since the Poisson bracket is. In an infinite-dimensional setting, a derivation is not necessarily given by a vector field. In this case, the starting point is the structure map itself. Definition 2.3. A Poisson structure is a map K : T ∗ M → T M, mapping each fiber T p∗ M linearly to T p M, such that the associated bracket {F ,G} = hd F ,K dGi is skew-symmetric and satisfies the Jacobi identity. Note that this bracket automatically satisfies Leibniz rule, hence is a Poisson bracket. A Poisson structure is called nondegenerate, if it has a trivial kernel. In this case we have an inverse K −1 : T M → T ∗ M, and we can define a bilinear form υ on vector fields by υ(X, Y ) = hK −1 X ,Y i. This form is skew-symmetric and nondegenerate, since K −1 is. A Hamiltonian vector field with respect to υ is then again given by X H = K d H , since υ B X H = d H . Moreover, υ(X G , X H ) = hdG,K d H i = {G, H }, so υ is also closed in view of (2.1) and Jacobi’s identity, hence is a symplectic form. Thus, a nondegenerate Poisson structure K gives rise to a symplectic structure with S = K −1 . Conversely, a symplectic structure S defines a nondegenerate Poisson structure with K = S −1 . Hence, these two notions are equivalent, and in the following we will not distinguish between them.

2 Hamiltonian Formalism

23

Lie Brackets and Integrals The Lie bracket of two vector fields X and Y , considered as derivations, is defined as [X ,Y ] = Y X − X Y. Alternatively, one has [X ,Y ] = L Y X = −L X Y = − [Y , X ], where L X denotes the Lie-derivative with respect to the vector field X . This bracket is clearly bilinear and skew-symmetric. Moreover, the Lie bracket of two Hamiltonian vector fields is again Hamiltonian: Proposition 2.4. [X G , X H ] = X {G, H } for any two Hamiltonians G and H on a symplectic manifold. Proof. With {G, H } = dG(X H ) = X H G we have υ B X {G, H } = d {G, H } = d L XH G = L X H dG = L X H (υ B X G ) = (L X H υ) B X G + υ B (L X H X G ) = υ B [X G , X H ], since L X H υ = 0 by the Hamiltonian character of X H . u t A smooth non-constant function G is called an integral of a Hamiltonian system with Hamiltonian H , if {G, H } = 0. Since {G, H } = X H G, this means that G is constant along the flow lines of X H , which justifies the terminology. By skew symmetry of the bracket, if G is an integral for X H , then H is an integral for X G , and one says that the two Hamiltonians G and H are in involution. There may be nontrivial functions C ∈ F, called Casimir functions, with {C , · } ≡ 0. They are integrals for any Poisson system and may exist if the Poisson structure is degenerate. In the nondegenerate case the preceding proposition implies that two Hamiltonians G and H are in involution, if and only if [X G , X H ] = 0. Since this is equivalent to the fact that the flows of X G and X H commute, one also says that the two vector fields X G and X H commute.

24

II Classical Background

Canonical Transformations To preserve the Hamiltonian nature of vector fields a diffeomorphism of a symplectic or Poisson manifold has to preserve the underlying structure. Definition 2.5. A diffeomorphism 8 of a symplectic manifold is called symplectic or a symplectomorphism, if it preserves the symplectic form: 8∗ υ = υ. A diffeomorphism 8 of a Poisson manifold is called canonical, if it preserves the Poisson bracket: {F ,G} B 8 = {F B 8,G B 8} for any F and G. We consider the two cases in detail. A symplectomorphism transforms Hamiltonian vector fields according to the rule 8∗ X H = X H B8 . Using that 8∗ d H = d(H B 8) this follows from the calculation υ B X H B8 = d(H B 8) = 8∗ d H = 8∗ (υ B X H ) = 8∗ υ B 8∗ X H = υ B 8∗ X H .

(2.2)

Moreover, 8 then also preserves the associated Poisson bracket, since {G, H } B 8 = 8∗ υ(X G , X H ) = υ(8∗ X G , 8∗ X H ) = υ(X GB8 , H H B8 ) = {G B 8, H B 8}. So 8 is also canonical. On the other hand, on a Poisson manifold a canonical transformation 8 also has to preserve the Poisson structure K . This follows by inspecting its definition. If K is nondegenerate and thus defines a symplectic structure as above, then also 8∗ X H = (8∗ K )(8∗ d H ) = K d(H B 8) = X H B8 , so 8 also transforms Hamiltonian vector fields properly. Traversing the calculation in (2.2) backwards, this implies that 8∗ υ B 8∗ X H = υ B 8∗ X H for all H . But this in turn implies that 8∗ υ = υ, so 8 is a symplectomorphism with respect to the induced symplectic structure. The concepts of a canonical transformation and a symplectomorphism generalize in an obvious way to transformations between two different Poisson or symplectic manifolds.

2 Hamiltonian Formalism

25

The Standard Example The standard example of a symplectic manifold is R2n = Rn × Rn with coordinates (q, p) = (q1 , . . . , qn , p1 , . . . , pn ), often referred to as positions and moments. The symplectic form is n X υ0 = dqi ∧ d pi , i=1

which in terms of the standard scalar product h · , · i0 on Euclidean space is given as   0 I J0 = , υ0 = h · , J0 · i0 , −I 0 where I denotes the n-dimensional identity matrix. Note that J0−1 = J0T = −J0 ,

h · , J0 · i0 = hJ0−1 · , · i0 ,

so the symplectic structure is given by J0−1 , interpreted as a map T R2n → T ∗ R2n . Expressing d H = h∇ H , · i0 in terms of the h · , · i0 -gradient of H , the identity υ B X H = d H amounts to h∇ H , · i0 = d H = υ B XH = hX H , J0 · i0 = hJ0−1 X H , · i0 . We obtain X H = J0 ∇ H and the equations of motion q˙i =

∂H , ∂ pi

p˙ i = −

∂H . ∂qi

Moreover, for the induced Poisson bracket we find

 {F ,G} = J0 ∇ F , J02 ∇G 0

 = ∇ F , J0 ∇G 0



 = Fq ,G p 0 − F p ,G q 0 , and a transformation 8 is symplectic, if hD8 · , J0 D8 · i0 = h · , J0 · i0 , or D8T J0 D8 = J0 . So everything comes out as it should.

26

II Classical Background

Similarly, the standard example of a Poisson manifold is the same manifold R2n = Rn × Rn with Poisson bracket



 {F ,G}0 = Fq ,G p 0 − F p ,G q 0 . Given a Hamiltonian H , the coordinate functions evolve by q˙i = {qi , H }0 = H pi ,

p˙ i = { pi , H }0 = −Hqi ,

as before. Moreover, {F ,G}0 = hd F , J0 ∇Gi, so the Poisson structure is J0 , interpreted as a map from T ∗ R2n to T R2n . This map is invertible, and the associated symplectic form is υ(X, Y ) = hJ0−1 X ,Y i = hX , J0 Y i, as it ought to be. A variant of the standard example is obtained when the position coordinates are identified modulo 2π and are thus angular coordinates. The phase space is then Tn × Rn , where Tn = Rn /2πZn denotes the n-dimensional torus. The coordinates are usually denoted by (θ, I ) = (θ1 , . . . , θn , I1 , . . . , In ) and are called angle-action coordinates. The standard symplectic two-form is then υ0 =

n X

dθi ∧ d Ii .

i=1

Darboux Coordinates In the linear case all symplectic vector spaces of the same dimension are symplectically isomorphic [55, p. 6]. This, of course, is no longer true for nonlinear symplectic manifolds. But Darboux’s theorem states that this is still true locally around every point of a symplectic manifold. Theorem 2.6 (Darboux). Locally a symplectic manifold (M, υ) of dimension 2n is symplectomorphic to an open subset of (R2n , υ0 ). That is, given any point p in M, there is a neighbourhood W of p in M and a diffeomorphism 8 : V → W of an open set V in R2n onto W such that 8∗ υ = υ0 . The coordinates provided by 8 are called Darboux coordinates. For a proof see for example [55, p. 10–11].

3 Liouville Integrable Systems

27

3 Liouville Integrable Systems Integrable systems are particular Hamiltonian systems that can be solved for any initial data by quadratures – whence the name. For this the system has to admit sufficiently many conserved quantities in involution. It turns out that for a system of n degrees of freedom, n independent integrals in involution suffice. From now on, all manifolds, mappings and functions are assumed to be real analytic, unless otherwise stated. Lagrangian Foliations A family of m functions F1 , . . . , Fm on M is called independent, if their 1-forms d F1 , . . . , d Fm are linearly independent at every point in M. Definition 3.1. A Hamiltonian system on a symplectic manifold M of dimension 2n is called integrable (in the sense of Liouville), if its Hamiltonian H admits n independent integrals F1 , . . . , Fn in involution. That is, (i) {H ,Fi } = 0 for 1 ≤ i ≤ n, (ii) {Fi ,F j } = 0 for 1 ≤ i, j ≤ n, and (iii) d F1 ∧ · · · ∧ d Fn 6= 0 everywhere on M. Example A. In standard angle-action coordinates (θ, I ) on Tn ×Rn any Hamiltonian of the form H = H (I ) is integrable with integrals Fi = Ii ,

1 ≤ i ≤ n.

They are everywhere in involution and independent, and the Hamiltonian is in fact a function of these integrals. This example will be discussed below. Example B. In standard cartesian coordinates (q, p) on Rn × Rn any Hamiltonian of the form H = H (q12 + p12 , . . . , qn2 + pn2 ) is integrable with integrals Fi = qi2 + pi2 ,

1 ≤ i ≤ n.

More precisely, these functions are integrals in involution everywhere on Rn × Rn , but are independent only on the dense open subset, where none of them vanishes. This example will be discussed in the next section. To give a geometric description of an integrable system consider first an arbitrary number of smooth independent functions F1 , . . . , Fm on M. Let F = (F1 , . . . , Fm ) : M → Rm .

28

II Classical Background

This map is a submersion, and every value is a regular value. Every nonempty leaf M c = F −1 (c) = { p ∈ M : F( p) = c } is therefore a smooth submanifold of M of codimension m, and the whole manifold M is foliated into these leaves. To give a symplectic description of the tangent space of M c , define the skeworthogonal complement of a set V of vectors to be the linear space V ∠ of all vectors X such that υ B X vanishes on V . Then, by the definition of Hamiltonian vector fields, \ \ ∠ ker d Fi = X∠ Tp M c = Fi ( p) = V F ( p), 1≤i≤m

1≤i≤m

where VF = span (X F1 , . . . , X Fm ). If the functions Fi are also in involution, then the following result applies. Lemma 3.2. Suppose the functions F = (F1 , . . . , Fm ) define a foliation of M with leaves M c = F −1 (c). Then the following statements are equivalent. (i) The functions F1 , . . . , Fm are in involution: {Fi ,F j } = 0 for 1 ≤ i, j ≤ m. (ii) The Hamiltonian vector fields X Fi are everywhere tangent to the leaves of F. That is, X Fi ( p) ∈ T p M c for 1 ≤ i ≤ m and p ∈ M c . (iii) The leaves M c are co-isotropic submanifolds of M. That is, T p∠ M c ⊆ T p M c at every point p in M c . Proof. All these statements are different interpretations of the condition that {Fi ,F j } = d Fi (X F j ) = υ(X Fi , X F j ) vanishes for 1 ≤ i, j ≤ m. T p∠ M c

t u

Mc

From ⊆ Tp and the dimension formula for orthogonal complements in general we conclude that m ≤ n. So on a symplectic manifold M of dimension 2n there are at most n independent functions in involution. Moreover, if their number is indeed n, then T p∠ M c = T p M c everywhere, so each leaf is a so called Lagrangian submanifold of M. The latter is, by definition, a submanifold of maximal possible dimension such that the restriction of the symplectic form to it vanishes. Corollary 3.3. If F1 , . . . , Fn are n independent functions in involution on M, then the map F = (F1 , . . . , Fn ) defines a foliation of M into Lagrangian submanifolds M c = F −1 (c). The converse is also true. If F = (F1 , . . . , Fn ) defines a foliation of M into Lagrangian submanifolds, then the Fi are independent and in involution. This follows immediately from the preceding lemma.

3 Liouville Integrable Systems

29

Now suppose the Hamiltonian H admits F1 , . . . , Fn as independent integrals. From {H ,Fi } = 0 we conclude that 0 = {Fi , H } = d Fi (X H ), so its Hamiltonian vector field X H is tangent to the leaves M c . It follows that these are invariant manifolds with respect to its flow, and we obtain the following geometric picture of an integrable Hamiltonian system. Corollary 3.4. A Hamiltonian system is integrable in the sense of Liouville if and only if it admits a foliation of its phase space into invariant Lagrangian submanifolds. It is clear from the preceding discussion that there is nothing intrinsic about a particular integrable Hamiltonian admitting F1 , . . . , Fn as integrals. Any other Hamiltonian in involution with them is integrable as well. This leads one to consider the Poisson algebra associated with those integrals. Definition 3.5. The Poisson algebra associated with an integrable Hamiltonian system with independent integrals F = (F1 , . . . , Fn ) is the space A(F) = { G ∈ F : {G,Fi } = 0 for 1 ≤ i ≤ n } of all functions in F = C ω (M) in involution with F1 , . . . , Fn . This is an algebra with multiplication given by the Poisson bracket, since by the Jacobi identity, 0 = {{G, H },Fi } + {{H ,Fi },G} + {{Fi ,G}, H } = {{G, H },Fi } for any G and H in A(F) and any Fi . In fact, we have {G, H } = 0,

G, H ∈ A(F),

since X G and X H are tangent to any leaf of the associated foliation of M, and the latter are Lagrangian manifolds. Thus, {G, H } = υ(X G , X H ) = 0. We also note that any smooth function of functions in A(F) is again in A(F). For if G 1 , . . . , G m are constant along the flow lines of any X Fi , then so is any compound function f (G 1 , . . . , G m ), whence it is in involution with F1 , . . . , Fn . The Liouville-Arnold-Jost Theorem Suppose the Hamiltonian H is integrable in the sense of Liouville with integrals F1 , . . . , Fn in involution. Liouville showed that locally around every point one can introduce standard symplectic coordinates (q, p) in such a way that H = H ( p).

30

II Classical Background

Thus, the coordinates p1 , . . . , pn become integrals. Under an additional topological assumption there is a global version of Liouville’s result due to Arnold [4] in the case of the standard symplectic space. Jost [58] generalized it to arbitrary symplectic manifolds and also removed an unnecessary assumption from Arnold’s result. The following is therefore known as the Liouville-Arnold-Jost Theorem, while Arnold himself refers to it as Liouville’s Theorem. First we consider n commuting integrals by themselves. Theorem 3.6 (Liouville-Arnold-Jost). Let (M, υ) be a symplectic manifold of dimension 2n, and let F = (F1 , . . . , Fn ) be n independent functions in involution on M. Suppose one of the leaves of F, say M 0 = F −1 (0), is compact and connected. Then (i) M 0 is an n-dimensional embedded torus, and (ii) there exist an open neighbourhood U of M 0 in M, an open neighbourhood D of 0 in Rn , and a diffeomorphism 9 : Tn × D → U introducing angle-action coordinates with 9 ∗ υ = υ0 ,

9 ∗ M 0 = Tn × {0},

such that the functions Fi B 9 are independent of the angular coordinates. There is nothing intrinsic about a particular set of integrals F = (F1 , . . . , Fn ), and we may always replace them by other integrals in the Poisson algebra A(F). Using this freedom we have the following addendum to the Liouville-Arnold-Jost theorem. Addendum to Theorem 3.6. Let B be the range of F on U . Restricting the neighbourhoods D and U , if necessary, there also exists a diffeomorphism 4 : B → D, such that the functions 4 B Fi B 9 are the coordinate functions on D. Now assume the Hamiltonian H is integrable with integrals F1 , . . . , Fn satisfying the assumptions of the Liouville-Arnold-Jost Theorem. Replacing them by other integrals if necessary and denoting the coordinates by (θ, I ), we can assume that F˜i = Fi B 9 = Ii . The transformed Hamiltonian H˜ = H B 9 then satisfies 0 = { H˜ , F˜i } =

∂ H˜ , ∂θi

1 ≤ i ≤ n.

3 Liouville Integrable Systems

31

Hence we find that H˜ = H˜ (I ) is independent of the angular coordinates as in example A above. Thus we have the following corollary, which is also often referred to as the Liouville-Arnold-Jost Theorem. Corollary 3.7 (Liouville-Arnold-Jost). Suppose the Hamiltonian H is integrable in the sense of Liouville. If one of its leaves is compact and connected, then a neighbourhood U of it is completely foliated into invariant n-dimensional Lagrangian tori, and one can introduce angle-action coordinates by a transformation 9 : Tn × D → U,

9 ∗ υ = υ0 ,

such that the transformed Hamiltonian H B 9 is a function of the actions alone. Thus example A is typical for an integrable system with compact leaves, at least locally around any given leaf. For the question of global existence of angle-action coordinates see for example [37]. It is evident from the preceding discussion that one set of angle-action coordinates works not only for a particular integrable Hamiltonian, but indeed for all Hamiltonians in the associated algebra A. Thus, one coordinate transformation introduces angle-action coordinates simultaneously for all Hamiltonians in A. Moreover, in these coordinates, A consists of all functions in involution with I1 , . . . , In , so we have A = { G ∈ F : G = G(I ) }. In an arbitrary coordinate system, A consists of all analytic functions that are functions of the integrals F1 , . . . , Fn . Kronecker Tori Consider now an integrable Hamiltonian H = H (I ) in angle-action coordinates. The equations of motion are θ˙i = ωi (I ), I˙i = 0, where ωi (I ) =

∂H (I ), ∂ Ii

1 ≤ i ≤ n.

They are easily integrated, whence the name integrable system. Their general solution is θ(t) = θ o + ω(I o )t, I (t) = I o . Every solution curve is a straight line, which, due to the identification of the angular coordinates θ modulo 2π, is winding around the underlying invariant torus TI o = Tn × { I o } with constant angular velocities, or frequencies ω(I o ) = (ω1 (I o ), . . . , ωn (I o )).

32

II Classical Background

They completely determine the dynamics on this torus which consists of parallel translations. Such tori play an eminent role in dynamics and have a variety of special names. They are called Kronecker tori, or parallel or rotational tori. One also speaks of tori with parallel, linear, or rotational flow, among others. The associated frequencies are often simply called the frequencies of the invariant torus. In other coordinates a Kronecker torus usually does not look that simple. We therefore make the following definition, which applies to vector fields in general. Definition 3.8. Let X be a smooth vector field on a manifold M of arbitrary dimension. An invariant n-torus T of X is called a Kronecker torus, or torus with linear flow, if there exists a diffeomorphism 8 : Tn → T, such that 8∗ X is a constant n-vector ω on Tn , the frequency vector of the Kronecker torus. The pair (T, ω) is called a Kronecker torus with frequencies ω. Put differently, 8 : Tn → T is an embedding into the manifold M, which is oneto-one and onto and introduces angular coordinates θ in such a way that the equations of motion on T reduce to θ˙ = ω. Consequently, through 8 the flow of X on T is conjugate to the family of parallel translations by ωt on Tn : 8tω

Tn −−−−→   8y

Tn   y8

Xt

T −−−−→ T with 8tω (θ) = θ + ωt. We point out that the frequencies ω of a Kronecker torus are not uniquely determined. Applying a torus automorphism A : Tn → Tn defined by a matrix A ∈ SL(n, Z), a so called unimodular matrix, we can always replace ω by Aω. What is uniquely determined, however, is the frequency module  M(ω) = hk,ωi : k ∈ Zn consisting of all integer combinations of the frequencies ω = (ω1 , . . . , ωn ) with integer coefficients. From a geometrical point of view an integrable Hamiltonian system around a compact connected leaf is thus completely foliated into an n-parameter family of invariant tori with linear flow, the parameters being provided by suitable integrals of the Hamiltonian system. Moreover, all these tori are Lagrangian submanifolds. From an analytical point of view all solution curves on an invariant Kronecker torus T with frequencies ω are represented as 8(θ o + ωt),

θ o ∈ Tn .

3 Liouville Integrable Systems

33

Each coordinate function, and more generally every observable along such a solution curve, is therefore what is called a quasi-periodic function of t. Definition 3.9. A continuous function q : R → R is called quasi-periodic with frequencies ω = (ω1 , . . . , ωn ), if there exists a continuous function Q : Tn → R, called the hull of q, such that q(t) = Q(ωt) for all t ∈ R. In other words, there exists a continuous function Q on Rn with period 2π in each of its arguments θ = (θ1 , . . . , θn ) such that q(t) = Q(ω1 t, . . . , ωn t) for all t. If n = 1, then q is a periodic function of period 2π/ω1 . So the notion of a quasi-periodic function generalizes that of a periodic function. If Q is sufficiently smooth, say of class C 2n+1 , then it is represented by its multidimensional Fourier series as X Q(θ ) = Qˆ k eihk,θi , k∈Zn

with its Fourier coefficients Qˆ k given by Z 1 ˆ Q(θ )e−ihk,θi dθ. Qk = (2π)n Tn We then obtain q(t) =

X

qˆk eihk,ωit ,

qˆk = Qˆ k .

k∈Zn

Thus a quasi-periodic function has a frequency spectrum, which is precisely the frequency module M(ω) of ω. Hence, from an analytical point of view the solutions of an integrable Hamiltonian system near a compact connected leaf are all quasi-periodic functions of t, which can be represented in the form X ˆ k (I o )eihk,θ o i eihk,ω(I o )it , 8(θ o + ω(I o )t, I o ) = 8 k∈Zn

where θ o ∈ Tn , I o varies over some domain in Rn , and 8 is a smooth map into the symplectic manifold M with period 2π in its first n arguments. Resonant and Nonresonant Tori The flow on a Kronecker torus is rather simple. Yet its topological and metrical properties differ sharply depending on arithmetical properties of its frequencies ω. There are essentially two cases.

34

II Classical Background

Case 1. The frequencies ω are nonresonant, or rationally independent: for all 0 6= k ∈ Zn .

hk,ωi 6= 0

Then, on this torus, each orbit is dense, the flow is ergodic, and the torus itself is minimal. Case 2. The frequencies are resonant, or rationally dependent: there exist integer relations hk,ωi = 0 for some 0 6= k ∈ Zn . The prototype is ω = (ω1 , . . . , ωm , 0, . . . , 0) with n − m ≥ 1 trailing zeroes and nonresonant (ω1 , . . . , ωm ). In this case the torus decomposes into an n−m-parameter family of identical invariant m-tori. Each orbit is dense on such a lower dimensional torus, but not in the entire Kronecker torus. A special case arises when there exist n − 1 independent resonant relations. Then each frequency ω1 , . . . , ωn is an integer multiple of a fixed non-zero frequency ω∗ , and the whole torus is filled by periodic orbits with one and the same period 2π/ω∗ . In an integrable system the frequencies on the tori may or may not vary with the torus, depending on the nature of the frequency map I 7 → ω(I ). If it is nondegenerate in the sense that det

∂2 H ∂ω = det 6 = 0, ∂I ∂I2

then this map is a local diffeomorphism, and “the frequencies ω effectively depend on the amplitudes I ”. Nonresonant and resonant tori of all types then each form dense subsets in phase space. Indeed, the resonant ones sit among the nonresonant ones like the rational numbers among the irrational numbers.

4 Birkhoff Integrable Systems Another type of integrable system arises in the study of equilibria of Hamiltonian systems, the so called Birkhoff integrable systems. On one hand, this type is more special than the Liouville type, as it suffices to look at a neighbourhood of a single point. On the other hand, it is also more general, as this neighbourhood is foliated into invariant tori of any dimension between 0 and n. Birkhoff Normal Forms Consider an isolated equilibrium of a Hamiltonian system on some symplectic manifold, that is, an isolated singular point of the Hamiltonian vector field. Choosing

4 Birkhoff Integrable Systems

35

Darboux coordinates we may transfer this equilibrium to the origin in the standard symplectic space Rn × Rn with symplectic form υ0 = h · , J0 · i and coordinates u = (q, p). Dropping an irrelevant additive constant the Hamiltonian then starts with quadratic terms, 1 H = hAu,ui + . . . , 2 where A is the symmetric 2n × 2n-Hessian of H at 0. The equations of motion are u˙ = J0 Au + . . . , where here and in the following, the dots stand for terms of higher order in u. We focus on the case of an elliptic equilibrium. That is, the spectrum of the linearized system u˙ = J0 Au is purely imaginary: spec(J0 A) = { ±iλ1 , . . . , ±iλn }, with real numbers λ1 , . . . , λn . Assuming this spectrum to be simple, there exists a linear symplectic change of coordinates that brings the quadratic part of the Hamiltonian into the normal form hAu,ui =

n X

λi (qi2 + pi2 ),

i=1

where for simplicity we denote the new coordinates by the same symbols. The associated equations of motion are q˙i = λi pi ,

p˙ i = −λi qi .

This is a system of n uncoupled harmonic oscillators, each moving in its own invariant plane with frequencies λ1 , . . . , λn , respectively. The total motion of all oscillators combined is quasi-periodic in time. Incidentally, the same normal form can be obtained, if the Hamiltonian is definite around the origin, without assuming the spectrum to be simple [55, section 1.7]. Having put the quadratic terms into normal form, this normalization process can be pushed to higher order, if additional nonresonance conditions are satisfied. The result is known as the Birkhoff normal form of a Hamiltonian. Definition 4.1. The frequencies λ1 , . . . , λn are nonresonant up to order m, if n X i=1

ki λi 6 = 0 whenever 1 ≤

n X

|ki | ≤ m,

i=1

where k1 , . . . , kn are arbitrary integers and m ≥ 1. They are nonresonant, if they are nonresonant of any finite order, or equivalently, if k1 λ1 + · · · + kn λn vanishes only when all the integer coefficients vanish.

36

II Classical Background

Definition 4.2. A Hamiltonian H is in Birkhoff normal form up to order m, if it is of the form H = N2 + N4 + · · · + Nm + Hm+1 + . . . , where the Nk , 2 ≤ k ≤ m, are homogeneous polynomials of order k, which are actually functions of q12 + p12 , . . . , qn2 + pn2 , and where Hm+1 + . . . stands for arbitrary terms of order strictly greater than m. If this holds for any m, the Hamiltonian is simply said to be in Birkhoff normal form. Note that if m is odd, then Nm is zero, and the last nontrivial term in the normal form is at most of order m − 1. Theorem 4.3 (Birkhoff normal form of order m). Let H = N2 + . . . be a real analytic Hamiltonian around the origin in standard symplectic coordinates with quadratic part n 1X N2 = λi (qi2 + pi2 ). 2 i=1

If the frequencies λ1 , . . . , λn are nonresonant up to order m ≥ 3, then there exists a real analytic symplectic transformation 8 = id + . . . such that H B 8 = N2 + N4 + . . . Nm + Hm+1 + . . . is in Birkhoff normal form up to order m. Moreover, the normal form terms are uniquely defined as long as the normalizing transformation is of the form id + . . . . The theorem is proven by applying a succession of symplectic transformations, which from H = N2 + H3 + H4 + . . . eliminate H3 , normalize H4 , eliminate H5 , and so on, until finally Hm is normalized. See the proof of Theorem 14.2 for the first two steps of this procedure, and for example [26, 98] for the general result. It is clear from this construction, that the normalizing transformation is by no means uniquely determined. In contrast, the normal form is, as long as the quadratic terms are kept fixed. If the frequencies λ1 , . . . , λn are nonresonant to any order, then this normalization process can be carried to any order as well. The resulting symplectic transformation, however, is in general no longer convergent in any neighbourhood of the origin and can only be given a meaning as a formal power series. Indeed, Siegel showed that for this transformation to be convergent, infinitely many analytic conditions have to be satisfied [123]. Theorem 4.4 (Formal Birkhoff normal form). Let H = N2 + . . . be a real analytic Hamiltonian as in the preceding theorem. If the frequencies λ1 , . . . , λn are nonresonant, then there exists a symplectic transformation 8 = id + . . . , represented by a formal power series, such that H B 8 = N2 + N4 + . . . is in Birkhoff normal form as a formal power series.

4 Birkhoff Integrable Systems

37

The R¨ussmann-Vey-Ito Theorem If some transformation into Birkhoff normal form were convergent, then the resulting Hamiltonian would be integrable in a neighbourhood of the origin, the integrals in involution being q12 + p12 , . . . , qn2 + pn2 or any functions of these, as in example B in the previous section. These integrals are functionally independent in the sense, that their 1-forms are linearly independent on a dense open subset. It turns out that a certain converse is also true. If a Hamiltonian with a nonresonant elliptic equilibrium admits n functionally independent integrals in involution, then the formal transformation into Birkhoff normal form is convergent, hence the Hamiltonian itself is integrable. Such a result was first proven by R¨ussmann [115] for systems of two degrees of freedom. It was later extended by Vey [132] to higher degrees of freedom. Finally Ito [56] removed a certain nondegeneracy assumption from Vey’s theorem. See also [140] for further results. Again, as in the case of the Liouville-Arnold-Jost Theorem, this is not so much a statement about a particular Hamiltonian, but rather a statement about a Poisson algebra of Hamiltonians. To formulate this result, we make the following definition. Definition 4.5. A Poisson algebra A of Hamiltonians on a symplectic manifold M is said to be nonresonant at a point m in M, if it contains a Hamiltonian with a nonresonant elliptic equilibrium at m. It turns out that then m is an equilibrium for any Hamiltonian in the algebra. ¨ Theorem 4.6 (Russmann-Vey-Ito [56, 115, 132]). Let F = (F1 , . . . , Fn ) be n functionally independent functions in involution in a neighbourhood of a point m on a symplectic manifold M of dimension 2n. If their associated Poisson algebra A(F) is nonresonant at m, then one can introduce symplectic coordinates (q, p) around m so that A(F) consists of all functions in F = C ω (M), which are actually functions of q12 + p12 , . . . , qn2 + pn2 . Thus, one symplectic transformation puts all Hamiltonians simultaneously into Birkhoff normal form, which are in involution with F1 , . . . , Fn . Among those, many even have resonant frequencies at the equilibrium. Incidentally, the assumption that the functions F1 , . . . , Fn are in involution can be dropped. This property follows if all other conditions are satisfied [56]. For a particular Hamiltonian we have the following corollary. But first we make a definition. Definition 4.7. A 2n-dimensional Hamiltonian is called Birkhoff integrable near an equilibrium, if it admits n functionally independent integrals in involution in some neighbourhood of the equilibrium. Corollary 4.8. Suppose the Hamiltonian H is Birkhoff integrable near an elliptic equilibrium. If the frequencies at the equilibrium are nonresonant, then one can introduce symplectic coordinate (q, p) around it so that the Hamiltonian is a function of q12 + p12 , . . . , qn2 + pn2 alone. We refer to coordinates of this type as Birkhoff coordinates.

38

II Classical Background

Orbit Structure Consider now an integrable Hamiltonian in convergent Birkhoff normal form, H = H (I1 , . . . , In ) with 2Ii = qi2 + pi2 for 1 ≤ i ≤ n. The equations of motions are q˙i = ωi (I ) pi , where ωi (I ) =

p˙ i = −ωi (I )qi , ∂H (I1 , . . . , In ) ∂ Ii

are constant along each orbit, as the Ii are integrals. Again, each orbit is the superposition of the motions of n oscillators, each moving in its own invariant plane with fixed frequencies ω1 , . . . , ωn , which now depend on I1 , . . . , In in a usually nonlinear fashion. This orbit is winding around the underlying invariant torus  T I o = (q, p) : qi2 + pi2 = 2Iio for 1 ≤ i ≤ n . Hence, the entire motion is again quasi-periodic. But in contrast to a Liouville integrable Hamiltonian, the dimension of these tori is not fixed, but varies. Indeed,  dim T I o = card Iio : Iio > 0 is the number of positive amplitudes, or oscillators actually in motion. For example, on the dense open subset where all oscillators are excited the tori have maximal dimension, and we may introduce angle-action coordinates (θ, I ) by the symplectic transformation p p qi = 2Ii cos θi , pi = 2Ii sin θi for 1 ≤ i ≤ n to obtain a Liouville integrable Hamiltonian as in the previous section. If, however, only a smaller number of oscillators is excited, say I1o , . . . , Imo > 0,

o Im+1 , . . . , Ino = 0,

with 1 ≤ m < n, then the motion is confined to an m-dimensional torus, and we may introduce angle-action coordinates only for the first m modes. We obtain an integrable Hamiltonian of the form 2 2 H = H (I1 , . . . , Im , qm+1 + pm+1 , . . . , qn2 + pn2 )

=

m X i=1

λi Ii +

n 1 X λi (qi2 + pi2 ) + . . . . 2 i=m+1

5 KAM Theory

39

The equations of motion are θ˙i = ωi (I ),

I˙i = 0

for 1 ≤ i ≤ m, and q˙i = ωi (I ) pi ,

p˙ i = −ωi (I )qi

for m + 1 ≤ i ≤ n, with frequencies ωi (I ) defined as before. This system features a family of m-dimensional invariant tori Tm × { I o } × { 0 } × { 0 }, depending on m parameters I o = (I1o , . . . , Imo , 0, . . . , 0). On each torus the motion is described by m internal frequencies ω1 (I o ), . . . , ωm (I o ). Their normal space is described by the remaining cartesian coordinates, whose origin is an elliptic equilibrium with characteristic frequencies ωm+1 (I o ), . . . , ωn (I o ), the external frequencies of the torus. These so called lower dimensional elliptic invariant tori will play a central role in extending the results of the classical KAM theory described in the next section to an infinite dimensional system such as the perturbed KdV equation.

5 KAM Theory Integrable systems are the exception, not the rule. But many interesting Hamiltonian systems may be viewed as small perturbations of an integrable system. Examples are the planetary system we live in, the motion of a free particle on a slightly deformed surface of revolution, or a Hamiltonian in Birkhoff normal form up to a finite order. Thus the question arises: What happens to a foliation of invariant tori with their quasi-periodic motions under small perturbations of the Hamiltonian? The Classical KAM Theorem Consider a Hamiltonian in angle-action coordinates (θ, I ) of the form H = H0 (I ) + Hε (θ, I ), where H0 is the unperturbed Liouville integrable Hamiltonian, and Hε is a general perturbation. For simplicity we assume the latter to be of the form Hε = ε H1 (θ, I ), so that ε measures the size of the perturbation. We recall that the unperturbed system is said to be nondegenerate, if the frequency map I 7→ ω(I ) =

∂ H0 (I ) ∂I

40

II Classical Background

is a local diffeomorphism everywhere. This is sometimes called Kolmogorov’s condition. Under this assumption, the frequencies vary with the actions in a locally oneto-one manner. The first result goes back to Poincar´e and is of a negative nature. He observed that the resonant tori are in general destroyed by an arbitrarily small perturbation. In particular, out of a torus with an n − 1-parameter family of periodic orbits, usually only finitely many periodic orbits survive a perturbation, while the others disintegrate and give way to chaotic behavior. So in a nondegenerate system a dense set of tori is usually destroyed. This, in particular, implies that a generic Hamiltonian system is not integrable [85]. A dense set of tori being destroyed there seems to be little hope for other tori to survive. Indeed, until the fifties it was a common belief that arbitrarily small perturbations can turn an integrable system into an ergodic one on each energy surface. In the twenties there even appeared an – erroneous – proof of this “ergodic hypothesis” by Fermi. But in 1954 Kolmogorov [65] observed that the converse is true – the majority of tori survives. He stated the persistence of those Kronecker systems, whose frequencies ω are not only nonresonant, but are strongly nonresonant or diophantine in the sense that there exist constants α > 0 and τ > 0 such that α |hk,ωi| ≥ τ for all 0 6 = k ∈ Zn . |k| Such a condition is called a diophantine or small divisor condition, as the expressions hk,ωi enter into the denominators of formal series expansion of quasi-periodic solutions of the perturbed system also known as Lindstedt series. The existence of such frequencies is easy to verify. Fix τ , and let 1α denote the set of all ω ∈ Rn satisfying these infinitely many conditions with given α > 0. If τ > n − 1, then for any bounded subset  ⊂ Rn one has the straightforward Lebesgue measure estimate meas( X 1α ) = O(α). Hence, almost every ω in Rn belongs to some set 1α with α > 0. But although almost all frequencies are strongly nonresonant, it is not true that almost all tori survive a given perturbation, no matter how small ε. The reason is that the parameter α in the small divisor condition limits the size of the perturbation through the condition ε  α2. Conversely, under a given small perturbation of size ε, only those Kronecker tori with frequencies ω in 1α with √ α  ε, do survive. Thus, we can not allow α to vary, but have to fix it in advance. To state the KAM theorem, we therefore single out from a bounded domain  in Rn the subsets α ⊂ , α > 0,

5 KAM Theory

41

whose frequencies belong to 1α and also have at least distance α to the boundary of . These, like 1α , are Cantor sets: they are closed, perfect and nowhere dense, hence of first Baire category. But they also have large Lebesgue measure: meas( X α ) = O(α), provided the boundary of  is piecewise smooth. The main theorem of Kolmogorov, Arnold and Moser can now be stated as follows. Theorem 5.1 (The Classical KAM Theorem [2, 65, 96]). Suppose the Hamiltonian H = H0 + Hε is real analytic on the closure of Tn × D, where D is a bounded domain in Rn . If the integrable Hamiltonian H0 is nondegenerate and its frequency map a diffeomorphism D → , then there exists a constant δ > 0 such that for |ε| < δα 2 all Kronecker tori (Tn , ω) of the unperturbed system with ω ∈ α persist as Lagrangian tori, being only slightly deformed. Moreover, they depend in a Lipschitz continuous way on ω and fill the phase Tn × D up to a set of measure O(α). √ Note that given a perturbation of size ε one may choose √α ∼ ε, so the deformed tori fill the phase space at least up to a set of measure O( ε). It is an immediate and important consequence of the KAM theorem that small perturbations of nondegenerate Hamiltonians are not ergodic, as the Kronecker tori form an invariant set, which is neither of full nor of zero measure. Thus the ergodic hypothesis of the twenties was wrong. Since its conception the KAM theorem has been generalized and extended in numerous ways, and all its assumptions have been relaxed. We mention just a few of these developments, for more extensive discussions and references we point for example to [13, 20, 79, 81] and the references therein. First, neither the perturbation nor the integrable Hamiltonian need to be real analytic. It suffices that they are differentiable of class C l with l > 2τ + 2 > 2n to prove the persistence of individual tori [99, 105, 120]. For their Lipschitz dependence some more regularity is required [106]. For optimal regularity results, in particular for the closely related problem of the existence of invariant curves for area preserving twist maps, we refer to the work of Herman [53, 54]. Unfortunately, most of his deep results appeared only in the form of manuscripts. The nondegeneracy condition may also be relaxed. Originally, Kolmogorov’s condition was used to completely control the frequencies, so that their diophantine estimates can be preserved under perturbation. But in fact, it suffices that the intersection

42

II Classical Background

of the range of the frequency map with any hyperplane has measure zero. Then, after perturbation, one can still find sufficiently many diophantine frequencies, albeit they are now not known a priori. For example, if it happens that ∂ H0 = (ω1 (I1 ), . . . , ωn (I1 )) ∂I is a function of I1 alone and thus completely degenerate in the above sense, it suffices to require that ! ∂ j ωi 6= 0, det j ∂ I1 1≤i, j≤n see Xiu, You & Qiu [137]. For further results see for example the papers of Cheng & Sun, R¨ussmann and Sevryuk [22, 118, 119, 122]. Finally, the Hamiltonian nature of the equations of motion is almost indispensable for a large family of tori to persist. Analogous results are true for reversible systems [20, 100, 106]. But in any event the system has to be conservative. Any kind of dissipation immediately destroys the Cantor family of tori, although isolated ones may persist. Lower Dimensional Tori The classical KAM theorem is concerned with the persistence of Kronecker tori with strongly nonresonant frequencies in a nondegenerate system. It does not apply to resonant tori, which are foliated into lower dimensional tori and usually break up, giving rise to chaotic motions. Those lower dimensional tori do not vanish entirely, however. Consider a resonant Kronecker torus, decomposing into an n − m -parameter family of identical m-dimensional minimal tori with 1 ≤ m ≤ n − 1. In suitable coordinates its frequencies are ω = (ω1 , . . . , ωm , 0, . . . , 0), where ωˆ = (ω1 , . . . , ωm ) is nonresonant. For m = 1 in particular, such a torus is foliated into identical closed orbits. Bernstein & Katok [8] showed that in a convex system at least n of them survive any sufficiently small perturbation. For m = n − 1 > 1, the analogous result for nondegenerate systems is due to Cheng [21]. He showed that at least two tori survive, one elliptic and one hyperbolic generically, provided ωˆ is strongly nonresonant. For the intermediate cases with 1 < m < n−1, only partial result are known such as [23, 129]. These require additional assumptions for the unperturbed system or its perturbation. The long standing conjecture is that at least n − m + 1, and generically 2n−m , invariant m-tori always survive in a nondegenerate system, when ωˆ is strongly nonresonant. That is, their number should be equal to the number of critical points of smooth functions on the torus Tn−m .

5 KAM Theory

43

But in a Birkhoff integrable system, one also encounters another type of lower dimensional tori: the elliptic tori described in the previous section. Families of such tori of dimension m form smooth 2m-dimensional submanifolds, and one may ask as well about the persistence of such families under small perturbations. A special case arises for m = 1, where we encounter families of periodic orbits filling two-dimensional discs containing the equilibrium. Already Lyapunov showed that such discs persist, being only slightly deformed, if at the equilibrium the frequency of the periodic orbit is not in resonance with all other frequencies. More precisely, if λ1 , . . . , λn are these frequencies, and the disc of the unperturbed periodic orbits is contained in the plane associated with the frequency λ j , then this disc persists slightly deformed if λi ∈ / Z, λj

i 6 = j.

This condition is also known as Lyapunov’s condition. Here, no small divisors occur, and standard analytic methods suffice [124]. For 2 ≤ m ≤ n − 1, however, small divisors do occur. The first results for lower dimensional elliptic tori were announced by Melnikov [93] in the sixties of the last century, but no proofs were given. Meanwhile, Moser [97] proved the existence of quasi-periodic solutions for a large class of parameter-dependent ordinary differential equations that included the case of a Hamiltonian elliptic invariant torus of dimension m = n − 1. The case m < n − 1, however, was not accessible by his method, since it required all frequencies to be kept fixed. Finally, Eliasson [38] established the general result by allowing the frequencies to vary with the perturbation. To motivate the formulation of this result consider a Birkhoff integrable Hamiltonian in standard coordinates, H = H0 (I1 , . . . , In ),

Ii =

1 2 (q + pi2 ). 2 i

Fixing 1 ≤ m ≤ n − 1, we are interested in the persistence of the family of mdimensional tori  T I o = (q, p) : qi2 + pi2 = 2Iio for 1 ≤ i ≤ n o , . . . , I o = 0. with, say, I1o , . . . , Imo > 0 and Im+1 n Using an approach first used by Moser [97] we introduce angle-action coordinates around each individual torus T I o by setting p p pi = 2(ξi + Ii ) sin θi qi = 2(ξi + Ii ) cos θi ,

for 1 ≤ i ≤ m depending on the amplitudes ξ = (ξ1 , . . . , ξm ) = (I1o , . . . , Imo ), while keeping the other cartesian coordinates. This way, we introduce its position as an m-dimensional parameter ξ . Expanding H0 up to terms of first order in the Ii and

44

II Classical Background

second order in the qi , pi , we obtain an integrable Hamiltonian N0 =

m X

ωi (ξ )Ii +

i=1

with ωi (ξ ) =

n 1 X ωi (ξ )(qi2 + pi2 ) 2 i=m+1

∂ H0 o (I , . . . , Imo , 0, . . . , 0). ∂ Ii 1

It turns out that the discarded higher order terms may be regarded as just another perturbation. Thus, the new starting point of the perturbation theory is the family of integrable Hamiltonians N0 above on the phase space Tm × Rm × Rn−m × Rn−m with coordinates (θ, I, q, p), depending on the m-dimensional parameter ξ . Its equations of motion admit for each ξ the invariant m-torus T0 = Tm × { 0 } × { 0 } × { 0 } with internal and external frequencies ω = (ω1 , . . . , ωm ) and  = (ωm+1 , . . . , ωn ), respectively, all depending on ξ . We are interested in the persistence of this torus under small perturbations of the Hamiltonian N0 for a large set of parameters. Thus, instead of proving the existence of a large family of invariant tori in one Hamiltonian system, we establish the existence of one invariant torus for a large family of Hamiltonian systems. These two approaches are essentially equivalent, but the latter has a number of advantages. Among others, the unperturbed system is very simple, and the control of the frequencies is separated from the actions. As usual, we need some nondegeneracy to obtain a KAM theorem. Definition 5.2. The parameter-dependent family of Hamiltonians N0 given above is nondegenerate, if the map ξ 7 → ω(ξ ) is a local diffeomorphism everywhere on its domain of definition, and if ξ 7 → hk,ω(ξ )i + hl,(ξ )i 6 ≡ 0 for all (k, l) ∈ Zm × Zn−m with 1 ≤ |l| ≤ 2. The first condition is tantamount to the usual Kolmogorov condition and makes sure that there is complete control over the internal frequencies ω. The second condition, also known as Melnikov’s condition, makes sure that the additional divisors hk,ω(ξ )i + hl,(ξ )i arising in the perturbation theory are not locked in complete resonance.

5 KAM Theory

45

Consider now a perturbation N0 + Pε of N0 that is real analytic in the space coordinates as well as the parameters and is of the order of ε, say, Pε = ε P1 (θ, I, q, p, ξ ). Theorem 5.3. Suppose the Hamiltonian H = N0 + Pε is real analytic in a fixed neighbourhood of T0 × 5, where 5 is a closed bounded m-dimensional parameter domain of positive Lebesgue measure. If N0 is nondegenerate, then for all sufficiently small ε there exists a Cantor subset 5ε of parameters, such that for each parameter value in 5ε the perturbed system admits an elliptic invariant Kronecker torus close to T0 . Moreover, meas(5 X 5ε ) → 0 as ε → 0. A much more general and detailed formulation of this theorem is given and proven in chapter V, so we will not go into this here. We point out that in general we can not make explicit the rate with which the measure of 5 X 5ε tends to zero. The reason is that the external frequencies are not completely under control due to the smaller numbers of parameters – see section 22 for these matters. But if the perturbation problem does arise from considering a Birkhoff normal form of finite order, then one indeed has √ meas(5 X 5ε ) = O( ε), see [107]. Recent significant extension of this theorem are due to R¨ussmann [119] and You [138]. In particular, the latter relaxed Melnikov’s condition to the requirement that hk,ω(ξ )i − i (ξ ) 6≡ 0 for all k ∈ Zm and m + 1 ≤ i ≤ n. This is the analogue for quasi-periodic solutions to Lyapunov’s condition for periodic solutions. This way, one can also allow for multiple external frequencies, which is not possible with Melnikov’s condition. Infinite Dimensional Systems Since its conception there has been a great interest in extending the classical KAM theory to infinite dimensional systems in order to apply it to certain nonlinear partial differential equations in Hamiltonian form and models in mathematical physics, among others. The objective is to find quasi-periodic, or more generally, almostperiodic solutions. The latter can be thought of as “quasi-periodic solutions with an infinite dimensional frequency module”. More formally, they can be characterized as the uniform limits of trigonometric polynomials – see for example [63]. In the seventies some special results were obtained by Nikolenko, Ware, Zehnder and others extending to infinite dimensions Siegel’s famous result about the linearization of complex analytic vector fields in the neighbourhood of an elliptic equilibrium – see [102, 123, 134, 139] and in particular [103] and the references therein.

46

II Classical Background

Nikolenko applied these results to evolution equations in Banach spaces including nonlinear heat and Schr¨odinger equations. However, these results are restricted to perturbations of linear systems by higher order terms in a sufficiently small neighbourhood of the equilibrium, with the frequencies of the linear system satisfying infinitely many small divisor conditions. Moreover, some results such as [102] do not apply to the real analytic case. In general the difficulties arise from the fact that for any n-vector of frequencies ω = (ω1 , . . . , ωn ), |ω| min |hk,ωi| ≤ n−1 0<|k|∞ ≤K K by Dirichlet’s theorem. Hence, as the number n of frequencies tends to infinity, the small divisor bounds tend to zero rapidly even when the frequencies ωn are allowed to grow at some power of n. As a result, the bound on admissible perturbations tends to zero exponentially in n at best, and a simple minded extension of the classical KAM theorem to infinite dimensions does not apply to any perturbation different from zero. The situation is different, if the frequencies ωn tend to infinity very rapidly, say, hyperexponentially with n. In this case the classical diophantine estimates still hold, and the classical KAM proof works also in this case, when norms are properly chosen. Such a result was obtained by Chierchia & Perfetti [24] for Hamiltonians of the form “kinetic energy + potential energy” in the so called isochronous case, where ωn = In , establishing the existence of almost-periodic solutions on invariant tori of maximal dimension. But the strong growth condition on the frequencies obviously excludes applications to realistic problems. In mathematical physics one is interested in lattices of harmonic oscillators which are subject to anharmonic coupling forces of finite or short range, or of hierarchical structure. Their frequencies are usually assumed to be independent, identically distributed random variables with bounded or unbounded range. Such infinite dimensional systems are well approximable by finite dimensional ones, and one can show that for sufficiently rapidly decaying amplitude distributions the majority of frequency distributions gives rise to almost-periodic motions of the entire ensemble of oscillators. These developments were initiated by the works of Vittot & Bellissard [133] and Fr¨ohlich, Spencer & Wayne [45], and a unified approach and improved results were given in [108]. The idea is to exploit the strongly localized structure of the perturbations affected by the couplings, which decay very rapidly with the distance of the oscillators involved. This allows to employ small divisor estimates of the form |hk,ωi| ≥

α , 1(|k|)1([k])

where 1 is a R¨ussmann approximation function [19, 116, 117] such as   t 1(t) = exp , β > 0, log1+β t

5 KAM Theory

47

and [k] represents the “weight” of k such as the cardinality or diameter of its support, or something similar. See [108] for details and examples. The upshot is that again the classical KAM theorem extends to this situation establishing the existence of infinite dimensional, albeit strongly localized invariant tori with almost-periodic motions. Moreover, these tori fill some nontrivial part of the phase space. Neither of these results, however, applies to nonlinear partial differential equations of Hamiltonian type, such as nonlinear Schr¨odinger or wave equations, or perturbed KdV equations. On one hand, the perturbations are not sufficiently localized to allow a rapid approximation by finite dimensional systems. On the other hand, the frequencies grow only polynomially at most, which is far too slow for the other approach. Essentially, up to now there is no general KAM theorem to handle the effects of small divisors including arbitrary combinations of infinitely many frequencies in systems arising from pde’s. But in such systems there are also families of finite dimensional elliptic invariant tori with quasi-periodic motions forming real analytic submanifolds which fill the underlying phase space densely. A KAM theorem for such tori along the lines of Theorem 5.3 involves a substantially smaller set of small divisor conditions. Besides the usual Kolmogorov conditions |hk,ωi| ≥

α |k|τ

for k 6 = 0 for the finitely many internal frequencies ω = (ω1 , . . . , ωn ), there are the Lyapunov conditions α |hk,ωi − i | ≥ 1 + |k|τ and the Melnikov conditions hk,ωi − i ±  j ≥

α 1 + |k|τ

involving the infinitely many external frequencies  = (1 , 2 , . . . ). Thus, at most two of them enter these conditions at a time, making them essentially finite dimensional and manageable. The first results in this direction are due to Kuksin and Wayne [70, 135]. In subsequent developments Kuksin applied them for example to nonlinear wave and Schr¨odinger equations on intervals with Dirichlet boundary conditions and perturbed KdV equations. We point to his books [72, 75] for a comprehensive presentation and further references. In this book we will develop this theory in full detail following the exposition in [109]. Therefore we will not go into further details here. It turns out, however, that in order to meet all three conditions above the external frequencies should grow asymptotically like i ∼ i d ,

d ≥ 1.

48

II Classical Background

A linear growth rate i ∼ i seems to be the limiting case, and a slower growth rate with d < 1 does not suffice, since resonances are no longer sufficiently separated. This restricts the theory essentially to nonlinear pde on one-dimensional domains. Another restriction arises from Melnikov’s condition, which in particular requires the external frequencies to be separated from each other: i −  j ≥ α, i 6= j. As a consequence one can not deal with periodic boundary conditions, where the frequencies are asymptotically double. This restriction, however, was recently alleviated by Chierchia & You [25], who observed that it suffices to require that the i asymptotically form finite clusters of the same size and structure. Melnikov’s condition, either in the form given above or in the form formulated by Chierchia & You, is required to put the variational equations of motion along a lower dimensional invariant torus into constant coefficient form. This allows the classical KAM procedure of successive transformations of the entire system of equations to proceed. It does not, however, enter directly into the construction of the invariant torus with quasi-periodic motions itself. For example, it does not show up in Lyapunov’s theorem about the existence of families of periodic solutions near an elliptic equilibrium. Indeed, in the nineties Craig & Wayne [28] extended Lyapunov’s classical result and constructed Cantor discs of periodic solutions for a nonlinear wave equation with periodic boundary conditions. They used a Lyapunov-Schmidt reduction scheme for the external components together with small divisor estimates in the spirit of KAM theory. This way they could replace Melnikov’s condition by a somewhat implicit, but more general condition about the distribution of all frequencies. Their approach was considerably extended by Bourgain, who not only obtained quasi-periodic solutions for Schr¨odinger equations in this way, but also periodic and quasi-periodic solutions for some two-dimensional Schr¨odinger equations to which the standard KAM theory does not apply. See [14, 15, 16] and in particular [17]. In contrast to the latter, however, this approach does not establish the linear stability of the solutions so constructed. So far, all these extensions and generalizations are restricted to perturbations which are given by bounded nonlinear operators, as explained in section 16 on page 137. They do not apply to perturbations affected by unbounded operators, as is the case for perturbed KdV equations. In this book we therefore choose to develop the underlying KAM theory along the classical lines. All these results concern the existence of quasi-periodic solutions filling finite dimensional invariant tori in an infinite dimensional phase space. Almost nothing is known, however, about the existence of almost-periodic solutions for nonlinear pde. It seems that the nonlinearities affect a strong, long range coupling which is beyond the control of the current techniques. There are, however, some existence results for a simplified problem. Bourgain [16] considered the Schr¨odinger equation iu t = u tt − V (x)u − |u|2 u

5 KAM Theory

49

on [0, π] with Dirichlet boundary conditions, depending on an analytic potential V . Given an almost-periodic solution of the linear equation with small and very rapidly decaying amplitudes and nonresonant frequencies he showed that the potential can be modified so that this solution persists in the nonlinear equation. A similar result was obtained in [111]. In both cases the construction of quasi-periodic solutions is iterated infinitely often, increasing at each step the number of frequencies involved. The potential V plays the role of infinitely many external parameters which can be adjusted along this process. This renders the example somewhat academic.

III Birkhoff Coordinates

6 Background and Results In this chapter we consider the KdV equation qt = −qx x x + 6qqx on the space L 2 (S 1 ) of 1-periodic functions on the real line. Our aim is to show that the subspace  L 20 (S 1 ) = q ∈ L 2 (S 1 ) : [q] = 0 R of functions with vanishing mean value [q] = S 1 q(x) dx admits global Birkhoff coordinates which linearize the KdV equation. More precisely, our aim is to show that there are canonical coordinates (xn , yn )n≥1 on L 20 (S 1 ) in which the KdV equation for smooth initial data takes the form x˙n = ωn yn , y˙n = −ωn xn , with frequencies (ωn )n≥1 , which depend only on the actions In = 21 (xn2 + yn2 ). In particular, the latter are preserved quantities. Indeed, the same coordinates are Birkhoff coordinates for any Hamiltonian in the KdV hierarchy. Integrals for KdV To state the result we review some basic facts about the KdV equation and fix some notations. For any integer N ≥ 0 we introduce the Sobolev space  H N = q ∈ L 2 (S 1 ; R) : kqk N < ∞

52

III Birkhoff Coordinates

of real valued functions q on S 1 = R/Z with finite norm kqk N , where X 2 2 kqk2N = |q(0)| |k|2N |q(k)| ˆ ˆ + k∈Z

P 2πikx . is defined in terms of the discrete Fourier transform qˆ of q, q(x) = k∈Z q(k)e ˆ 0 2 1 N In particular, we have H = L (S ) with norm k · k = k · k0 . We endow H with the Poisson structure proposed by Gardner, Z ∂ F d ∂G {F ,G} = dx, (6.1) S 1 ∂q(x) dx ∂q(x) where F and G are differentiable functions on H N with L 2 -gradients in H 1 . The KdV equation with periodic boundary conditions can then be expressed as a Hamiltonian system, d ∂H ∂q = , ∂t dx ∂q with the KdV Hamiltonian H (q) =

Z S1

1 2 2 qx


+ q 3 dx.

The mean value function M = [ · ] is a Casimir function for the Poisson structure d/dx, that is, d ∂M =0 dx ∂q everywhere on L 2 (S 1 ). So M commutes with every Hamiltonian on L 2 (S 1 ). Therefore, L 20 (S 1 ) is an invariant subspace for every flow defined in terms of this Poisson structure, so in particular for the KdV equation. Besides the mean value, the evolution of the KdV equation admits infinitely many independent conserved quantities, or integrals. An elegant way to construct such a family is by way of the so called Lax pair formalism. Lax [77] observed that under sufficient regularity assumptions, q is a solution of the KdV equation if and only if d L = [B,L], dt where d2 +q dx 2 denotes the Schr¨odinger operator with potential q, considered on [0, 2] with periodic boundary conditions, and [B,L] = B L − L B denotes the commutator of L with the anti-symmetric operator L=−

B = −4

d3 d d + 3q + 3 q. dx dx dx 3

6 Background and Results

53

It follows by an elementary calculation that the flow of d U = BU, dt

U (0) = I,

defines a family of unitary operators U (t) such that U ∗ (t)L(t)U (t) = L(0), where L(t) denotes the Schr¨odinger operator with potential q(t, · ). Consequently, the spectrum of L(t) is independent of t, and the flow of the KdV equation defines an isospectral deformation on the space of sufficiently smooth functions q in L 2 when considered as potentials of the Schr¨odinger operator L. Therefore, the corresponding spectral data of q provide conserved quantities of the KdV equation. Let us recall the basic properties of the Schr¨odinger operator L with potential q on [0, 2] with periodic boundary conditions – see also appendix B. It is well known [80, 82, 84] that its periodic spectrum, denoted spec(q), is pure point and consists of an unbounded sequence of periodic eigenvalues λ0 (q) < λ1 (q) ≤ λ2 (q) < λ3 (q) ≤ λ4 (q) < . . . . Equality or inequality may occur in every place with a ‘≤’-sign. The asymptotic behavior of the periodic eigenvalues for q ∈ L 2 is λ2n−1 (q), λ2n (q) = n 2 π 2 + [q] + `2 (n), where `2 (n) stands for the n-th term of a square summable sequence. By Floquet theory, the periodic spectrum also determines the spectrum of L considered on the real line R with L 2 -integrability condition. The latter is the union [λ0 , λ1 ] ∪ [λ2 , λ3 ] ∪ [λ4 , λ5 ] ∪ . . . of the spectral bands [λ2n , λ2n+1 ]. Between two adjacent bands one has the spectral gaps (λ2n−1 , λ2n ). By the above inequalities, these gaps are pairwise disjoint. But each of them may degenerate to an empty set, when λ2n = λ2n−1 , in which case one speaks of a collapsed gap. Its length, γn = λ2n − λ2n−1 ,

n ≥ 1,

is called the n-th gap length of the spectrum. By the asymptotic behavior of the eigenvalues, γn = `2 (n). In view of the Lax pair formalism the periodic eigenvalues λn (q) and hence the gap lengths γn (q) are conserved quantities under the evolution of the KdV equation. However, λ2n , λ2n−1 and γn are not smooth functions of q whenever the corresponding gap collapses. In contrast, the squared gap lengths γn2 = (λ2n − λ2n−1 )2 ,

n ≥ 1,

54

III Birkhoff Coordinates

are real analytic integrals of the KdV equation. Moreover, Garnett & Trubowitz [48] showed that together with the average [q], they uniquely determine the periodic spectrum of q. The space L 20 (S 1 ) thus decomposes into the isospectral sets  Iso(q) = p ∈ L 20 (S 1 ) : spec( p) = spec(q) , which are invariant under the KdV flow and may also be characterized by the gap lengths,  Iso(q) = p ∈ L 20 (S 1 ) : gap length( p) = gap length(q) . As shown by McKean & Trubowitz [89], these are homeomorphic to compact connected tori, whose dimension equals the number of positive gap lengths and is infinite generically – see also Proposition B.14. Moreover, the asymptotic behavior of the gap lengths characterizes the regularity of a potential in exactly the same way as its Fourier coefficients do [84]. Therefore, for any q in L 20 (S 1 ), q ∈ H0N

iff

Iso(q) ⊂ H0N

for each N ≥ 1, where H0N = H N ∩ L 20 . Hence also the phase space H0N decomposes into a collection of tori of varying dimension which are invariant under the KdV flow. In classical mechanics the existence of a foliation of the phase space into Lagrangian invariant tori is tantamount, at least locally, to the existence of angle-action coordinates. This is the content of the Liouville-Arnold-Jost theorem as described in section 3. In the infinite dimensional setting of the KdV equation, however, such coordinates can not exist due to the ubiquity of finite gap potentials. Recall that a potential q ∈ L 20 is called a finite gap potential, if only a finite number of its spectral gaps are open, while all others are collapsed. It is a fact that finite gap potentials are real analytic and dense in H0N for each N ≥ 0 – see [49] or [84, Theorem 3.4.3]. As a consequence, the dimension of the foliation of H0N into invariant tori is nowhere constant. In particular, arbitrarily close to an infinite dimensional invariant torus Iso( p), there are finite dimensional tori of arbitrary dimension. Consequently, there are no angle-action coordinates in any neighbourhood of Iso( p). Nevertheless, we are going to show that the spectral data of a potential can be used to construct actions In and angles θn on dense open subsets of L 20 in such a way that the associated Birkhoff coordinates xn =

p 2In cos θn ,

yn =

p 2In sin θn ,

extend real analytically to all of L 20 and give rise to a global symplectic coordinate system. The actions can be viewed as rescalings of the squared gap lengths, and we

6 Background and Results

55

will prove all of the above statements about squared gap lengths for these actions instead. For example,  Iso(q) = p ∈ L 20 (S 1 ) : actions( p) = actions(q) , that is, isospectral sets may be characterized by these actions. Actions and Angles We now try to motivate the definitions of actions and angles given in the next two sections. To this end we introduce some more concepts and notations. Some more details are given in appendix B. Let y1 (x, λ, q) and y2 (x, λ, q) be the standard fundamental solution of −y 00 + qy = λy, defined by the initial conditions y1 (0, λ, q) = 1, y10 (0, λ, q) = 0,

y2 (0, λ, q) = 0, y20 (0, λ, q) = 1.

Let 1(λ, q) = y1 (1, λ, q) + y20 (1, λ, q) be its associated discriminant. The periodic spectrum of q is precisely the zero set of the entire function 12 (λ, q) − 4, and we have the product representation 12 (λ) − 4 = 4(λ0 − λ)

Y (λ2n − λ)(λ2n−1 − λ) , n4 π4

n≥1

see for example [89]. Hence, this function is also a spectral invariant. The square root of 12 (λ) − 4 is defined on the Riemann surface  6(q) = (λ, z) : z 2 = 12 (λ, q) − 4 ⊂ C2 , which is a hyperelliptic surface whose genus is precisely the number of open gaps of q. It may be viewed as two copies of the complex plane slit open along (−∞, λ0 ) and each open gap (λ2n−1 , λ2n ) and then glued together crosswise along the slits. This Riemann surface is another spectral invariant associated with q. To define angles we also need to consider the spectrum of the differential operator L = −d 2 /dx 2 + q with Dirichlet boundary conditions on the interval [0, 1]. This spectrum consists of an unbounded sequence of Dirichlet eigenvalues µ1 (q) < µ2 (q) < µ3 (q) < . . . , which satisfy λ2n−1 (q) ≤ µn (q) ≤ λ2n (q) for all n ≥ 1. Thus, the n-th Dirichlet eigenvalue µn is always contained in the n-th interval [λ2n−1 , λ2n ].

56

III Birkhoff Coordinates

By the Wronskian identity, 2

12 (µn ) − 4 = (y1 (1, µn ) − y20 (1, µn )) . With any Dirichlet eigenvalue µn we can therefore uniquely and analytically associate a sign by defining q ∗ 12 (µn ) − 4 = y1 (1, µn ) − y20 (1, µn ). (6.2) This in turn defines the point µ∗n

  q ∗ 2 = µn , 1 (µn ) − 4

(6.3)

on the Riemann surface 6(q), which is sometimes referred to as a Dirichlet divisor. McKean and Trubowitz [89] observed that these Dirichlet divisors provide coordinates on isospectral sets. Each set Iso(q) is homeomorphic to the product of two-fold coverings of the intervals [λ2n−1 (q), λ2n (q)], n ≥ 1, with endpoints identified, and each point on Iso(q) is uniquely described by the √ positions of its Dirichlet eigenvalues µn within these intervals and the signs of ∗ 12(µn ) − 4. See Proposition B.14 for the details. The Dirichlet eigenvalues (µn )n≥1 can be complemented to a symplectic coordinate system on L 20 by introducing the quantities κn (q) = 2 log (−1)n y20 (1, µn (q), q),

n ≥ 1,

[43, 130]. Then q 7 → (µˆ n (q), κn (q))n≥1 , where µˆ n = µn − n 2 π 2 , defines a real analytic diffeomorphism of L 20 into a suitable Hilbert space of sequences, and one computes that {µn ,µm } = 0, {κn ,µm } = δnm , {κn ,κm } = 0, for all n, m ≥ 1. See [112, Theorem 2.8] for a detailed proof. Hence, the new variables are canonical, and the induced symplectic structure is given by dα, where X α= κm dµm . m≥1

We may now define actions in analogy to Arnold’s formula in the finite dimensional case [3, section 50]. Set Z Z 1 1 X In = α= κm dµm , 2π cn 2π cn m≥1

6 Background and Results

a1

57

a2

λ0 Figure 1 a-cycles

where cn is a cycle on the invariant torus Iso(q) corresponding to µn . As dµm = 0 for m 6 = n along cn , actually Z Z 1 1 In = κn dµn = − µn dκn . 2π cn 2π cn Along cn , µn moves from λ2n−1 to λ2n and back to λ2n−1 , with ing its sign at the turning point. With y20 (1, µn )

p ∗

12 (µn ) − 4 chang-

  q 1 ∗ 2 = 1(µn ) − 1 (µn ) − 4 , 2

one calculates dκn = 2

d y20 (1, µn ) y20 (1, µn )

p 1 − 1(µn )/ ∗ 12 (µn ) − 4 p dµn 1(µn ) − ∗ 12 (µn ) − 4 ˙ n) 21(µ =−p dµn . ∗ 12 (µn ) − 4 ˙ n) = 21(µ

(6.4)

So altogether we may write In =

2 π

Z

λ2n

˙ 1(µ) µp dµ, 2 λ2n−1 1 (µ) − 4

with the sign of the root to be chosen properly. Now Flaschka & McLaughlin [43] observed that the last integral can be rewritten as a contour integral on the Riemann surface 6(q), In =

1 π

˙ 1(λ) dλ, λp an 12 (λ) − 4

Z

where an denotes a contour around the lift of [λ2n−1 , λ2n ] on the canonical sheet of 6(q) as indicated in figure 1. This way, the actions In are determined entirely by the spectral data of q. See also [44, 131].

58

III Birkhoff Coordinates

Now let us pretend that we can complement the actions In by canonically conjugate angles θn . Then the canonical one-form α can also be written as X α= Im dθm + d S m≥1

with some exact one-form d S. Let us also assume that the restriction of d S to any isospectral torus Iso(q) vanishes, that is, S is a function of the actions alone. Then, formally, ∂ def = α = αn . dθn Iso(q) ∂In Iso(q) Integrating along any path on Iso(q) from some fixed starting point q0 we thus get Z q Z q θn = dθn = αn . q0

q0

This integral is independent of the path chosen since dα = 0 on Iso(q). To make this more explicit we look again at the (µ, κ)-coordinate system. Fix as q0 the unique potential on Iso(q) with µk (q0 ) = λ2k−1 (q0 ),

k ≥ 1.

Then we can move from q0 to q by moving µk from λ2k−1 to µk (q) successively for k = 1, 2, . . . . For each move, the proper √ sheet of the two-folded covering of [λ2k−1 , λ2k ] is determined by the sign of ∗ 12(µk ) − 4, that is, by the point µ∗k . This way we obtain, somewhat formally, X Z µk θn = αn . (6.5) k≥1

λ2k−1

We still have to get hold of the integrals in this identity. Expressing α again in terms of the (µ, κ)-coordinates we can proceed as above for the actions. Formally, Z µk Z µk 1 ∂κk 1 αn = dµk . 2π λ2k−1 2π λ2k−1 ∂ In Similarly to (6.4) one computes 2∂1/∂ In |µk ∂κk =−p ∗ ∂ In 12 (µk ) − 4 and again notes that the last integral can be rewritten as a path integral on the Riemann surface 6(q). That is, 1 2π

Z

µk

λ2k−1

∂κk 1 dµk = ∂ In 2π

Z

µ∗k

λ2k−1

βn ,

6 Background and Results

59

with the on 6(q) holomorphic one-form 2∂1/∂ In βn = − p dλ. 12 (λ) − 4 To identify βn note that by the definition of Im , Z ∂ Im 1 = αn = δmn , ∂ In 2π cm and hence 1 2π

Z

βn = δmn

am

for all m, n ≥ 1. But these properties uniquely characterize a holomorphic one-form on 6(q), and it is actually fairly straightforward to construct them in the first place. More precisely, using the implicit function theorem we will construct entire functions ψ1 , ψ2 , . . . such that Z ψn (λ) 1 p dλ = δmn 2π am 12 (λ) − 4 R µk for all m, n ≥ 1 – see appendix D. Then we can rewrite each integral λ2k−1 αn in R µ∗ k (6.5) as a Riemann surface integral λ2k−1 βn , and we define the n-th angle as θn =

XZ k≥1

µ∗k λ2k−1

ψn (λ) p dλ mod 2π 12 (λ) − 4

for each open gap (λ2n−1 , λ2n ). By a slight abuse of terminology, we may refer to the map q 7→ θ(q) = (θn (q)) as the Abel map [101]. It turned out that these angles linearize the KdV equation [32, 33, 35, 36, 57, 88, 89, 90]. That is, if t 7 → q t is a solution curve of the KdV equation with q o = q, then q t evolves on Iso(q o ) in such a way that for every n corresponding to an open gap, θn (q t ) = θn (q o ) + ωn t with a certain frequency ωn . We will reprove this fact by verifying that the In and θn are indeed angle-action coordinates for the KdV equation. This will also justify the assumption on d S made above. Birkhoff Coordinates The action variables In are real analytic on all of L 20 , while each angle θn is real analytic modulo 2π on the dense open domain L 20 X Dn , with Dn = { q : γn (q) = 0 }. We show that the associated Birkhoff coordinates p p xn = 2In cos θn , yn = 2In sin θn

60

III Birkhoff Coordinates

are real analytic on all of L 20 and give rise to a global coordinate system, in which the KdV Hamiltonian is a function of the actions alone. Indeed, the same holds simultaneously for any Hamiltonian in the KdV hierarchy on suitable subspaces of L 20 . This may be considered as an instance of an infinite dimensional version of the R¨ussmann-Vey-Ito theorem of section 4 concerning the existence of transformations into integrable Birkhoff normal forms [56, 115, 132]. To state this result, we introduce the model space of real sequences hr = `r2 × `r2 with elements (x, y), where o n X n 2r |xn |2 < ∞ . `r2 = x ∈ `2 (N; R) : kxkr2 = n≥1

We endow this space with the standard Poisson structure, for which {xn ,ym } = δnm , while all other brackets vanish. Theorem 6.1. There exists a diffeomorphism  : L 20 → h1/2 with the following properties. (i)  is one-to-one, onto, bi-analytic, and preserves the Poisson bracket. (ii) For each N ≥ 0, the restriction of  to H0N , denoted by the same symbol, is a map  : H0N → h N +1/2 , which is one-to-one, onto, and bi-analytic as well. (iii) The coordinates (x, y) in h3/2 are global Birkhoff coordinates for the KdV equation. That is, the transformed KdV Hamiltonian H B −1 depends only on xn2 + yn2 , n ≥ 1, with (x, y) being canonical coordinates in h3/2 . (iv) The same holds for any other Hamiltonian in the KdV hierarchy, considered on a subspace H0N with appropriate N . Further properties of the map  are formulated in Theorems 9.8, 11.9 and 11.10 below. They are all collected in a single theorem, Theorem 12.6 on page 108, at the end of this chapter. Given that (x, y) are Birkhoff coordinates for each Hamiltonian H in the KdV hierarchy on the appropriate subspace of h1/2 , the corresponding equations of motions are simply x˙n = ωn (I)yn , y˙n = −ωn (I)xn , where ωn (I) =

∂ H0 (I), ∂ In

I = (In )n≥1 =

1 2

(xn2 + yn2 )n≥1 ,

are the frequencies associated with the transformed Hamiltonian H 0 = H B −1 . Thus, the KdV equations are rewritten as infinite-dimensional systems of integrable ordinary differential equations.

6 Background and Results

61

Furthermore, using Riemann bilinear relations we obtain explicit formulas and asymptotic expansions of the frequencies of the first three Hamiltonians in the KdV hierarchy, which we need for the KAM results in chapter IV. Outline of Proof The proof of the theorem splits into four parts. First we define actions In and angles θn as indicated above. We show that each In is real analytic on L 20 , while each θn is real analytic only on the dense open domain L 20 X Dn , when taken modulo 2π. Next, we define the associated cartesian coordinates xn and yn as usual. Although defined originally only on L 20 X Dn , we show that they extend real analytically to a complex neighbourhood W of L 20 . Surely, the angle θn blows up when γn collapses, but this blow up is compensated by the rate at which In vanishes in the process. In particular, for real q the resulting limit will vanish. For complex potentials, however, this limit will typically not vanish, since the associated Schr¨odinger operator is no longer self-adjoint. As a consequence, it may happen that λ2n = λ2n−1 , but µn 6 = λ2n . In such a case, xn and yn will not vanish. Then we show that the thus defined map  : q 7→ (x, y) is a diffeomorphism between L 20 and h1/2 . The main problem here is to verify that dq  is a linear isomorphism at every point q. This is done with the help of orthogonality relations among the coordinates, which are nothing but their Poisson brackets. For the nonlinear Schr¨odinger equation the corresponding orthogonality relations have first been established by McKean & Vaninsky [91, 92]. It turned out that many of their ideas can also be used in the case of the KdV equation. We also verify that each Hamiltonian in the KdV hierarchy becomes a function of the actions alone, using their characterization in terms of the asymptotic expansion of the discriminant 1 as λ → −∞ and hence as spectral invariants. Finally, we verify that  preserves the Poisson bracket. To this end it is more convenient to look at the associated symplectic structures. This way, we only need to establish the regularity of the gradient of θn at special points, not everywhere. Thus, we equivalently show that  is a symplectomorphism. This will complete the proof of the main results of Theorem 6.1. Some Notations and Notions In the sequel we will need to consider various root functions, and it will be important to fix the proper branch in each √ case. The principal branch of the square root on C X (−∞, 0] is denoted by + λ , so √ + λ > 0 for λ > 0. √ If the radicand λ is obviously real and nonnegative,√ however, we simply write λ. s On C X [−1, 1] we define a “standard” branch 1 − λ2 by p p s + 1 − λ2 = iλ 1 − λ−2 for |λ| > 1. (6.6)

62

III Birkhoff Coordinates

‘−1’

‘−i’ −1

‘+1’

Figure 2 Signs of

‘+i’ 1 p s

1 − λ2

Thus, p s 1 − λ2 > 0

for

iλ > 0,

as illustrated in figure 2. We then extend this definition continuously to more general quadratic radicands (b − λ)(λ − a) with a 6= b. To avoid ambiguities we require that a ≺ b in the lexicographic ordering, where    Re a < Re b a≺b ⇔ (6.7) or   Re a = Re b and Im a ≤ Im b. Setting γ = b − a 6 = 0 and τ = (b + a)/2 we then define p γ p s s 1 − w2 (b − λ)(λ − a) = 2 p λ−τ + = i(λ − τ ) 1 − w −2 , w= , γ /2

(6.8)

for λ not on the segment from a to b. The last expression also makes sense when a = b and gives p s (a − λ)(λ − a) = i(λ − a). (6.9) Typically, we have a = λ2n−1 and b √ = λ2n . S We define a “canonical” branch c 12(λ) − 4 on CX n≥0 (λ2n−1 , λ2n ), the complex plane slit open along the interval (−∞, λ0 ) and each open gap (λ2n−1 , λ2n ), by requiring that for real q, p c i 12 (λ) − 4 > 0 for λ ∈ (λ0 , λ1 ). (6.10) For complex q this root is defined √ by continuous extension. See figure 3 for an illustration. Finally, we denote by 12(λ) − 4 the root on the hyperelliptic surface 6(q). The L 2 -gradient of a differentiable function F : L 2 → C is denoted by ∂F =

∂F , ∂q

and is that function on L 2 satisfying Z d F(v) = 0

1

∂ F ·v dx.

7 Actions

‘+1’ ‘−1’

‘−1’

‘−i’ λ0

λ1

‘+i’

63

‘+1’

λ2 λ3 ‘−1’ p c Figure 3 Signs of 12 (λ) − 4 for real q ‘+1’

λ4

The functions we consider, such as the actions and angles, naturally extend from L 20 to L 2 , so their gradients in this sense are also well defined. However, those functions are invariant under translations of q. Consequently, Z 1 d ∂ F dx = h∂ F ,1i = F(q + c) = 0, (6.11) [∂ F] = dc 0 0 that is, their gradients belong to L 20 . So their gradients as functions on L 20 and on L 2 are actually the same, and we need not distinguish between them. For standard notations concerning spectra and eigenfunctions of Schr¨odinger operators we refer to appendix B. Finally, all neighbourhoods are assumed to be connected without saying so explicitly.

7 Actions In this section we define action variables In by the formula of Flaschka & McLaughlin [43] derived in the previous section and prove some regularity and asymptotic properties. In particular, we are interested in analyticity properties. This requires that we not only consider real, but also complex potentials q in some small complex neighbourhood W of L 20 within L 20,C = { q ∈ L 2C : [q] = 0 }. For q ∈ L 20,C , denote by y1 (x, λ, q) and y2 (x, λ, q) the usual fundamental solution of the Schr¨odinger equation −y 00 + qy = λy, and let 1(λ, q) = y1 (1, λ, q) + y20 (1, λ, q) be its associated discriminant. The spectrum of the operator −d 2 /dx 2 + q with periodic boundary conditions on the interval [0, 2] is the zero set of the entire function 12 (λ, q)−4. It is a sequence of complex numbers, which we order lexicographically as in (6.7) so that λ0 ≺ λ1 ≺ · · · ≺ λ2n−1 ≺ λ2n ≺ . . . . For real q, the periodic eigenvalues λn are real, and this ordering amounts to the usual ordering of the periodic eigenvalues, in which case they are continuous functions of q. For non-real q, however, λn is not a continuous function of q, since eigenvalues may change their position in the ordering discontinuously as indicated in figure 4.

64

III Birkhoff Coordinates λ2n

λ2n−1

λ2n

λ2n−1

λ2n

λ2n−1 Figure 4 Labeling of periodic eigenvalues as q varies

Restricting ourselves to a sufficiently small complex neighbourhood W of L 20 , however, we can always ensure that the closed intervals G n = { (1 − t)λ2n−1 + tλ2n : 0 ≤ t ≤ 1 },

n ≥ 1,

as well as G 0 = { (1 − e−t ) + et λ0 : − ∞ < t ≤ 0 } are disjoint from each other. Due to the asymptotic behavior of the periodic eigenvalues, they admit mutually disjoint neighbourhoods Un ⊂ C, called isolating neighbourhoods. Moreover, inside each Un we may choose a circuit 0n around G n with counterclockwise orientation as illustrated in figure 5. Both Un and 0n may be chosen to be locally independent of q. Following Flaschka & McLaughlin [43] we now define actions for the KdV equation by Z ˙ 1(λ) 1 λp dλ, n ≥ 1, In = π an 12 (λ) − 4 p where an is the lift of the circuit 0n to that sheet of 6(q) on which 12 (λ) − 4 is given by the c-root. Hence, this definition is equivalent to Z ˙ 1 1(λ) In = λp dλ, n ≥ 1. c π 0n 12 (λ) − 4 By Cauchy’s theorem, In does not depend on the choice of 0n as long as it stays inside Un . In the following we may need to shrink the complex neighbourhood W of L 20 within L 20,C several times, but we will denote it by the same symbol throughout. Theorem 7.1. There exists a complex neighbourhood W of L 20 such that for each n ≥ 1, the function In is analytic on W with L 2 -gradient Z 1 ∂1(λ) p ∂ In = − dλ. (7.1) π 0n c 12 (λ) − 4 This L 2 -gradient has mean value zero. Moreover, on the real subspace L 20 , each function In is real, nonnegative, and vanishes if and only if γn vanishes.

7 Actions

01 G0

G1

λ0

λ1

U0

U1

65

02 λ2

λ3

G2 λ4

U2

Figure 5 Isolating neighbourhoods

Proof. Choose the complex neighbourhood W of L 20 as above. Locally on W , the contours of integration 0n can p be chosen to be independent of q. As 1 is an analytic function of λ and q, and c 12 (λ) − 4 is analytic in a neighbourhood of 0n , the function In is clearly analytic in q. Since In is invariant under translations of q, the gradient of In has mean value zero in view of (6.11). To obtain its representation we observe that on the interval [λ2n−1 , λ2n ], we have (−1)n 1(λ) ≥ 2 and hence p (−1)n 1(λ) ± 12 (λ) − 4 > 0 for real q. Therefore, on a sufficiently small complex neighbourhood Wn of L 20 and a circuit 0n sufficiently close to [λ2n−1 , λ2n ], the principle branch of the logarithm   p c φ(λ) = log (−1)n 1(λ) − 12 (λ) − 4 is well defined along 0n . Since p c ˙ ˙ 12 (λ) − 4, φ(λ) = −1(λ)/ partial integration gives In =

1 π

Z 0n

  p c log (−1)n 1(λ) − 12 (λ) − 4 dλ.

(7.2)

Again keeping 0n fixed and taking the gradient with respect to q we obtain the above formula for ∂ In on Wn . As both sides of (7.1) are analytic on W , this identity holds everywhere on W . To prove the remaining assertions we observe that in view of the existence of the primitive φ(λ), Z ˙ 1(λ) p dλ = 0. c 0n 12 (λ) − 4

66

III Birkhoff Coordinates

˙ near λ2n−1 and λ2n , we can therefore also write With λ˙ n denoting the root of 1 Z ˙ 1 1(λ) In = (λ − λ˙ n ) p dλ. c π 0n 12 (λ) − 4 For real q we then obtain 2 In = π

Z

λ2n

˙ 1(λ) (−1)n−1 (λ − λ˙ n ) +p dλ λ2n−1 12 (λ) − 4

by shrinking the contour of integration to the real interval and taking into account ˙ the definition of the c-root. Since sign (λ − λ˙ n )1(λ) = (−1)n−1 on [λ2n−1 , λ2n ], the integrand is non-negative, and the claimed results follow. u t Next we show that each action is not only a continuous, but even a compact function on L 20 in the following sense: if q converges weakly to p, then In (q) converges to In ( p) for each n. Lemma 7.2. Each function In , n ≥ 1, is compact on L 20 . Proof. The periodic eigenvalues λm are compact functions on L 20 – the proof is the same as for the Dirichlet eigenvalues in [112, p. 32]. The same holds for the dis˙ considered as maps from q to functions on the criminant 1 and its λ-derivative 1, complex plane with the topology of uniform convergence on bounded subsets of C [112, p. 18]. If q converges weakly to p in L 20 , then eventually we may choose the contour 0n to be independent of q, and conclude that Z ˙ 1 1(λ, q) lim In (q) = lim λp dλ c q* p q* p π 0 2 1 (λ, q) − 4 n Z ˙ 1(λ, q) 1 = λ lim p dλ π 0n q* p c 12 (λ, q) − 4 Z ˙ 1(λ, p) 1 λp = dλ π 0n c 12 (λ, p) − 4 = In ( p). as required. u t Let  Dn = q ∈ W : γn (q) = 0 be the subvariety of potentials with collapsed n-th gap in the complex neighbourhood W of L 20 of Theorem 7.1. This subvariety is analytic, as it may also be written as  ˙ n (q)) = 0 . Dn = q ∈ W : 1(τn (q)) = 2(−1)n , 1(τ As In and γn2 are analytic on W , their quotient is analytic on W X Dn . We show that this quotient extends analytically to all of W to a nonvanishing function.

7 Actions

67

Theorem 7.3. There exists a complex neighbourhood W of L 20 such that the quotient In /γn2 extends analytically to W for all n ≥ 1 and satisfies 8πn

  In log n = 1 + O n γn2

locally uniformly on W . Moreover, s    8In 1 log n + ξn = = 1 + O √ n γn2 πn is well defined as√ a real analytic, nonvanishing function on W . In particular, at q = 0, we have ξn = 1/ πn for all n ≥ 1. Remark. The theorem also implies that the squared gap lengths are real analytic functions on L 20 . Proof. Let W be the complex neighbourhood of L 20 of the preceding theorem. We show that In /γn2 extends continuously from W XDn to all of W and is weakly analytic when restricted to Dn . By Theorem A.6 we may then conclude that In /γn2 is analytic on all of W . Recall the product expansions 12 (λ) − 4 = 4(λ0 − λ)

Y (λ2n − λ)(λ2n−1 − λ) , n4 π4

n≥1

˙ 1(λ) =−

Y λ˙ n − λ , n2 π2

n≥1

˙ Their asymptotics are where λ˙ n are the roots of 1. λ˙ n = τn + O(γn2 log n/n) by Proposition B.13, where τn = (λ2n + λ2n−1 )/2. Along the circuit 0n we then can write ˙ 1(λ) 1 λ − λ˙ n p p = χn (λ) c 2πn s (λ2n − λ)(λ − λ2n−1 ) 12 (λ) − 4 with Y πn λ˙ m − λ p χn (λ) = (−1)n−1 +√ , λ − λ0 m6=n + (λ2m − λ)(λ2m−1 − λ) where the principle branches in the last expression are well defined for λ near G n . To check the correctness of the sign in these expansions note that χn is nonnegative for real q and refer to the definitions (6.8) and (6.10) of the s-root and the c-root.

68

III Birkhoff Coordinates

With the formula for In in the preceding proof we then get, for q in W X Dn , ˙ 1(λ) (λ − λ˙ n ) p dλ c 0n 12 (λ) − 4 Z (λ − λ˙ n )2 1 p χn (λ) dλ = 2 s 2π n 0n (λ2n − λ)(λ − λ2n−1 ) Z γ2 dζ = n2 (ζ − δn )2 χn (τn + ζ γn /2) p s 8π n 0n0 1 − ζ2

In =

1 π

Z

upon the substitution λ = τn + ζ γn /2 with δn = 2(λ˙ n − τn )/γn , where 0n0 is some circuit in C around [−1, 1]. Thus, Z dζ In 1 8πn 2 = (ζ − δn )2 χn (τn + ζ γn /2) p s π 0n0 γn 1 − ζ2 on W X Dn . The right hand side of the last identity is continuous on all of W including Dn , since when γn tends to 0, also δn tends to zero by Lemma B.13, so it tends to 1 π

ζ 2 dζ 2 χn (τn ) p = χn (τn ) s π 0n0 1 − ζ2

Z

1

t 2 dt = χn (τn ). √ −1 1 − t 2

Z

But χn and τn are analytic on Dn and even on W . Thus, we may apply Theorem A.6 and conclude that In /γn2 extends analytically to all of W . By Lemma L.2, χn (λ) = 1 + O(log n/n) for λ near G n locally uniformly on W . Together with the last two displayed identities and in view of the asymptotics of δn we thus conclude that   In log n 8πn 2 = 1 + O n γn locally uniformly on W . For q = 0 we have γn = 0, δn = 0, and χn (τn ) = 1 for all n ≥ 1. Hence, for the analytic extension of In /γn2 we actually have 8πn

In = 1, γn2

n ≥ 1,

at q = 0. Finally, on the real subspace L 20 , we have locally uniformly as n → ∞ In 2 0 < 8πn 2 = π γn

Z

1

(t − δn )2 χn (τn + tγn /2) √

−1

→ lim χn (τn ) = 1. n→∞

dt 1 − t2

8 Angles

69

Therefore, by choosing the complex neighbourhood W sufficiently small we can assure that the real part of n In /γn2 is positive and locally uniformly bounded away from zero for all n ≥ 1. Consequently, s 8In + ξn = γn2 is well defined, real analytic, and positive for real q in W for all n ≥ 1. u t

8 Angles Next we define angular coordinates θn for potentials q in W . More precisely, the n-th angle θn is defined for those q whose n-th gap is not collapsed. The starting point is the following theorem, which we will prove in appendix D and is due to [6] in this form. For an earlier version see [90]. Theorem 8.1. There exists a complex neighbourhood W of L 20 such that for each q in W there exist entire functions ψn , n ≥ 1, satisfying Z ψn (λ) 1 p dλ = δmn (8.1) c 2π 0m 12 (λ) − 4 for all m ≥ 1. These functions depend analytically on λ and q and admit a product representation 2 Y σmn − λ ψn (λ) = , πn m 2 π2 m6=n

whose complex roots σmn depend real analytically on q and satisfy σmn = τm + O



γm2 m



locally uniformly on W and uniformly in n, where τm = (λ2m + λ2m−1 )/2. We now turn to the definition of the angles formally derived in section 6. Recall that Dn = { q ∈ W : γn (q) = 0 }. For q ∈ W X Dn define X θn (q) = ηn (q) + βkn (q), k6 =n

where ηn (q) =

Z

µ∗n

λ2n−1

ψn (λ) p dλ mod 2π 12 (λ) − 4

70

III Birkhoff Coordinates

and βkn (q)

Z

µ∗k

= λ2k−1

ψn (λ) p dλ, 12 (λ) − 4

and where µ∗n

is given by (6.3). As we will show, the paths of integration are arbitrary on the Riemann surface 6(q) as long as their projections onto C stay inside an isolating neighbourhood of the corresponding interval G n as described above on page 64. We call such paths admissible. Note that the function ηn is considered as a function on W X Dn taking values in the cylinder C/2πZ rather than C, whereas the βkn are considered as functions taking values in C. We begin by showing that these functions are well defined in the sense that they are independent of the path of integration. Indeed, the βkn are well defined on all of W . Lemma 8.2. (i) For any k 6 = n, the function βkn is well defined on all of W and vanishes at q = 0. (ii) For n ≥ 1, the function ηn is well defined on W X Dn . Proof. Consider βkn for k 6 = n. By the product expansions for 12 (λ) − 4 and ψn (λ), we have, for λ near G k , σkn − λ ψn (λ) p =p ζkn (λ) (λ2k − λ)(λ − λ2k−1 ) 12 (λ) − 4 with ζkn (λ) =

(8.2)

Y πn σmn − λ p , √ (σnn − λ) + λ − λ0 m6=k + (λ2m − λ)(λ2m−1 − λ)

p p where we set σnn = τn , and where 12 (λ) − 4 and (λ2k − λ)(λ − λ2k−1 ) are understood as functions on a neighbourhood around the lift of [λ2k−1 , λ2k ] onto the Riemann surface 6(q). If γk 6 = 0, then the factor σkn − λ p (λ2k − λ)(λ − λ2k−1 ) is integrable along any admissible path. If γk = 0, then λ2k = σkn = λ2k−1 , and this factor equals ±i. So the integrand of βkn is an analytic function on each sheet of 6(q) around [λ2k−1 , λ2k ]. Consequently, the function βkn is well defined in both cases. The integral is independent of any admissible path of integration, since by (8.1), Z λ2k ψn (λ) p dλ = 0, k 6= n. (8.3) λ2k−1 12 (λ) − 4 It vanishes when µk = λ2k−1 , so in particular when q = 0. This proves part (i). As to ηn the integral exists along an admissible path as long as λ2n−1 6 = λ2n . It is well defined modulo 2π by (8.1) for k = n. This proves part (ii). u t

8 Angles

71

Next we prove analyticity. Lemma 8.3. (i) The functions βkn , k 6= n, are real analytic on W . (ii) The functions ηn , n ≥ 1, are real analytic on W X Dn , if taken modulo π. Remark. The values of ηn have to be taken modulo π due to the discontinuities of the periodic eigenvalues as functions of q when q is not real, which in turn is due to their lexicographic ordering. Along a curve in W X Dn , the eigenvalues λ2n and λ2n−1 may jump discontinuously as indicated in figure 4, causing ηn to jump by ±π. This, of course, does not happen in the real space L 20 X Dn . There, ηn is real analytic when taken modulo 2π. Proof. In W consider the two subsets Dk = { q ∈ W : γk (q) = 0 }, E k = { q ∈ W : µk (q) ∈ { λ2k (q), λ2k−1 (q) } }. Like the former, the latter is an analytic subvariety of W , since it may also be written in the form  E k = q ∈ W : 1(µk (q)) = 2(−1)k . Our plan is to prove that βkn is analytic on W X (Dk ∪ E k ) and continuous on W , and that its restrictions to Dk and E k are each weakly analytic there. In view of Theorem A.6 it then follows that βkn is analytic on W . Outside of Dk , λ2k and λ2k−1 are simple eigenvalues, and locally there exist − + − analytic functions λ+ k and λk such that the sets { λk , λk } and { λ2k , λ2k−1 } are equal. − In view of (8.3) and the substitution λ = λk + z we then can write βkn

Z

µ∗k

= λ− k

ψn (λ)

p dλ = 12 (λ) − 4

where D(z) =

µ∗k −λ− k

Z 0

ψn (λ− + z) dz, √ √k z D(z)

12 (λ− k + z) − 4 z

is analytic near z = 0 with D(0) 6 = 0. If we integrate along an √admissible path not − going through λ+ , then D(z) does not vanish and ψ (λ + z)/ D(z) is smooth and n k k locally analytic on W X (Dk ∪ E k ). As µ∗k and λ− are analytic on this set as well, we k conclude that the latter integral representing βkn is analytic on W X (Dk ∪ E k ). n Next we show that βk is weakly analytic when restricted to either Dk or E k . In n view of (8.3), βk E k = 0, so on E k this is obvious. On Dk , λ2k = λ2k−1 = τk = σkn , and with (8.2) we can write Z µ∗ Z µk k ψn (λ) p βkn = dλ = ±i ζkn (λ) dλ. 2 τk τk 1 (λ) − 4

72

III Birkhoff Coordinates

As µk is analytic, this integral is an analytic function on Dk . Altogether we conclude that these restrictions of βkn are weakly analytic on Dk and on E k . It remains to prove that βkn is continuous on all of W . Clearly, βkn is continuous on W X (Dk ∪ E k ). One shows that it is continuous on E k X Dk and Dk X E k . The continuity on Dk ∩ E k follows from (8.2) and the estimate σkn − τk = O(γk2 /k). By Theorem A.6, βkn is then analytic on W . Obviously, it is real valued on L 20 . The proof for ηn is analogous and even simpler, since we only need to consider the domain W X Dn . In view of (8.1) we have Z λ2n ψn (λ) p dλ = ±π (8.4) λ2n−1 12 (λ) − 4 for the straight line integral, so as above we can write Z µ∗n ψn (λ) p ηn = dλ mod π. − λn 12 (λ) − 4 We conclude that modulo π, the function ηn is analytic outside of Dn ∪ E n , and continuous outside Dn . Since ηn mod π vanishes on E n , it is weakly analytic on E n , and we are done. u t Lemma 8.4. βkn = O



|γk | + |µk − τk | k |n − k|



uniformly in k and n with k 6 = n on bounded subsets of W . Proof. By (8.3), βkn =

Z

µ∗k

λ2k−1

ψn (λ) p dλ = 12 (λ) − 4

Z

µ∗k λ2k

ψn (λ) p dλ. 12 (λ) − 4

The following argument is not affected if one interchanges the roles of λ2k−1 and λ2k . Therefore we may assume in the following that |µk − λ2k−1 | ≤ |µk − λ2k |. For λ near G k we have σkn − λ ψn (λ) p =p ζkn (λ) (λ2k − λ)(λ − λ2k−1 ) 12 (λ) − 4 by equation (8.2). The infinite product in the representation of the function ζkn is uniformly bounded in view of the asymptotic behavior of the σmn and Lemma L.2, so we immediately obtain, uniformly in k and n with k 6 = n,     n 1 n ζ (λ) = O = O k k |n − k| k n 2 − k 2

8 Angles

73

for λ near G k . Moreover, if we integrate along a straight line ` from λ2k−1 to µk on the sheet of 6(q) determined by µ∗k , then we have s

σkn − λ = O(1), λ2k − λ

since |µk − λ2k−1 | ≤ |µk − λ2k | and σkn = τk + O(γk2 ). Thus, it remains to show that Z µ∗ s k λ − σkn dλ = O(|γk | + |µk − τk |), λ2k−1 λ − λ2k−1 when integrating along the straight line `. But this follows with the simple substitution λ = λ2k−1 + t (µk − λ2k−1 ). Setting ε = |σkn − λ2k−1 | and δ = |µk − λ2k−1 | we obtain the bound Z 0

1

√ √ √ ε+δ √ √ δ dt = 2 ε + δ δ ≤ ε + 2δ. δ t

As ε = O(|γk |) and δ = O(|γk | + |µk − τk |), the claim follows. u t P It is immediate by the preceding estimates that the sum k6=n βkn converges locally uniformly on W to a real analytic function on W , which is locally uniformly of order 1/n. Altogether we obtain the following result. Theorem 8.5. The function βn =

X

βkn

k6=n

is real analytic on W and locally uniformly of order O(1/n). The angle function θn = ηn +

X

βkn

k6 =n

is real analytic on the real space L 20 X Dn and extends to a real analytic function on W X Dn when taken modulo π. The gradients of βkn , ηn and θn all have mean value zero. Proof. It remains to prove the statement about the mean value. Inspecting the formula for βkn one verifies that βkn , ηn and hence θn are invariant under translations of the potential when considered on L 2 X Dn . Hence, their gradients have mean value zero by (6.11). u t

74

III Birkhoff Coordinates

9 Cartesian Coordinates In the preceding two sections we defined actions and angles, In =

1 2 2 ξ γ , 8 n n

θn = ηn +

X

βkn ,

k6=n

for potentials q in L 20 and L 20 X Dn , respectively. We showed that these are real analytic on a complex neighbourhood W of L 20 and on W X Dn , respectively. We did not show yet, however, that they are coordinates, nor did we show that they are canonical variables. This will be done in sections 11 and 12. Here we introduce and study the associated rectangular variables xn and yn . As usual, on the real space L 20 X Dn they are defined as xn =

p 2In cos θn ,

yn =

p 2In sin θn ,

where the choice of sin and cos is made so that dxn ∧d yn = d In ∧dθn . This definition extends to the complex domain W X Dn by setting xn =

ξn γn cos θn , 2

yn =

ξn γn sin θn . 2

The main result of this section is that these functions are in fact well defined and real analytic on all of W . This holds even in those points, where the angles θn are not well defined. To attack this problem, recall that the functions ξn and βn = θn − ηn have already been shown to be real analytic on W . Therefore we focus our attention on the complex functions z n± = γn e±iηn , which so far are defined on W X Dn . We first show that z n± are analytic on W X Dn . This is not entirely obvious, since γn has discontinuities. Lemma 9.1. The functions z n± = γn e±iηn are analytic on W X Dn . Proof. Locally around every point in W X Dn there exist analytic functions λ+n and λ−n such that as sets, { λ−n (q), λ+n (q) } = { λ2n−1 (q), λ2n (q) }. Let γn = λn − λn , −

+

−

ηn = −

Z

µ∗n

λ− n

ψn (λ) p dλ. 12 (λ) − 4

Depending on whether λ−n = λ2n−1 or λ−n = λ2n , respectively, we then have γn = γn− ,

ηn = ηn− ,

9 Cartesian Coordinates

75

or γn = −γn− ,

ηn =

Z

µ∗n

λ+ n

ψn (λ) p dλ = ηn− + π mod 2π 2 1 (λ) − 4

in view of (8.4). In either case we obtain −

γn e±iηn = γn− e±iηn . The right hand side is analytic, which proves the lemma. u t In analogy with the representation (8.2) we write ψn (λ) ζn (λ) p =p 2 (λ − λ)(λ − λ2n−1 ) 1 (λ) − 4 2n

(9.1)

for λ near G n , where Y πn σmn − λ p ζn (λ) = (−1)n−1 +√ , + λ − λ0 m6=n (λ2m − λ)(λ2m−1 − λ) and the two roots in (9.1) are understood as functions on a neighbourhood of the lift of [λ2n−1 , λ2n ] onto 6(q) related to each other by this identity. The principal branches in the last identity are well defined for λ near G n . Lemma 9.2. For µ ∈ G n , ζn (µ) = 1 + O



|γn | n



locally uniformly on W and uniformly in n ≥ 1. Proof. First let q be real with γn > 0. In view of Theorem 8.1 and the definition of the c-root in (6.10) we have Z λ2n ψn (λ) p (−1)n−1 dλ = π + λ2n−1 12 (λ) − 4 for the straight line integral from λ2n−1 to λ2n . On this line, both (−1)n−1 ψn and ζn are positive. With (9.1) we thus obtain, for every µ ∈ G n , Z λ2n ζn (λ) p π= dλ + (λ2n − λ)(λ − λ2n−1 ) λ2n−1 Z λ2n ζn (µ) p = dλ + (λ2n − λ)(λ − λ2n−1 ) λ2n−1 Z λ2n ζn (λ) − ζn (µ) p + dλ + (λ2n − λ)(λ − λ2n−1 ) λ2n−1
= πζn (µ) + O supλ∈G n |ζn (λ) − ζn (µ)| .

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III Birkhoff Coordinates

Hence,
ζn (µ) = 1 + O supλ∈G n |ζn (λ) − ζn (µ)| for µ ∈ G n . In view of Proposition L.2, ζn is uniformly bounded on a neighbourhood of radius O(n) around G n . This bound holds uniformly in n and locally uniformly on W , no matter if γn 6= 0 or not. Therefore, by Cauchy’s estimate, |ζn (λ) − ζn (µ)| ≤

M M |λ − µ| ≤ |γn | n n

for λ, µ ∈ G n with a uniform constant M. This proves the claim for real q. For complex q in W the preceding identities remain true at least up to a sign. Hence, by the continuity of ζn in λ and q, the claimed statement must also be valid for complex q . u t We now investigate the limiting behavior of z n± as the n-th gap collapses. This limit is different from zero when the limit potential is in the open set X n = { q ∈ W : µn (q) ∈ / G n (q) }. This set does not intersect the real space L 20 , since µn ∈ [λ2n−1 , λ2n ] for real q. Let Z µn ζn (λ) − ζn (τn ) dλ. χn (q) = λ − τn τn The integral exists due to the analyticity of ζn in λ and is analytic in q. To facilitate the statement of the following result we define a sign εn = ±1 for potentials q in X n in such a way that ψn (µn ) εn ζn (µn ) p = p . ∗ s 2 (λ2n − µn )(µn − λ2n−1 ) 1 (µn ) − 4

(9.2)

Note that the sign of the s-root is well defined, since µn ∈ / G n for q ∈ X n . Lemma 9.3. As q ∈ / Dn tends to p ∈ Dn ∩ X n , γn e±iηn → −2(1 ± εn )(µn − τn )e±εn χn , where εn is defined in (9.2). Proof. As X n is open and p ∈ X n ∩ Dn , we have q ∈ X n for all q sufficiently near p. Also, q ∈ / Dn by assumption. For q ∈ X n X Dn we can write, modulo 2π, Z µ∗n ψn (λ) p ηn = dλ λ2n−1 12 (λ) − 4 Z µn ζn (λ) p = εn dλ s (λ − λ)(λ − λ2n−1 ) λ2n−1 2n

9 Cartesian Coordinates

= εn

77

µn

ζn (λ2n−1 ) p dλ s (λ2n − λ)(λ − λ2n−1 ) λ2n−1 Z µn ζn (λ) − ζn (λ2n−1 ) p + εn dλ s (λ λ2n−1 2n − λ)(λ − λ2n−1 ) Z

= ηn0 + ηn00

mod 2π.

The limiting behavior of the second term ηn00 is straightforward. If q → p, then γn → 0 and so for λ 6 = τn , p s (λ2n − λ)(λ − λ2n−1 ) → i(λ − τn ) by (6.9). Hence, by the definition of χn , Z µn ζn (λ) − ζn (τn ) dλ = εn χn ( p) mod 2π. iηn00 → εn λ − τn τn Consequently, eiηn → eεn χn ( p) 00

as q → p.

Now consider ηn0 . By the substitution λ = τn + zγn /2 and the definition (6.8), Z ρn Z µn dλ dz p = = φ(ρn ) √ s s (λ2n − λ)(λ − λ2n−1 ) 1 − z2 λ2n−1 −1 with

Z w dz µn − τn , φ(w) = . √ s γn /2 1 − z2 −1 √ s It follows that eiφ(w) = −w + i 1 − w 2 , as both sides are analytic, univalent functions on CX[−1, 1], which have the same limit at −1 and satisfy the same differential equation i f 0 (w) = √ . s f (w) 1 − w2 ρn =

Hence, writing exp (iηn0 ) = exp (iεn φ(ρn )ζn (λ2n−1 )) = exp (iφ(ρn )εn ) exp (iεn φ(ρn )ζˆn ) with ζˆn = ζn (λ2n−1 ) − 1 we obtain for q ∈ X n X Dn p
εn iε φ(ρ )ζˆ 0 γn eiηn = γn −ρn + i s 1 − ρn2 e n n n. Now let q → p. Then γn → 0, while vn = µn − τn tends to a limit different from zero, and hence |ρn | → ∞. For the first factor on the right hand side of the

78

III Birkhoff Coordinates

above equation we then obtain by (6.6) p p
εn
γn −ρn + i s 1 − ρn2 = γn −ρn + iεn s 1 − ρn2 p
= γn −ρn − εn ρn + 1 − ρn−2 p = −2vn − 2vn εn + 1 − ρn−2 → −2vn (1 + εn ). As to the second factor we observe that for |ρn | large, Z ρn dz |φ(ρn )| = √ s 1 − z2 −1 Z |ρn | dt ≤c+ √ √ t − 1 t +1 1 p  = O |ρn | .

(9.3)

Since ζˆn = O(|γn |/n) by Lemma 9.2 we thus conclude that φ(ρn )ζˆn → 0 and so eiεn φ(ρn )ζn → 1 ˆ

as q → p.

00

Together with the result for eiηn we thus arrive at γn eiηn → −2vn (1 + εn )eεn χn ( p) as claimed. The limit of γn e−iηn is a simple variation of the preceding argument.

t u

In view of the preceding result it is natural to extend the functions z n± to Dn by defining ( −2(1 ± εn )(µn − τn )e±εn χn on Dn ∩ X n , ± (9.4) zn = 0 on Dn X X n . The main result is then the following. Theorem 9.4. The functions z n± = γn e±iηn , as extended above, are analytic on W . Moreover, z n± = O(|γn | + |µn − τn |) locally uniformly on W . Proof. We apply Theorem A.6 to the function z n± on the domain W with the analytic subvariety Dn . By Lemma 9.1 these functions are analytic on W X Dn . By a simple inspection of the formula for z n± they are also continuous in every point of Dn ∩ X n and of Dn X X n . Thus, the functions z n± are continuous on all of W .

9 Cartesian Coordinates

79

It remains to show that they are weakly analytic when restricted to Dn . Let D be a one-dimensional complex disc contained in Dn . If the center of D is in X n , then the entire disc D is in X n , if chosen sufficiently small. The analyticity of z n± = γn e±iηn on D is then evident from formula (9.4), the definition of χn , and the local constancy of εn on X n . If the center of D does not belong to X n , then consider the analytic function µn − τn on D. This function either vanishes identically on D, in which case z n± vanishes identically, too. Or it vanishes in only finitely many points. Outside these points, D is in X n , hence z n± is analytic there. By continuity and analytic continuation, these functions are analytic on all of D. We thus have shown that z n± are analytic on D. The result now follows with Theorem A.6. To prove the claimed estimate we continue with the notations of the previous proof and recall that on X n X Dn , γn eiηn = γn −ρn + iεn

p
00 ˆ s 1 − ρn2 eiεn φ(ρn )ζn eiηn

with ρn = 2(µn − τn )/γn and ηn00 = εn

Z

µn

λ2n−1

ζn (λ) − ζn (λ2n−1 ) p dλ mod 2π. s (λ2n − λ)(λ − λ2n−1 )

In view of the analyticity and local uniform boundedness of ζn by Lemma 9.2 we 00 have eiηn = O(1) locally uniformly. Similarly, φ(ρn )ζˆn = O(1) locally uniformly in view of (9.3). Finally, for |ρn | ≤ 1 we have p
γn −ρn ± i s 1 − ρn2 ≤ 2 |γn |, while for |ρn | > 1, using (6.8) and γn ρn = 2(µn − τn ), p p
γn −ρn ± i s 1 − ρn2 = 2 |µn − τn | 1 ± + 1 − ρn−2 ≤ 6 |µn − τn |. Altogether, on X n X Dn , γn −ρn + iεn

p
s 1 − ρn2 ≤ 6(|γn | + |µn − τn |).

This establishes the estimate for z n = γn eiηn on X n XDn . By continuity, it extends in a locally uniform fashion to all of W . The argument for γn e−iηn is completely analogous. u t Before returning to the definition of xn and yn , we determine the gradients of z n± at points in L 20 ∩ Dn , which exist by analyticity. For the definition of the normalized Dirichlet eigenfunctions gn , see appendix B.

80

III Birkhoff Coordinates

Theorem 9.5. At a point in L 20 ∩ Dn , ∂z n± = h 2n − gn2 ± 2ign h n , where gn denotes the normalized eigenfunction for the Dirichlet eigenvalue µn and h n that periodic eigenfunction orthonormal to gn which is positive at 0. These gradients are in H 2 , have mean value zero and satisfy the asymptotic estimates   1 ± ∂z n = 2 cos 2πnx ± 2i sin 2πnx + O . n These hold locally uniformly on L 20 in the sense that at each point q only those n with γn (q) = 0 are taken into account. At q = 0, ∂z n± = 2 cos 2πnx ± 2i sin 2πnx for all n ≥ 1. Proof. Consider z n+ . To compute its gradient at p ∈ L 20 ∩ Dn , we approximate p by potentials q in p  Bn = q ∈ L 20 X Dn : µn = τn and sign ∗ 12 (µn ) − 4 = (−1)n−1 . This set is not empty by Theorem B.14. For such q we have ψn (µn ) ζn (µn ) p = +p , ∗ (λ2n − µn )(µn − λ2n−1 ) 12 (µn ) − 4 as sign ψn (µn ) = (−1)n−1 and sign ζn (µn ) = 1. Going through the calculations in the proof of Lemma 9.3 with the latter expression in place of (9.2) we then have Z µn ζn (λ) p ηn = dλ mod 2π. + (λ − λ)(λ − λ2n−1 ) λ2n−1 2n Further on, all arguments remain valid for λ ∈ G n , if we set εn = 1 and replace 0 00 all s-roots by positive roots. So as before we can write z n+ = γn eiηn = γn eiηn · eiηn , where both factors on the right hand side are real analytic on W . It follows that   0 ∂z n+ = lim ∂ γn eiηn , Bn 3q→ p

0

00

since γn eiηn = 0 for p ∈ L 20 ∩ Dn by the preceding theorem and eiηn = 1. Moreover, with vn = µn − τn = ρn γn /2 and ζˆn = ζn (λ2n−1 ) − 1, p
ζn (λ2n−1 ) 0 γn eiηn = γn −ρn + i + 1 − ρn2 p p

ζˆn = −2vn + iγn + 1 − ρn2 · −ρn + i + 1 − ρn2 .

9 Cartesian Coordinates

81

The gradients of both factors have a limit as q → p and the product rule can be applied. For q ∈ Bn , we have µn = τn and hence vn = 0 as well as ρn = 0. Thus, in the limit the first factor vanishes, while the second converges to 1 by Lemma 9.2. As a consequence the identity 0
∂z n+ = lim ∂ γn eiηn Bn 3q→ p p
= lim ∂ −2vn + iγn + 1 − ρn2 Bn 3q→ p

= 2(∂τn − ∂µn ) + i = 2(∂τn − ∂µn ) + i

lim

∂γn

lim

2 2 f 2n − f 2n−1

Bn 3q→ p Bn 3q→ p

holds. 2 − f2 By what we have just proven, lim Bn 3q→ p f 2n 2n−1 exists. By the next lemma below, f 2n and f 2n−1 converge individually to solutions of the corresponding differential equation and thus to periodic eigenfunctions of p, again denoted f 2n and f 2n−1 . They are orthonormal and satisfy h f 2n ,gn i > 0 > h f 2n−1 ,gn i. For the limiting eigenfunctions we therefore must have f 2n = αh n + βgn , f 2n−1 = βh n − αgn ,

(9.5)

with α, β > 0 and α 2 + β 2 = 1, and gn and h n as stated in the theorem. Moreover, inserting this representation into equation (9.6), we obtain β/α = α/β, or α = β. Thus we have 2 2 − f 2n−1 = 2gn h n , f 2n 2 2 2∂τn = f 2n + f 2n−1 = gn2 + h 2n ,

while 2∂µn = 2gn2 as usual. Altogether, ∂z n+ = h 2n − gn2 + 2ign h n . Clearly, this gradient has mean value zero√and is in H 2 . With the asymptotic formulas gn √ = 2 sin πnx + O(1/n) and, by orthogonality and chosen normalization, h n = 2 cos πnx + O(1/n), the stated asymptotic formula for ∂z n+ then follows with standard trigonometric identities. The formula for ∂z n− is obtained analogously. Moreover, at q = 0 these formulas hold without the error terms. u t Lemma 9.6. As q ∈ Bn tends to p ∈ L 20 ∩ Dn , the periodic eigenfunctions f 2n and f 2n−1 of q do converge to normalized eigenfunctions of p, denoted by the same symbols, such that in the limit h f 2n ,gn i > 0 > h f 2n−1 ,gn i and h f 2n ,gn i f 2n (0) =− . h f 2n−1 ,gn i f 2n−1 (0)

(9.6)

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III Birkhoff Coordinates

Proof. As q tends to p, the initial values of the normalized eigenfunctions f 2n and f 2n−1 remain bounded in R2 , since each eigenfunction is represented in terms of its initial data and the fundamental solution y1 , y2 of −y 00 + qy = λy, and the angle between y1 and y2 as well as their L 2 -norms are locally uniformly bounded away from zero. Hence we can always choose a convergent subsequence of these initial values. As the fundamental solution depends analytically on λ and q, the corresponding eigenfunctions then converge uniformly to some eigenfunctions of p, which we temporarily denote by f¯2n and f¯2n−1 . We show that these limiting eigenfunctions are indeed uniquely determined. This will establish the convergence of the entire sequence. 2 − f2 We observed that lim Bn 3q→ p f 2n 2n−1 exists. By the analyticity of τn , also 2 2 2 and f 2 lim Bn 3q→ p f 2n + f 2n−1 exists. Hence, also the individual limits of f 2n 2n−1 exist, which fixes f¯2n and f¯2n−1 uniquely up to sign. This sign is also fixed, once we show that f¯2n (0) 6 = 0, since f¯2n (0) ≥ 0 by normalization. The same argument applies to f¯2n−1 (0). To establish that f¯2n (0) does not vanish we observe that (λ2n − µn ) h f 2n ,gn i = hλ2n f 2n ,gn i − h f 2n ,µn gn i 1 = f 2n g 0 n

0
= f 2n (0) (−1)n gn0 (1) − gn0 (0)

by the differential equation and partial integration. As we will show in a moment, lim

Bn 3q→ p

(−1)n gn0 (1) − gn0 (0) = κn > 0, λ2n − µn

(9.7)

that is, the limit exists and is positive. Hence, in the limit we have

 f¯2n ,gn = κn f¯2n (0).

But then f¯2n (0) can not be zero, since otherwise, f¯2n would be a nontrivial multiple of gn , and the left hand side could not be zero. By an analogous calculation,
(λ2n−1 − µn ) h f 2n−1 ,gn i = f 2n−1 (0) (−1)n gn0 (1) − gn0 (0) . Taking the quotient with the similar equation for f 2n and observing that by construction λ2n − µn = µn − λ2n−1 for q ∈ Bn , we also obtain (9.6). It remains to prove (9.7), or equivalently, lim

Bn 3q→ p

(−1)n y20 (1, µn ) − 1 > 0. λ2n − µn

9 Cartesian Coordinates

Use

p ∗

83

12 (µn ) − 4 = y1 (1, µn ) − y20 (1, µn ) to get for q ∈ Bn q ∗ 12 (µn ) − 4 q + = 1(µn ) + (−1)n 12 (µn ) − 4.

2y20 (1, µn ) = 1(µn ) −

As 2 = (−1)n 1(λ2n ), it thus remains to prove that   q 1 + n 2 2 lim (−1) (1(µn ) − 1(λ2n )) + 1 (µn ) − 1 (λ2n ) > 0. Bn 3q→ p λ2n − µn Concerning the first term in the sum, 1(µn ) − 1(λ2n ) ˙ 2n ) = 0, → 1(λ µn − λ2n as we are at a double eigenvalue. Concerning the second term in the sum, write 12 (µn ) − 12 (λ2n ) = −2

Z

λ2n

µn Z λ2n

= −2 µn

˙ 1(λ)1(λ) dλ 1(λ)

Z

λ

λ˙ n

¨ 1(µ) dµ dλ,


˙ in the n-th gap. With λ˙ n = τn + o γn2 it follows where λ˙ n is the unique root of 1 that 12 (µn ) − 12 (λ2n ) ¨ 2n ) > 0, → −1(λ2n )1(λ (µn − λ2n )2 which proves our claim. u t Let us now define the cartesian coordinates p eiθn + e−iθn , 2In cos θn = ξn γn 4 p eiθn − e−iθn yn = 2In sin θn = ξn γn . 4i

xn =

Expressed in terms of z n± this is equivalent with
ξn + iβn z n e + z n− e−iβn , 4
ξn + iβn yn = z n e − z n− e−iβn . 4i

xn =

The latter expressions apply to all complex potentials in W , while the former expressions apply to potentials in W X Dn only.

84

III Birkhoff Coordinates

Theorem 9.7. The functions xn and yn are real analytic on W with gradients having mean value zero. At a point in L 20 ∩ Dn their gradients are     2  ξn cos βn − sin βn ∂ xn h n − gn2 = . ∂ yn 2h n gn 2 sin βn cos βn They are in H 2 and satisfy the asymptotic estimates       1 log n ∂ xn cos 2πnx =√ +O ∂ yn n 3/2 πn sin 2πnx locally uniformly on L 20 in the sense that at each point q only those n with γn (q) = 0 are taken into account. At q = 0 the above formulas hold without the error terms for all n ≥ 1. Proof. Analyticity of xn and yn follows from Theorems 7.3 and 8.5 and what we have shown above. On L 20 X Dn their gradients are given in terms of ∂ In and ∂θn , hence have mean value zero by Theorems 7.1 and again 8.5. As to their gradients on L 20 ∩ Dn , we have xn + iyn =

ξn + iβn z e . 2 n

Since z n+ vanishes on L 20 ∩ Dn , ∂ xn + i∂ yn =

ξn iβn + e ∂z n . 2

Inserting ∂z n+ = h 2n − gn2 + 2ign h n and decomposing into the real and imaginary part we obtain the representations for ∂ xn and ∂ yn above. The asymptotics follow with Theorem 9.5 and     log n 1 1 +O ξn = √ , β = O (9.8) n n n 3/2 πn from Theorems 7.3 and 8.5. At q = 0 we have 1 1 ∂z n+ = √ ∂ xn + i∂ yn = √ (cos 2πnx + i sin 2πnx) πn 4πn for all n ≥ 1 by Theorems 7.3 and 9.5 and Lemma 8.2.

t u

Summarizing our construction so far we may associate with every potential q in L 20 infinitely many coordinates (q) = (x(q), y(q)), where x(q) = (x1 (q), x2 (q), . . . ), y(q) = (y1 (q), y2 (q), . . . ).

10 Orthogonality Relations

85

In view of the asymptotic estimates of Theorem 9.4 and equation (9.8),   |γn | + |µn − τn | |xn | + |yn | = O √ n locally uniformly on W . Thus,  maps L 20 into the space h1/2 = `21/2 × `21/2 . Theorem 9.8. The map  : L 20 → h1/2 q 7 → (q) = (x(q), y(q)) is real analytic and extends analytically to all of W . Its Jacobian at q = 0 is boundedly invertible, its inverse is the weighted Fourier transform d0 −1 : h1/2 → L 20 X√
(x, y) 7→ 2πn xn en + yn e−n , n≥1

√ √ where en = 2 cos 2πnx and e−n = 2 sin 2πnx for n ≥ 1. Proof. The map  is analytic as a map from L 20 into h1/2 by Theorem A.5 in view of the analyticity of its components and its local boundedness. Its Jacobian at 0 is given by
d0 (h) = (d0 xn (h))n≥1 , (d0 yn (h))n≥1 , √ with d0 xn (h) = h∂ xn ,hi = hen ,hi/ 2πn by Theorem 9.7, and similarly for d0 yn . This proves the theorem. u t Thus,  is a local diffeomorphism at q = 0. To show that  is a global diffeomorphism, we need to establish some orthogonality relations. 10 Orthogonality Relations In this section we establish orthogonality relations among the gradients of In , θn , xn and yn . They will enable us to show that  is a global diffeomorphism and indeed a symplectomorphism. These orthogonality relations are nothing but the Poisson brackets between In , θn , xn and yn . To avoid issues of regularity and anti-symmetry, however, we introduce the notation [8,9] := h∂8,∂x ∂9i, where we assume 8 and 9 to be differentiable functions on L 20 such that ∂8 ∈ L 2 ,

∂9 ∈ H 1 .

We refer to [8,9] as the bracket of 8 and 9. To start, we consider products of solutions of −y 00 + qy = λy. In this case, regularity is not an issue.

86

III Birkhoff Coordinates

Lemma 10.1. Let F and G be linear combinations of products of solutions of −y 00 + qy = λy for λ = α and λ = β, respectively, with coefficients depending only on q. If q ∈ H 1 and F G|10 = 0, then
1 4(α − β) hF ,G 0 i = F 00 G − F 0 G 0 + F G 00 0 . Remark. We assume q to be in H 1 in order that the boundary values of F 00 and G 00 make sense. Proof. Consider the operator L = − 12 D 3 + q D + Dq, where D = ∂x . One verifies that for q ∈ H 1 , the product Y = yi y j of any two functions of the fundamental solution satisfies the differential equation LY = 2λDY. It follows by partial integration that 2α hD F ,Gi + 2β hF , DGi = hL F ,Gi + hF ,LGi
1 = − 12 F 00 G − F 0 G 0 + F G 00 − 4q F G 0 . By another partial integration, 2α hD F ,Gi + 2β hF , DGi 1 = 2(α − β) hD F ,Gi + 2β F G 0

1 = (α − β) (hD F ,Gi − hF , DGi) + (α + β)F G 0

by symmetrization. Combining the two identities,
2(α − β) hF ,G 0 i − hF 0 ,Gi =
1 F 00 G − F 0 G 0 + F G 00 + 2(α + β − 2q)F G , 0

from which the result follows.

t u

We first apply this lemma to the discriminant 1. For each λ consider def

1λ = 1(λ, · ) as a function on L 20 . By Proposition B.3, its gradient is ∂1λ = m 2 y12 + (m 02 − m 1 )y1 y2 − m 01 y22 ,

(10.1)

where all functions are evaluated at λ and q, and where the convenient notation     m1 m2 y1 y2 (λ, q) = (1, λ, q) m 01 m 02 y10 y20

10 Orthogonality Relations

87

for the entries of the Floquet matrix for −y 00 + qy = λy at x = 1 is used. Hence, ∂1λ is a linear combination of y12 , y1 y2 and y22 and belongs to H 2 . Moreover, by Proposition B.3 one also has the representation ∂1λ = y2 (1, λ, Tt q),

Tt q = q( · + t).

(10.2)

This shows that ∂1λ is 1-periodic, and that y2 (1, λ, Tt q) is represented by products of the fundamental solution. Lemma 10.2. (i) For all complex α and β, [1α ,1β ] = 0. (ii) More generally, if 8 is a differentiable function on L 20 , which at every point q depends only on the periodic spectrum of q, then [8,1β ] = 0 for all complex β. Proof. (i) In view of the representation of ∂1λ in equation (10.1) the bracket [1α ,1β ] is continuous on L 20 . It therefore suffices to prove the claim for smooth q. But then ∂1λ is smoothly 1-periodic in t by (10.2), and so all boundary terms in Lemma 10.1 cancel each other. Consequently, [1α ,1β ] = 0 for α 6 = β. The case α = β is trivial. (ii) Fix β, and consider X = ∂x ∂1β as a vector field on L 20 . As def

L X 1α = h∂1α , X i = [1α ,1β ] = 0 for all α, we conclude that the function 1 and hence the periodic spectrum is constant along the flow lines of X . But then also 8 is constant along these flow lines, as 8 is assumed to depend only on the periodic spectrum. Therefore, also the Lie-derivative of 8 with respect to X vanishes, so L X 8 = [8,1β ] = 0 for all β. u t Lemma 10.3. On L 20 , 1 m 2 (λ) [µn ,1λ ] = 2 m˙ 2 (µn )

p ∗ 12 (µn ) − 4 λ − µn

for all n ≥ 1 and all λ 6 = µn . Proof. By continuity, it suffices to prove the lemma for q in H 1 . By Proposition B.7, we have m 1 (µn ) 2 ∂µn = y ( · , µn ). m˙ 2 (µn ) 2

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III Birkhoff Coordinates

Thus, with equation (10.2), [µn ,1λ ] =

 m 1 (µn ) 2 y2 ( · , µn ),∂x y2 (1, λ, Tx q) . m˙ 2 (µn )

Applying Lemma 10.1 with F = y22 ( · , µn ) and observing that all boundary terms of F, F 0 and F 00 vanish except the contribution from (y20 )2 we obtain
1 m 1 (µn ) 0 y2 (x, µn )2 y2 (1, λ, Tx q) 0 m˙ 2 (µn )
m 1 (µn ) 0 = m (µn )2 m 2 (λ) − m 2 (λ) m˙ 2 (µn ) 2
m 2 (λ) m 02 (µn ) − m 1 (µn ) , = m˙ 2 (µn )

2(µn − λ) [µn ,1λ ] =

since m 1 (µn )m 02 (µn ) = p 1 by the Wronskian identity. The claim now follows with m 1 (µn ) − m 02 (µn ) = ∗ 12 (µn ) − 4 by (6.2). u t Lemma 10.4. 2 [θn ,1λ ] = ψn (λ) for all n ≥ 1 and all λ on L 20 X Dn .   Proof. We first consider the bracket βkn ,1λ for k 6 = n under the assumption that λ2k−1 < µk < λ2k . Recall that βkn =

Z

µ∗k

λ2k−1

ψn (λ) p dλ. 12 (λ) − 4

The lower bound of the integral and the integrand are spectral invariants, hence invariant under the flow of the vector field X = ∂x ∂1λ . Thus we have 

 ψn (µk ) βkn ,1λ = L X βkn = p L X µk ∗ 12 (µk ) − 4 ψn (µk ) = p [µk ,1λ ] ∗ 12 (µk ) − 4 1 m 2 (λ) ψn (µk ) = 2 m˙ 2 (µk ) λ − µk

by the previous lemma. This holds indeed on all of L 20 , since the first and the last expression are continuous on L 20 as long as λ 6= µk , and since λ2k−1 < µk < λ2k holds on an open and dense subset of L 20 . The same expression holds for [ηn ,1λ ] on L 20 X Dn , since the formula for ηn is analogous to the formula for βkn modulo 2π, and additive constants vanish upon differentiation.

10 Orthogonality Relations

89

P Now consider θn = ηn + k6=n βkn . By Lemma 8.4, this series converges locally uniformly. By Cauchy’s theorem, the same holds true for its gradient. Thus we obtain X  2 [θn ,1λ ] = 2 [ηn ,1λ ] + 2 βkn ,1λ k6=n

X m 2 (λ) ψn (µk ) = ψn (λ) = m˙ 2 (µk ) λ − µk k≥1

by Proposition D.11. u t Proposition 10.5. [In ,Im ] = 0,

[θn ,Im ] = −δnm

on L 20 and L 20 X Dn , respectively. Moreover, [xn ,Im ] = δnm yn ,

[yn ,Im ] = −δnm xn ,

on L 20 . Finally, [8,Im ] = 0 for every differentiable function 8 on L 20 , which at every point q depends only on the periodic spectrum of q. Proof. By Theorem 7.1 and the representation (10.1) of ∂1λ , Z ∂1λ 1 p ∂ Im = − dλ π 0m c 12 (λ) − 4 is in H 2 . The first identity as well as [8,Im ] = 0 then follow with Lemma 10.2 by integration. The second identity follows from Z 1 [θn ,1λ ] p dλ [θn ,Im ] = − π 0m c 12 (λ) − 4 Z ψn (λ) 1 p dλ = −δnm , =− c 2π 0m 12 (λ) − 4 using Lemma 10.4 for the second for the last identity. Finally, on √ identity and (8.1) √ L 20 X Dn the definitions xn = 2In cos θn and yn = 2In sin θn give [xn ,Im ] = √

p 1 cos θn [In ,Im ] − 2In sin θn [θn ,Im ] = yn δnm , 2In

and similarly [yn ,Im ] = −xn δnm . Since both sides are continuous on L 20 , this holds on all of L 20 . u t To determine the brackets among xn , xm , yn , ym at points in Dn ∩ Dm , we first consider the functions z n± . Recall that ∂z n± = h 2n − gn2 ± 2ign h n on L 20 ∩ Dn by Theorem 9.5, where gn denotes the normalized eigenfunction for the Dirichlet eigenvalue µn , and h n that periodic eigenfunction orthonormal to gn which is positive at 0. In particular, the gradients ∂z n± are in H 2 .

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III Birkhoff Coordinates

Lemma 10.6. On L 20 ∩ Dn , 

 z n+ ,z n− = −4i gn0 h n 0 6= 0,

± while on L 20 ∩ Dn ∩ Dm with n 6 = m, all brackets among z n± and z m vanish.

Proof. By Theorem 9.5,  ± ± 2  2 z n ,z m = h n − gn2 ± 2ign h n ,(h 2m − gm ± 2igm h m )0 . As both sides are continuous in q, it suffices to consider q in H 1 . For n 6= m, the right hand side vanishes by Lemma 10.1, since each product is periodic. For n = m, we obtain 

 1  + −

z n ,z n = gn h n ,(h 2n − gn2 )0 − h 2n − gn2 ,(gn h n )0 2i Z 1
= gn h 2n h 0n − gn2 gn0 h n − gn0 h 3n + gn3 h 0n dx 0

1

Z

W (gn , h n )(gn2 + h 2n ) dx = 2W (gn , h n ) = −2gn0 h n , =

0

0

where W (gn , h n ) = denotes the constant Wronskian of gn and h n . The function h n does not vanish at 0, since it is linearly independent of gn . u t gn h 0n

− gn0 h n

This lemma immediately carries over to a corresponding result for the functions xn and yn . Proposition 10.7. On L 20 ∩ Dn , [xn ,yn ] =

ξn2 0 g h n 6= 0, 2 n 0

while on L 20 ∩ Dn ∩ Dm with n 6 = m, all brackets among xn , xm , yn , ym vanish. Proof. Recall from the preceding section that
ξn + iβn z n e + z n− e−iβn , 4
ξn + iβn yn = z n e − z n− e−iβn . 4i

xn =

Hence, ∂ xn and ∂ yn are in H 2 on Dn . As z n± vanishes on Dn , the preceding lemma gives
ξ2 ξ2 [xn ,yn ] = n [z n− ,z n+ ] − [z n+ ,z n− ] = n gn0 h n . 0 16i 2 The other identities follow analogously. u t Eventually we will see that [xn ,yn ] = 1. But the far simpler result stated in Proposition 10.7 is sufficient to show in the next section that  is a global diffeomorphism.

11 The Diffeomorphism Property

91

11 The Diffeomorphism Property We are going to show that  : L 20 → h1/2 is a global diffeomorphism onto h1/2 . This is done in two steps. First we show that  is a local diffeomorphism not only at 0 by Theorem 9.8, but at every point in L 20 . Then we show that  is globally one-to-one and onto. To establish  as a local diffeomorphism it is more convenient to consider the map 8 = d0 −1 B  : L 20 → L 20 X√
q 7→ 2πn xn (q)en + yn (q)e−n n≥1

with en =

√ 2 cos 2πnx,

√ e−n =

2 sin 2πnx

(11.1)

for n ≥ 1. Clearly,  is a local diffeomorphism if and only if 8 is a local diffeomorphism. Consider the Jacobian of 8 at q, dq 8 : L 20 → L 20 h 7 → dq 8(h) =

X

hh,dn (q)ien ,

n6 =0

where for n ≥ 1, dn (q) =

√

2πn ∂ xn ,

d−n (q) =

√

2πn ∂ yn .

(11.2)

To verify that dq 8 is a linear isomorphism we make use of the following general result. First two definitions. A system of vectors (vn ) in a Hilbert space is said to be linearly independent, if none of the vectors vn is contained in the closed linear span of all the other vectors vm , m 6 = n. A system of vectors (vn ) is a basis for a Hilbert space H , if there exists a Hilbert space h of sequences α = (αn ) such that the correspondence X α 7→ αn vn n≥1

is a linear isomorphism between h and H . Proposition 11.1. Let (en )n∈N , N ⊂ Z, be an orthonormal basis of a Hilbert space H . Suppose (dn )n∈N is another sequence of vectors in H that (i) either spans or is linearly independent and (ii) satisfies X kdn − en k2 < ∞. n∈N

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III Birkhoff Coordinates

Then (dn )n∈N is also a basis of H , and the map X hh,dn ien h 7→ n∈N

is a linear isomorphism of H . Moreover, if the vectors dn , n ∈ N , depend continuously on a parameter such that (i) holds for every parameter, (ii) holds on a dense subset of parameters, and the above map depends continuously in the operator norm on the parameter, then the conclusion holds for every parameter. Proof. Define an operator A on H by Ah =

X

hh,en idn .

n∈N

By item (ii) A is a well defined, bounded operator, which maps en into dn . Since X X k(A − I )en k2 = kdn − en k2 < ∞, n∈N

n∈N

A − I is Hilbert-Schmidt, hence A is a compact perturbation of the identity. If the system (dn )n∈N is linearly independent, then A is one-to-one, for if Ah = 0, then hh,en i = 0 for all n ∈ N and hence h = 0. If, on the other hand, (dn )n∈N spans, then the range of A is dense in H . But the range of a compact perturbation of the identity is closed, so A is onto. So in either case, A is boundedly invertible by the Fredholm alternative. Hence (dn )n∈N is also a basis. Next, we note that hh,dn i = hh, Aen i = hA∗ h,en i. Since also A∗ is boundedly invertible, the map h 7→ (hA∗ h,en i)n∈N = (hh,dn i)n∈N is a linear isomorphism between H and `2 . Consequently, the map X hh,dn ien h 7→ n∈N

is a linear isomorphism of H . Finally, suppose the system (dn )n∈N depends on a parameter as stipulated. Then A − I is compact for a dense set of parameters. By assumption, A∗ is continuous in the parameter, hence A − I is compact for all parameters, and the argument applies as before. u t We apply the preceding proposition to the system of vectors (dn )n6=0 defined by (11.2) in the Hilbert space L 20 and compare them to the orthonormal basis (en )n6=0 defined by (11.1). The point q is considered as a parameter on which the dn depend continuously.

11 The Diffeomorphism Property

93

Lemma 11.2. For q in the dense subset of finite gap potentials in L 20 , X kdn − en k2 < ∞. n6=0

Proof. This is an immediate consequence of Theorem 9.7. For the density of finite gap potentials in L 20 , see Theorem B.16. u t We go on to show that the system of vectors (dn )n6=0 is linearly independent at every point q in L 20 . We fix q, and let A(q) = { n ∈ N : γn (q) > 0 }. The argument is given in two steps. Lemma 11.3. At every point q in L 20 , the vectors (dn )n6=0 are linearly independent in L 20 , if the vectors (∂ In )n∈A(q) are linearly independent in L 20 . Proof. Fix q, and let A = A(q). Suppose (αn )n6=0 are real numbers such that X f = αn dn = 0. n6 =0

For m ∈ A, Proposition 10.5 gives √ 0 = h f ,∂x ∂ Im i = 2πm (αm [xm ,Im ] + α−m [ym ,Im ]) √ = 2πm (αm ym − α−m xm ). It follows that the 2-vector (αm , α−m ) is parallel to the 2-vector (xm , ym ). Hence 2 = α 2 + α 2 , and (αm , α−m ) = am (cos θm , sin θm ) with am m −m 1 √ (αm dm + α−m d−m ) = αm ∂ xm + α−m ∂ ym 2πm = am (cos θm ∂ xm + sin θm ∂ ym ) am =√ ∂ Im 2Im by inserting the definitions of xm and ym . Consider now m ∈ / A, so that γm = 0. In view of the above representation and Propositions 10.7 and 10.5 we find that √ 0 = h f ,∂x ∂ ym i = 2πm αm [xm ,ym ], √ 0 = h f ,∂x ∂ xm i = 2πm α−m [ym ,xm ]. Hence α±m = 0 for m ∈ / A. We thus conclude that X X an 0= αn d n = √ ∂ In . 2In n∈A n6 =0 2 and hence If the vectors (∂ In )n∈A are linearly independent, then the an = αn2 + α−n also the α±n must vanish, since both are real. This proves the claim. u t

94

III Birkhoff Coordinates

Lemma 11.4. At every point q in L 20 the vectors (∂ In )n∈A(q) are linearly independent in L 20 . Proof. Fix q, and let A = A(q). Suppose X

αn ∂ In = 0

n∈A

with some real coefficients αn . We have to show that they are all zero. This would be straightforward if we were allowed to take inner products with ∂x ∂θm and make use of the orthogonality relations of Proposition 10.5. However, we do not know yet that P the gradient of θm is in H 1 , nor can we assume that n∈A αn ∂ In converges in H 1 . Therefore, we can not apply Proposition 10.5 right away. Instead we redo part of the argument that led to these orthogonality relations. Define the functions  ψm (µk )   ∂µk for λ2k−1 < µk < λ2k ,  − p ∗ 12 (µk ) − 4 m hk =  ψm (µk )   y1 y2 µ for µk ∈ { λ2k−1 , λ2k }.  k ˙ k) 1(µ In the case λ2k−1 < µk < λ2k , one obtains with Lemma 10.3

 ψm (µk ) ∂1λ ,∂x h m [µk ,1λ ] k = p ∗ 12 (µk ) − 4 1 m 2 (λ) ψm (µk ) = . 2 m˙ 2 (µk ) λ − µk

In the case µk ∈ { λ2k−1 , λ2k }, we have with equation (10.2)

  ψm (µk )

∂1λ ,∂x h m y1 y2 |µk , ∂x y2 (1, λ, Tx q) . k =− ˙ 1(µk ) To evaluate the right hand side, we first assume q ∈ H 1 and apply Lemma 10.1 with F = y1 y2 |µk and G = y2 (1, λ, Tx q). Since y2 = 0 and, by the Wronskian identity, y1 y20 = 1 at the boundary of [0, 1], and since G is periodic, we have 1 F 0 G 0 0 = 0,

F G 00 |10 = 0.

Moreover, since y200 = 0 at the boundary by the differential equation, 1 1 F 00 G 0 = 2y10 y20 G 0 = 2y10 (1, µk )y20 (1, µk )y2 (1, λ) = 2m 01 (µk )m 02 (µk )m 2 (λ),

11 The Diffeomorphism Property

95

where for the second equality we used that y10 (0) = 0. Hence we obtain 1

 ψm (µk ) 1 ∂1λ ,∂x h m F 00 G 0 k = ˙ 1(µk ) 4(λ − µk ) 1 ψm (µk ) m 01 (µk )m 02 (µk ) = m 2 (λ) ˙ k) 2 λ − µk 1(µ for q ∈ H 1 by Lemma 10.1, and then for general q in L 20 by continuity. Differentiating the Wronskian identity m 1 m 02 − m 01 m 2 = 1 with respect to λ and using that m 2 = 0 and m 1 = m 02 = (−1)k when µk is a periodic eigenvalue, one finds that ˙ k ) = 1(µ ˙ k )/m 02 (µk ), m 01 (µk )m˙ 2 (µk ) = m 02 (µk )1(µ or

m 01 (µk )m 02 (µk ) 1 . = ˙ m˙ 2 (µk ) 1(µk )

Together with the preceding identity we then arrive at the same result as in the case λ2k−1 < µk < λ2k . By Theorem 7.1 we then get Z

 m 2 (λ) ψm (µk ) dλ 1 p ∂ In ,∂x h m = . k 2π 0n m˙ 2 (µk ) λ − µk c 12 (λ) − 4 By standard asymptotic estimates,   ψm (µk ) m =O , m˙ 2 (µk ) |m 2 − k 2 |

k 6 = m.

We may therefore take the sum over all k to obtain Z X X

 1 m 2 (λ) ψm (µk )  dλ p ∂ In ,∂x h m = k 2π 0n m˙ 2 (µk ) λ − µk c 12 (λ) − 4 k≥1 k≥1 Z 1 ψm (λ) p dλ = 2π 0n c 12 (λ) − 4 = δnm , using Proposition D.11 in the second line and PTheorem 8.1 in the third line. We can now prove the lemma. Suppose n∈A αn ∂ In = 0. Taking the inner product with ∂x h m k and summing over all k we obtain X X

 0= αn ∂ In ,∂x h m k n∈A

=

X n∈A

Thus, all coefficients αn vanish. u t

k≥1

αn δnm = αm .

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III Birkhoff Coordinates

As the last three lemmas hold and the Jacobian dq 8 depends continuously on q, the vectors (dn )n6=0 satisfy the assumptions of Proposition 11.1. Thus, dq 8 is a linear isomorphism at every point q in L 20 , and we conclude the following intermediate result. Proposition 11.5. The map  : L 20 → h1/2 is a local diffeomorphism everywhere. We now show that this map is one-to-one and onto. This is a topological argument and boils down to verifying that the map  is proper. That is, the preimage of compact sets is compact. Lemma 11.6. The map  : L 20 → h1/2 is proper. Proof. It suffices to show that for any sequence (qm ) in L 20 with the property that (qm ) converges strongly in h1/2 there exists a subsequence that converges strongly in L 20 . So let (qm ) be such a sequence. If (qm ) converges in h1/2 , then also the norms k(qm )kh1/2 converge. By Proposition E.1 we know that X
k(qm )k2h1/2 = k xk2 (qm ) + yk2 (qm ) k≥1

=

X

2k Ik (qm ) =

k≥1

1 kqm k2L 2 . 2π

(11.3)

Hence, (qm ) is bounded in L 20 and thus admits a weakly convergent subsequence again denoted by (qm ). Let q be the weak limit of (qm ). By Lemma 7.2, each Ik is a compact function on L 20 , hence Ik (qm ) → Ik (q) for each k ≥ 1. With (11.3) we then conclude that also kqm k L 2 → kqk L 2 . This together with the weak convergence of qm to q implies that (qm ) converges to q strongly in L 20 . u t Proposition 11.7. The map  : L 20 → h1/2 is globally one-to-one and onto. Proof. Consider the set  M = z ∈ h1/2 : # −1 (z) = 1 . M is open and closed, since  is a local diffeomorphism everywhere and proper. M is not empty, since   −1 (0) = q ∈ L 20 : γn (q) = 0 for all n = 0 ∈ L 20 , q ≡ 0 being the only real valued potential with (q) = 0 by Proposition E.1. It follows that M = h1/2 , so  is globally one-to-one and onto. u t

11 The Diffeomorphism Property

97

Thus we have established the following result. Theorem 11.8. The map  : L 20 → h1/2 is a global real analytic diffeomorphism. The map  also respects certain subsets of L 20 in a natural way. Recall that  G A = q ∈ L 20 : γk (q) > 0 iff k ∈ A denotes the set of A-gap potentials in L 20 , and that  h A = (x, y) ∈ h0 : xk2 + yk2 > 0 iff k ∈ A . Theorem 11.9. For each A ⊂ N, G A = −1 (h A ). In particular, G A is a real analytic submanifold of L 20 . Proof. This is an immediate consequence of Theorem 7.3, which implies that for any n ≥ 1, we have γn = 0 iff In = 0 iff xn2 + yn2 = 0. u t Remark. It follows that also the union of all A-gap potentials with A = { 1, . . . , M }, the so called M-gap potentials, is dense in L 20 . In addition,  maps each isospectral set Iso(q) ⊂ L 20 onto a corresponding torus  Tor(I) = (x, y) : xn2 + yn2 = 2In for n ≥ 1 ⊂ h1/2 , where I = I(q) = (In (q))n≥1 . Theorem 11.10. The diffeomorphism  maps each isospectral set Iso(q) in L 20 oneto-one onto the torus Tor(I(q)) in h1/2 . Proof. Fix q and I = I(q). As any p ∈ Iso(q) has the same periodic spectrum as q, it also has the same 1-function as q in view of its product representation. The actions In are defined entirely in terms of this 1-function. Consequently, I( p) = I and ( p) ∈ Tor(I). Thus, (Iso(q)) ⊂ Tor(I). Conversely, take any point w = (x, y) in Tor(I). Set A = { n : In 6= 0 }, and for any n ∈ A consider the curve wn in Tor(I), which moves around the n-th circle { xn2 + yn2 = 2In } with unit speed, while all other coordinates stay put. Its preimage under  is a smooth curve q n in L 20 with nonvanishing velocity vector q˙ n everywhere. This velocity vector is perpendicular to ∂ Ik for k ≥ 1, to ∂θk for n 6= k ∈ A, and to ∂ xk , ∂ yk for k ∈ / A, since all these coordinates are constant along this curve. By comparison with Proposition 10.5 it follows that in every point, q˙ n = α∂x ∂ In

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III Birkhoff Coordinates

with some real α. Again with Proposition 10.5 this implies that d 1 B q n = L q˙ n 1 = α [1,In ] = 0. dt That is, the 1-function is invariant along the curve q n . Consequently, all potentials along this curve have the same periodic spectrum, and so image q n ⊂ Iso(q). Since this holds for each n ∈ A, we conclude by induction that −1 (Tor(I)) ⊂ Iso(q). This completes the proof. u t Finally, Theorem 11.8 is sharpened to the extent that  respects higher regularity. This was first proven in [6] with a quite different approach. Denote by  N the restriction of  to H0N . In particular, 0 = . Theorem 11.11. For each N ≥ 0, the map  N is a global, real analytic diffeomorphism between H0N and h N +1/2 . Remark. Thus, the union of all M-gap potentials is also dense in H0N for any N ≥ 0. Proof. For N = 0, this is the content of the preceding theorem. So fix N ≥ 1. We already noticed in section 9 that   |γn | + |µn − τn | |xn | + |yn | = O √ n locally uniformly on W . Moreover, there exists a complex neighbourhood W N of H0N in H0,NC such that X


n 2N |γn |2 + |µn − τn |2 = O(1)

n≥1

locally uniformly by Proposition B.9. Hence,  N maps W N ∩ W into h N +1/2,C and is locally bounded. Since each coordinate function is real analytic, the entire map  N : H0N → h N +1/2 is real analytic by Theorem A.5. This map is clearly one-to-one, since  is one-to-one. To show that  N is onto, let (x, y) ∈ h N +1/2 . Then q = −1 (x, y) is well defined in L 20 . As q is real valued, we may apply Theorem 7.3 to obtain for all large n that |xn |2 + |yn |2 = 2In ≥

1 γn2 . 8π n

11 The Diffeomorphism Property

99

Hence, X

n 2N γn2 < ∞,

n≥1

and consequently q ∈ H0N , as q is real [84]. This shows that  N is onto. It remains to show that  N is a local diffeomorphism everywhere. As before, this is done by considering the map N N 8 N = d0 −1 N B  N : H0 → H0 , −1 where d0 −1 N = (d0  ) h N +1/2 is the restriction of the inverse discrete Fourier transform d0 −1 to h N +1/2 . Clearly, 8 N = 8|H0N , and

dq 8 N : H0N → H0N ,

h 7→

X

hh,dn (q)ien

n6 =0

is the restriction of dq 8 to H0N , where the dn are given by (11.2). Thus, dq 8 N is one-to-one. Its deviation from the identity is the map Aq : h 7 → Aq h =

X

hh,dn (q) − en ien ,

n6=0

and it suffices to show that Aq is compact at any finite gap potential q, which are dense in H0N – see appendix B. By continuity, Aq is then compact everywhere, and dq 8 N = I + Aq is a linear isomorphism by the Fredholm alternative. To show that Aq is a compact operator on H0N , we consider the N -th derivative of Aq h, X ∂xN Aq h = (2π|n|) N hh,dn (q) − en ie˜n , n6=0

where e˜n is either ±en or ±e−n with the choice of sign depending on N . In view of Lemma 11.12 below, X |n|2N |hh,dn (q) − en i|2 < ∞, sup n6=0

h∈H0N ,khkH N ≤1

when q is a finite gap potential. Hence, Aq is a bounded operator on H0N , and for each ε > 0 there exists an integer M such that

X

hh,dn (q) − en ien N ≤ ε

|n|≥M

H

uniformly for khkH N ≤ 1. From this, it follows that Aq maps the unit ball in H N into a relatively compact subset of H N . Hence, Aq is compact. u t

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III Birkhoff Coordinates

Lemma 11.12. At a finite gap potential, the estimate   log n |hh,dn − en i| = O N +3/2 khkH N n holds uniformly for h ∈ H0N for each N ≥ 0. Proof. The proof consists in verifying the statement for N = 0 and N = 1 and in proving an induction step. We start with the latter and assume that h ∈ H0N with N ≥ 2. Given q, consider n sufficiently large so that γn = 0. It follows from Theorem 9.5 that ∂ xn and ∂ yn are linear combinations of squares of eigenfunctions of q for the eigenvalue λ2n−1 = λ2n . Hence they are smooth functions of x, and as in the proof of Lemma 10.1 we have Ldn = 2λ2n Ddn , where L = − 12 D 3 + q D + Dq. As the mean values of ∂ xn and ∂ yn vanish on Dn by Theorem 9.7, we thus have dn =

1 D −1 Ldn , 2λ2n 0

where D0−1 denotes the inverse of the restriction of D to H01 . It follows that E 1 D −1 hdn ,hi = D0 Ldn ,h 2λ2n E 1 D dn ,L D0−1 h = 2λ2n  1

=− dn ,h ∗ 4λ2n with h ∗ = h 00 − 4qh − 2q 0 D0−1 h ∈ H N −2 . On the other hand, by partial integration, hen ,hi = −

 1

00 e ,h . n 4π 2 n 2

Hence we obtain hdn − en ,hi = −

 1

dn − en ,h ∗ 4λ2n  

 1 1 + − en ,h ∗ 2 2 4λ2n 4π n  1

+ en ,h 00 − h ∗ . 2 2 4π n

11 The Diffeomorphism Property

101

By the induction hypotheses,

 dn − en ,h ∗ = O



log n n N −1/2



∗

h

H N −2

,

and furthermore   1 1 1 − = O , λ 2 2 π n n4 2n  

∗  1 en ,h ∗ = O

h N −2 , H N −2 n  

 en ,h 00 − h ∗ = O 1 khk N , H nN ∗ N where the latter estimate holds as h 00 −
h is in H , and q is smooth as a finite gap ∗ potential. As kh kH N −2 = O khkH N , the claimed result follows. It remains to verify the statement for N = 0 and N = 1. The case N = 0 is an immediate consequence of Theorem 9.7. The case N = 1 is proved similarly to the induction step. Performing another integration by parts, one gets

hdn ,hi =

1 0 −1 ∗  d ,D h , 4λ2n n 0

hen ,hi =

1 0 0 e ,h . 4π 2 n 2 n

From here on, one proceeds as before to estimate hdn − en ,hi. Using asymptotic formulas for the derivatives of ∂ xn and ∂ yn , which are obtained from their representations in Theorem 9.7 and standard asymptotic estimates √ of the derivatives of the fundamental solution one sees that dn0 − en0 = O(log n/ n). This proves the claim for N = 1 as well. u t The last two theorems allow us to conclude that each Hamiltonian in the KdV hierarchy, and more generally in the KdV algebra, is a function of the actions In alone, when expressed in terms of the coordinates (x, y). Theorem 11.13. Each Hamiltonian in the KdV hierarchy, expressed in terms of the coordinates (x, y) in a sufficiently regular subspace of h 1/2 , is a function of the actions alone. Proof. Each KdV Hamiltonian H is defined on some subspace H0N ⊂ L 20 with N sufficiently large. Then H B −1 is defined on h N +1/2 by Theorem 11.11. Moreover, each KdV Hamiltonian H is a spectral invariant due to its representation in terms of the asymptotic expansion of the 1-function for λ → −∞. Hence, Hˆ = H B −1 is constant on each torus  Tor(I) = (x, y) : xn2 + yn2 = 2In for n ≥ 1 ⊂ h N +1/2 by Theorem 11.10. But this means that Hˆ is a function of I1 , I2 , . . . alone. u t It remains to show that  preserves the Poisson bracket in order to establish (x, y) as Birkhoff coordinates. This is done in the following section.

102

III Birkhoff Coordinates

12 The Symplectomorphism Property Finally we show that  : L 20 → h1/2 is not only a diffeomorphism between these two spaces, but that it also preserves the associated Poisson brackets. It turns out, however, that it is more convenient to work with symplectic structures rather than Poisson structures, since this way we can avoid the issue of establishing the regularity of the gradient ∂θn everywhere on L 20 X Dn . We therefore show equivalently that  is a symplectomorphism. The symplectic form on h1/2 induced by the standard Poisson bracket is X dxn ∧ d yn . ω0 = n≥1

The symplectic form on L 20 induced by the Gardner bracket is ω = h∂x−1 · , · i, as the associated Poisson structure ∂x is nondegenerate on H01 – see section 2 on page 22. Theorem 12.1. The diffeomorphism  : L 20 → h1/2 is symplectic with respect to the symplectic forms ω on L 20 and ω0 on h1/2 . The proof of this theorem requires that we establish the regularity and behavior of ∂θn at certain finite gap potentials. This is done in the following three statements. Lemma 12.2. If λ2k−1 = µk < λ2k , then for any n, ∂βkn =

ψn (µk ) g h , ˙ k) k k 1(µ

with βkk = ηk , where gk denotes the k-th normalized Dirichlet eigenfunction and h k the unique solution of −y 00 + qy = µk y orthogonal to gk with Wronskian W (gk , h k ) = 1. In particular, ∂βkn is in H 2 . Proof. As to the gradient, there is no difference between the βkn for k 6= n and ηk . So it suffices to consider the former. The gradient of βkn exists by analyticity. To compute it at a potential p with λ2k−1 = µk < λ2k , we use a trick p of McKean & Vaninsky [91] and approach p by isospectral potentials q with ∗ 12 (µk ) − 4 > 0 and λ2k−1 < µk < τk . As on page 71, we can write Z µ∗ Z µk −λ2k−1 k ψn (λ) ψn (λ2k−1 + z) p dλ = βkn = dz, √ √ z D(z) 0 λ2k−1 12 (λ) − 4 where D(z) = (12 (λ2k−1 + z) − 4)/z is analytic and bounded away from zero near z = 0 for q near p. Taking the gradient with respect to q and undoing the substitution, ψn (µk ) ∂βkn = p (∂µk − ∂λ2k−1 ) ∗ 12 (µk ) − 4   Z µk −λ2k−1 ∂ ψn (λ2k−1 + z) dz + √ √ . ∂q z D(z) 0

12 The Symplectomorphism Property

103

The gradient under the integral is bounded, so the integral vanishes for µk → λ2k−1 . Hence, at p, ψn (µk ) ∂βkn = lim p (∂µk − ∂λ2k−1 ). ∗ q→ p 12 (µk ) − 4 In particular, the latter limit exists. To compute this limit we follow [91]. Recall that ∂1 ∂m 2 ∂λ2k−1 = − . , ∂µk = − ˙ λ m˙ 2 µ 1 k

2k−1

To write the gradient of µk in a useful way, differentiate the Wronskian identity m 01 m 2 = m 1 m 02 − 1 with respect to λ and q and use m 2 = 0 at µk to obtain ˙ + (m 1 − m 02 )m˙ 02 , m 01 m˙ 2 = m˙ 1 m 02 + m 1 m˙ 02 = m 02 1

(12.1)

m 01 ∂m 2 = ∂m 1 m 02 + m 1 ∂m 02 = m 02 ∂1 + (m 1 − m 02 )∂m 02 . Substituting these expressions into ∂µk = −∂m 2 /m˙ 2 we then have ∂µk − ∂λ2k−1

∂1 = ˙ λ 1

2k−1

m 02 ∂1 + (m 1 − m 02 )∂m 02 − . ˙ + (m 1 − m 0 )m˙ 0 m0 1 2

2

2

µk

p Cross multiplying and dividing by ∗ 12 (µk ) − 4 = (m 1 − m 02 ) µ , which is of the √ k order of µk − λ2k−1 , we arrive in the limit at m˙ 0 ∂m 1 − m˙ 1 ∂m 02 ∂µk − ∂λ2k−1 lim p = 2 ∗ ˙ 2m0 q→ p 1 12 (µk ) − 4 2 1 m˙ 02 ∂m 1 − m˙ 1 ∂m 02 = , ˙ m 01 m˙ 2 1 ˙ = m 0 m˙ 2 at µk = λ2k−1 by (12.1). where we used that m 02 1 1 By standard identities, ∂m 1 = m 2 y12 − m 1 y1 y2 and ∂m 02 = m 02 y1 y2 − m 01 y22 , see appendix B. Thus,
m˙ 02 ∂m 1 − m˙ 1 ∂m 02 = m 01 −m˙ 2 y1 y2 + m˙ 1 y22 , again using (12.1). Hence altogether we obtain ∂βkn =

ψn (µk )  m˙ 1 2  ψn (µk ) −y1 y2 + y y y = ˙ k) ˙ k) 0 2 m˙ 2 2 1(µ 1(µ

with y0 = −y1 + (m˙ 1 /m˙ 2 )y2 . Since ∂βkn has mean value zero, y0 and y2 are orthogonal to each other, and the result follows by setting h k = ky2 k y0 . u t

104

III Birkhoff Coordinates

Recall from Theorem 11.9 that G A = { u ∈ L 20 : γk (u) > 0 iff k ∈ A } denotes the submanifold of A-gap potentials in L 20 , for any A ⊂ N. Within G A we single out a submanifold of “normalized” potentials, namely GoA = { u ∈ G A : θk (u) = 0 for k ∈ A }. Equivalently, these A-gap potentials are characterized by µk = λ2k−1 for k ∈ A. Lemma 12.3. At a finite gap potential in GoA the series ∂θn = ∂ηn +

X

∂βkn ,

n ∈ A,

k6 =n

converges in H 1 to the L 2 -gradient of the function θn . Proof. Fix n ∈ A. For k ∈ A, ∂βkn is in H 2 by the preceding lemma. For k ∈ / A approximate a given q ∈ GoA by finite gap potentials in GoA∪{ k } with λ2k−1 = µk . By the same lemma and l’Hospital’s rule, ∂βkn =

ψ˙ n (µk ) g h . ¨ k) k k 1(µ

˙ – see Proposition B.13 – give The product expansions for ψn and 1 ψ˙ n (µk ) = −

Y σ n − µk 2 1 m πn k 2 π 2 m 2 π2 m6 =k,n

1 Y λ˙ m − µk ¨ k) = 1(µ . 2 k π2 m 2 π2 m6 =k

As σmn = λ˙ m for m ∈ / A, we thus have   2πn Y σmn − µk 1 ψ˙ n (µk ) =− =O 2 ¨ k) k 1(µ λ˙ n − µk m∈A λ˙ m − µk m6 =n

for n fixed. Finally, gk h k is in H 2 for each k, and by the asymptotics for gk from appendix B, the orthogonality of h k to gk and their Wronskian being one, we have   √ 1 gk = 2 sin πkx + O , k   1 1 hk = −√ cos πkx + O 2 . k 2πk

12 The Symplectomorphism Property

105

Hence gk h k = −

  1 1 sin 2πkx + O 2 . 2πk k

A corresponding estimate holds for the first derivative. From this the final result follows. u t Proposition 12.4. At a finite gap potential in GoA , [θn ,θm ] = 0,

[In ,θm ] = δnm ,

for n, m ∈ A. Moreover, [xk ,θm ] = 0,

[yk ,θm ] = 0,

for k ∈ / A and m ∈ A. Proof. By the preceding lemma the series ∂θn = ∂ηn + Thus we have [θn ,θm ] = 0 if we show that  n m βk ,βl = 0

P

k6=n

∂βkn converges in H 1 .

for all n, m ∈ A and all k, l ≥ 1, with βnn = ηn . In view of the representation of the gradients of βkn and βlm in Lemma 12.2 this reduces to showing that

 gk h k ,(gl h l )0 = 0 for all k and l, since only the factors involving the ψ-function depend on n and m. But this follows by a straightforward computation. There is nothing to do for k = l, so we may assume that k 6= l. Denoting by W (g, h) = gh 0 − g 0 h the Wronskian between g and h, we have





 2 gk h k ,(gl h l )0 = gk h k ,(gl h l )0 − (gk h k )0 ,gl h l Z 1
= gk h k gl h l0 + gk h k gl0 h l − gk h 0k gl h l − gk0 h k gl h l dx 0

Z =

1


gk gl W (h k , h l ) + h k h l W (gk , gl ) dx.

0

Since W (h k , h l )0 = (µk − µl )h k h l , and similarly for W (gk , gl )0 , this equals 1 µk − µl

Z

1


W (gk , gl )0 W (h k , h l ) + W (gk , gl )W (h k , h l )0 dx

0

= This proves that [θn ,θm ] = 0.

1 1 W (gk , gl )W (h k , h l ) = 0. 0 µk − µl

106

III Birkhoff Coordinates

The second identity [In ,θm ] = δnm follows from Proposition 10.5 by partial integration, using that ∂θm is in H 1 . Finally, given k ∈ / A consider first a potential in GoA∪{k} . The standard definitions √ √ xk = 2Ik cos θk and yk = 2Ik sin θk give p 1 cos θm [Ik ,θm ] − 2Ik sin θm [θk ,θm ] = 0, [xk ,θm ] = √ 2Ik and similarly [yk ,θm ] = 0. By continuity, the same holds for potentials in GoA . u t We are now in a position to show that  : L 20 → h1/2 is symplectic with respect to ω on L 20 and ω0 on h1/2 . More precisely, we show that def

ω˜ = ∗ ω = ω0 on h1/2 . By continuity, it suffices to verify this for the restrictions of ω˜ and ω0 to a family of submanifolds filling h1/2 densely. We consider the family of spaces  g N = (x, y) ∈ h0 : xn2 + yn2 > 0 iff 1 ≤ n ≤ N . By Theorem 11.9, their preimages under  are the manifolds G N of N -gap potentials in L 20 . On g N we may use standard angle-action coordinates (I, θ ) = (I1 , . . . , I N , θ1 , . . . , θ N ), in which the restriction of ω0 is again a symplectic form, namely X ω0 |g N = d In ∧ dθn . 1≤n≤N

So we have to show that ω|g ˜ N = ω0 |g N for each large N . Henceforth, we drop the ‘|g N ’ from the notation. Consider the standard tangent vectors en =

∂ , ∂ In

fn =

∂ , ∂θn

1 ≤ n ≤ N,

which form a basis of the tangent space to g N at any point of g N and are supplemented to a basis of h1/2 by the vectors ∂/∂ xk and ∂/∂ yk for k ≥ N + 1. Let E n = ∗ en ,

Fn = ∗ f n ,

1 ≤ n ≤ N,

be their preimages under . Lemma 12.5. For 1 ≤ n ≤ N , E n = ∂x ∂θn , everywhere on GoN and on G N , respectively.

Fn = −∂x ∂ In

12 The Symplectomorphism Property

107

Proof. Fix 1 ≤ n ≤ N . Expressing ∗ Fn = f n in the coordinates In , θn and xk , yk one obtains h∂ Im ,Fn i = 0, h∂θm ,Fn i = δmn , h∂ xk ,Fn i = 0, h∂ yk ,Fn i = 0, for 1 ≤ m ≤ N and k ≥ N + 1. In view of Lemma 10.5, exactly the same is true, when Fn is replaced by −∂x ∂ In , which is also in L 20 . Since  is a diffeomorphism, we thus must have Fn = −∂x ∂ In . The same reasoning applies to E n , using Lemma 12.4 and the fact that on GoN , t ∂θn is in H 1 for 1 ≤ n ≤ N . u Proof of Theorem 12.1. We can now prove that  is symplectic. Considering ω˜ = ∗ ω, where ω = h∂x−1 · , · i is the symplectic form on L 20 , we have ω(ξ, ˜ η) = ω(∗ ξ, ∗ η), where ξ , η are tangent vectors to g N at (q). We then find that for 1 ≤ m, n ≤ N , ω( ˜ f m , f n ) = ω(Fm , Fn ) = h∂x−1 Fm ,Fn i = − h∂ Im ,Fn i = 0 by Lemma 12.5, and similarly ω(e ˜ n , f m ) = ω(E n , Fm ) = h∂x−1 E n ,Fm i = h∂θn ,Fm i = δnm . Hence for the symplectic form restricted to g N we must have X X ω˜ = d In ∧ dθn + amn d Im ∧ d In , 1≤n≤N

1≤m,n≤N

with coefficients amn depending smoothly on I and θ in view of Theorem 11.8. However, since ω˜ is closed, L fn ω˜ = d(ω( ˜ f n , · )) = d(d In ) = 0, so the amn are independent of θ. It therefore suffices to consider points in g N with

108

III Birkhoff Coordinates

θ = 0, which correspond to N -gap potentials in GoN . But then, by Proposition 12.4, ω(e ˜ m , en ) = ω(E m , E n ) = h∂x−1 E m ,E n i = [θm ,θn ] = 0. That is, amn = 0 for 1 ≤ m, n ≤ N . This proves the theorem. u t Thus, we finally completed the proof of Theorem 6.1. Having established  as a canonical diffeomorphism we know that (x, y) are Birkhoff coordinates for each Hamiltonian H in the KdV hierarchy, or more generally in the Poisson algebra of KdV, when restricted to the appropriate subspace of h1/2 . That is, H 0 = H B −1 is a function of the actions alone, and in fact a real analytic one – see the Addendum to Theorem 15.1. Its equations of motions are given by x˙n = ωn (I)yn , y˙n = −ωn (I)xn , where ωn (I) =

∂ H0 (I) ∂ In

with I = (In )n≥1 = 12 (xn2 + yn2 )n≥1 are the associated frequencies. Thus, each KdV equation becomes an infinite-dimensional system of ordinary differential equations, describing the motion of infinitely many oscillators whose frequencies depend on their amplitudes in a nonlinear fashion. For the convenience of reference we collect the results stated in Theorems 6.1, 9.8, 11.9 and 11.10 again in one single statement, using the notations established there. Theorem 12.6. There exists a diffeomorphism  : L 20 → h1/2 with the following properties. (i)  is one-to-one, onto, bi-analytic, and preserves the Poisson bracket. (ii) For each N ≥ 0, the restriction of  to H0N , denoted by the same symbol, is a map  : H0N → h N +1/2 , which is one-to-one, onto, and bi-analytic as well. (iii) The coordinates (x, y) in h3/2 are global Birkhoff coordinates for the KdV equation. That is, the transformed KdV Hamiltonian H B −1 depends only on xn2 + yn2 , n ≥ 1, with (x, y) being canonical coordinates in h3/2 .

12 The Symplectomorphism Property

109

(iv) The same holds for any other Hamiltonian in the KdV hierarchy, considered on a subspace H0N with appropriate N . (v) The Jacobian of  at q = 0 is boundedly invertible, its inverse is the weighted Fourier transform d0 −1 : h1/2 → L 20 X√
(x, y) 7 → 2πn xn en + yn e−n . n≥1

(vi) For each finite index set A ⊂ N, G A = −1 (h A ). (vii)  maps each isospectral set Iso(q) in L 20 one-to-one onto the torus Tor(I(q)) in h1/2 . S (viii) M≥1 G{ 1,...,M } is dense in H0N .

IV Perturbed KdV Equations

13 The Main Theorems In this chapter we study small perturbations of the KdV equation u t = −u x x x + 6uu x on the real line with periodic boundary conditions. We consider this equation as an infinite dimensional, integrable Hamiltonian system and subject it to sufficiently small Hamiltonian perturbations. The aim is to show that large families of time-quasiperiodic solutions persist under such perturbations. This is true not only for this KdV equation, but in principle for all higher order KdV equations as well. As an example, the second equation in the KdV hierarchy will be considered in detail. Background To set the stage we introduce for any integer N ≥ 0 the phase space  H N = u ∈ L 2 (S 1 ; R) : kuk N < ∞ of real valued functions on S 1 = R/Z, where 2 X 2N 2 kuk2N = u(0) |k| u(k) ˆ + ˆ k∈Z

P 2πikx . In is defined in terms of the Fourier transform uˆ of u, u(x) = ˆ k∈Z u(k)e 0 2 1 N particular, we have H = L (S ) with norm k · k = k · k0 . We endow H with the Poisson structure proposed by Gardner, {F ,G} =

Z S1

∂ F d ∂G dx, ∂u(x) dx ∂u(x)

112

IV Perturbed KdV Equations

where F, G are differentiable functions on H N with L 2 -gradients in H 1 . The Hamiltonian corresponding to KdV is then given by Z
3 1 2 H (u) = 2 u x + u dx, S1

and

∂u d ∂H = ∂t dx ∂u is the KdV equation written in Hamiltonian form. The Poisson bracket { · , · } is degenerate and admits the average Z [u] = u(x) dx S1

as a Casimir function. Moreover, the Poisson structure induces a trivial foliation with leaves  HcN = u ∈ H N : [u] = c , c ∈ R. Instead of considering the restriction of the Hamiltonian H to each leaf HcN , it is more convenient to choose a fixed phase space, H0N , which is symplectomorphic to every other leaf HcN by translation. Writing u = v + c with [v] = 0 and c = [u] the Hamiltonian then takes the form H (u) = Hc (v) + c3 with Hc (v) =

Z S1

1 2 2 vx

+v

3




Z dx + 6c S1

1 2 2 v dx.

(13.1)

R Here, S 1 21 v 2 dx is the zero-th Hamiltonian of the KdV hierarchy and corresponds to translation. To describe the structure of the phase space we recall some facts from the spectral theory of Hill’s equation, with more details given in appendix B. For u in L 20 = H00 consider the differential operator L=−

d2 +u dx 2

on the interval [0, 2] with periodic boundary conditions. Its spectrum, denoted by spec(u), is pure point and consists of an unbounded sequence of periodic eigenvalues λ0 (u) < λ1 (u) ≤ λ2 (u) < λ3 (u) ≤ λ4 (u) < . . . . Equality or inequality may occur in every place with a ‘≤’-sign, and one speaks of the gaps (λ2n−1 (u), λ2n (u)) of the potential u and its gap lengths γn (u) = λ2n (u) − λ2n−1 (u),

n ≥ 1.

13 The Main Theorems

113

In the case of a double periodic eigenvalue, the gap is empty, and one speaks of a collapsed gap. Otherwise, the gap is said to be open. The space L 20 naturally decomposes into the isospectral sets  Iso(u) = v ∈ L 20 : spec(v) = spec(u) . Our particular interest is in the isospectral sets of so called finite gap potentials u, which are characterized by the fact that only a finite number of gaps are open. Such potentials are known to be real analytic and dense in any space H0N , see [49, 84]. In particular, for every finite index set A ⊂ N we may consider the set of A-gap potentials  G A = u ∈ L 20 : γn (u) > 0 ⇔ n ∈ A . Clearly, u ∈ GA

⇔

Iso(u) ⊂ G A ,

and G A ⊂ N ≥1 H0N , as all A-gap potentials are real analytic. It turns out that the isospectral sets of A-gap potentials are actually |A|-dimensional tori, which are uniquely parameterized by their positive gap lengths. As functions on L 20 , however, the periodic eigenvalues and their associated gap lengths are not differentiable at points where they are double. To avoid this difficulty, we may instead consider nonnegative actions T

In (u) ≥ 0,

n ≥ 1,

which are also defined entirely in terms of the periodic spectrum of u and closely related to the gap lengths γn (u). For instance, In vanishes precisely when γn vanishes. But unlike the latter, the former are real analytic on all of L 20 . See section 7 for the details. We then have the following picture. The set G A of A-gap potentials is a real analytic submanifold of L 20 , and there exists a bi-analytic diffeomorphism A  A : G A → T A × R+ ,

which maps each isospectral set T = Iso(u) in G A onto some torus TI = T A × { I }, where I = (In )n∈A are the nontrivial actions of u. See Theorems 6.1, 11.9 and 11.10 for the details. Incidentally, if u is not a finite gap potential, then Iso(u) is homeomorphic to a product of infinitely many circles endowed with the product topology, one circle for each open gap. It is not a manifold in the usual sense, since Iso(u) is compact in L 20 , whereas an infinite dimensional Hilbert manifold is never compact. Returning to the KdV equation, the relevance of these results stems from the fact that, considered as functions on L 20 , the periodic eigenvalues represent integrals for the evolution of the KdV equation, as follows from the Lax pair formalism for KdV. The same is then true for the gap lengths γn and the actions In , as they are defined in terms of the periodic spectrum, too. The latter are moreover real analytic, functionally independent and in involution on L 20 .

114

IV Perturbed KdV Equations

Thus, if u t is a solution curve of the KdV equation in some space H0N with initial value u o , then spec(u t ) = spec(u o ) for all t. So the entire solution is confined to Iso(u o ). Consequently, the whole phase space decomposes into a collection of tori of varying dimension which are invariant under the KdV flow. In particular, each manifold G A of A-gap potentials is completely foliated into invariant tori of the same dimension |A|, and the actions In , n ∈ A, form a complete set of independent integrals in involution on G A . Hence, by the Liouville-Arnold-Jost-Theorem, the flow on each torus is linear in suitable coordinates and completely characterized by |A| fixed frequencies: it consists of quasiperiodic motions winding around the torus in phase space. Analytically speaking, one can find a bi-analytic diffeomorphism A 8 A : T A × R+ → GA

and a real analytic frequency map A ϕ A : R+ → RA,

such that 8 A maps each fiber TI = T A × {I } onto an invariant torus T I = 8 A (TI ), and such that for each θ ∈ T A , 8 A (θ + tϕ A (I ), I ),

t ∈ R,

is a smooth, even real analytic solution of the KdV equation. On G A the KdV equation is thus completely integrable in the classical sense as a system of |A| degrees of freedom. In addition, all its invariant tori are linearly stable in the infinite dimensional ambient phase space, since all their Lyapunov exponents vanish. This follows from the fact that all nearby solutions are also confined to invariant tori. Results In the following theorem we are going to describe a perturbation theory for these A be a compact set of A-actions of positive Lebesgue A-gap solutions. Let 0 ⊂ R+ measure, and let [ T0 = TI ⊂ G A, T I = −1 A (TI ), I ∈0

T be the union of the corresponding tori in G A . Recall that G A ⊂ N ≥1 H0N . With H0,NC we denote the complexification of H0N with norm k · k N , and with sup

kFk N ;U = sup kF(u)k N u∈U

the usual sup-norm of a function F on a domain U ⊂ H0,NC . The following result was first proven by Kuksin for the case c = 0 [74].

13 The Main Theorems

115

A a compact Theorem 13.1. Let A ⊂ N be a finite index set of cardinality m, 0 ⊂ R+ subset of positive Lebesgue measure, and N ≥ 1. Assume that the Hamiltonian K is real analytic in a complex neighbourhood V of T0 in H0,NC and satisfies the regularity condition

∂ K sup ∂K

: V → H0,NC , ≤ 1.

∂u ∂u N ;V

Then, for any real c, there exists an ε0 > 0 depending only on A, N , c and the size of V such that for |ε| < ε0 the following holds. There exist (i) a nonempty Cantor set 0ε ⊂ 0 with meas(0 − 0ε ) → 0 as ε → 0, (ii) a Lipschitz family of real analytic torus embeddings 4 : Tm × 0ε → V ∩ H0N , (iii) a Lipschitz map χ : 0ε → Rm , such that for each (θ, I ) ∈ Tm × 0ε , the curve u(t) = 4(θ + χ (I )t, I ) is a quasiperiodic solution of   d ∂ Hc ∂K ∂u = +ε ∂t dx ∂u ∂u winding around the invariant torus 4(Tm × {I }). Moreover, each such torus is linearly stable. Remark 1. The set 0ε depends on the perturbation εK . Its measure estimate, however, depends only on its size ε and is independent of other properties of K . Remark 2. The “size of V ” refers to the diameter of a complex neighbourhood around a real domain. See section 17 for details. For further remarks see section 1 on page 9. Besides the KdV equation there is a whole hierarchy of so called higher order KdV equations, which are all in involution with each other – see appendix C. They also admit the same set of integrals as the KdV equation. Consequently, they all possess the same families of invariant tori. The difference is only in the frequencies of the quasi-periodic motions on each of these tori. Therefore, similar results should also hold for the higher order KdV equations. As an example we consider the second KdV equation, which reads ∂t u = ∂x5 u − 10u∂x3 u − 20∂x u∂x2 u + 30u 2 ∂x u. Its Hamiltonian is H 2 (u) =

Z S1

1 2 2 uxx


+ 5uu 2x + 52 u 4 dx,

which is defined on H 2 = H 2 (S 1 ; R). Again, with u = v + c and [v] = 0, 5 H 2 (u) = Hc2 (v) + c4 . 2

116

IV Perturbed KdV Equations

Here Hc2 (v) = H 2 (v) + 10cH 1 (v) + 30c2 H 0 (v), R R
with H 1 = S 1 12 vx2 + v 3 dx the KdV Hamiltonian, and H 0 = S 1 12 v 2 dx the Hamiltonian of translation. We study this Hamiltonian on H0N with N ≥ 2, considering c as a real parameter. A version of the following result was first stated by Kuksin [74]. A a compact subset of Theorem 13.2. Let A ⊂ N be a finite index set, 0 ⊂ R+ positive Lebesgue measure, and N ≥ 2. Assume that the Hamiltonian K is real analytic in a complex neighbourhood V of T0 in H0,NC and satisfies the regularity condition

∂ K sup ∂K N −2

: V → H0,C , ≤ 1.

∂u ∂u N −2;V

If c ∈ / E A2 , where the exceptional set E A2 is an at most countable subset of the real line not containing 0 and with at most |A| accumulation points, then the same conclusions as in Theorem 13.1 hold for the system with Hamiltonian Hc2 + εK . Remark. A more detailed description of the set E A2 is given in appendix J. It is likely that the theorem is true for all c ∈ R. Outline of Proof The proof of Theorems 13.1 and 13.2 is based on an infinite dimensional version of the KAM theory that is concerned with the persistence of finite dimensional invariant tori and is applicable to small perturbations of the KdV equation. A prerequisite for developing such a perturbation theory is the existence of coordinates with respect to which the linearized equations along the unperturbed motions on the invariant tori reduce to constant coefficient form. Often, such coordinates are difficult to construct even locally. In the case of the KdV equation, however, such coordinates exist globally. For any r ≥ 0 we introduce the model space of real sequences hr = `r2 × `r2 with elements (x, y), where n o X `r2 = x ∈ `2 (N, R) : kxkr2 = n 2r |xn |2 < ∞ . n≥1

We endow this space with the standard symplectic structure chapter III we construct a bi-analytic symplectomorphism 9 : h1/2 → L 20 ,

P

n≥1 dx n

∧ d yn . In

13 The Main Theorems

117

such that the KdV Hamiltonian on the model space h3/2 , again denoted Hc , is of the form 1 Hc = Hc (I1 , I2 , . . . ), In = (xn2 + yn2 ). 2 The equations of motion are thus x˙n = ωn (I)yn ,

y˙n = −ωn (I)xn ,

with frequencies ωn =

∂ Hc (I), ∂ In

I = (In )n≥1 ,

that are constant along each orbit. So each orbit is winding around some invariant torus T I = { (x, y) : xn2 + yn2 = 2In , n ≥ 1 }, where the parameters I = (In )n≥1 are the actions of its initial data. We are interested in a perturbation theory for families of finite-dimensional tori T I . So we fix an index set A ⊂ N of finite cardinality m = |A|, and consider tori with In > 0

⇔

n ∈ A.

The linearized equations of motion along any such torus have now constant coefficients and are determined by m internal frequencies ω = (ωn )n∈A and infinitely many external frequencies  = (ωn )n ∈A / . Both depend on the m-dimensional parameter ξ = (In )n∈A , since all other components of I vanish. The KAM theorem for such families of finite dimensional tori requires a number of assumptions, among which the most notorious and unpleasant ones are the so called nondegeneracy and nonresonance conditions. In this case, they essentially amount to the following. First, the map ξ 7→ ω(ξ ) from the parameters to the internal frequencies has to be a local homeomorphism, which is Lipschitz in both directions. This is known as Kolmogorov’s condition in the classical theory. Second, the zero set of any of the frequency combinations hk,ω(ξ )i + hl,(ξ )i has to be a set of measure zero in 5, for each k ∈ Zm and l ∈ Z∞ with 1 ≤ |l| ≤ 2. This is sometimes called Melnikov’s condition. To verify these conditions for the KdV Hamiltonian we need some knowledge of its frequencies. To this end we compute the first coefficients of the Birkhoff normal

118

IV Perturbed KdV Equations

form of the KdV Hamiltonian. Writing X u= γn qn e2πinx n6 =0

√ √ with weights γn = 2π |n| and complex coefficients q±n = (xn ∓iyn )/ 2, the KdV Hamiltonian becomes X X Hc = λn |qn |2 + γk γl γm qk ql qm n≥1

k+l+m=0

on h3/2 with λn = (2πn)3 + 6c · 2πn. Thus, at the origin we have an elliptic equilibrium with characteristic frequencies λ1 , λ2 , . . . . We then construct a real analytic, symplectic coordinate transformation 8 in a neighbourhood of the origin, which transforms Hc into Hc B 8 =

3X 2 1X λn (xn2 + yn2 ) − (xn + yn2 )2 + . . . . 2 4 n≥1

n≥1

The important fact about the non-resonant Birkhoff normal form is that its coefficient are uniquely determined, once the quadratic term is fixed, and do not depend on the normalizing transformation – see appendix G. Comparing the global and the local KdV Hamiltonians we may thus conclude that they agree up to terms of order four – that is, the local result provides us with the first terms of the Taylor series expansion of the globally integrable KdV Hamiltonian. We obtain X X Hc = Hc (I1 , I2 , . . . ) = λn In − 3 In2 + . . . . n≥1

n≥1

Consequently ωn (I) =

∂ Hc (I) = λn − 6In + . . . , ∂ In

where λn and ωn also depend on c. By further computing some additional terms of order six in the expansion above, we gain sufficient control over the frequencies ω to verify all nondegeneracy and nonresonance conditions for any c. This allows us to apply KAM theory and eventually prove Theorems 13.1 and 13.2.

14 Birkhoff Normal Form We begin by transforming the KdV Hamiltonians into their Birkhoff normal forms up to order four. Recall that after writing u = v + c with [v] = 0, our phase space is

14 Birkhoff Normal Form

119

H0N endowed with the Poisson structure {F ,G} =

Z S1

∂ F d ∂G dx. ∂u(x) dx ∂u(x)

The KdV Hamiltonian is Hc (v) =

Z S1

1 2 2 vx


+ v 3 dx + 3c

Z S1

v 2 dx,

(14.1)

where c is considered as a real parameter. To write this Hamiltonian system more explicitly as an infinite dimensional system we introduce infinitely many coordinates v = (vn )n6=0 by writing def

v = F (v) =

X

γn vn e2πinx ,

(14.2)

n6=0

where γn =

p 2π |n|

are fixed positive weights. The sequence v = (vn )n6=0 is an element of the Hilbert space hr• of all complex valued sequences w = (wn )n6=0 satisfying kwkr2 =

X

|n|2r |wn |2 < ∞,

w−n = wn .

n6=0

Due to the choice of the weights, we have an isomorphism F : h •N +1/2 → H0N for each N ≥ 1. The complex space hr• is canonically identified with the real space hr by setting √ wn = (xn − iyn )/ 2,

w−n = wn ,

n ≥ 1.

The minus sign in the definition of wn is chosen so that dwn ∧ dw−n = i dxn ∧ d yn . A function on hr• is said to be real analytic, if with this identification it is real analytic in xn and yn in the usual sense. The complexification of hr• is the same space of sequences, but with the condition w−n = w n dropped. The Hamiltonian expressed in the new coordinates v is determined by inserting the expansion (14.2) of v into the definition (14.1) of Hc . Using for simplicity the same symbol for the Hamiltonian as a function of v we obtain Hc (v) = 3c (v) + G(v) with 3c (v) =

X n≥1


γn6 + 6cγn2 |vn |2 ,

G(v) =

X k,l,m6=0 k+l+m=0

γk γl γm vk vl vm .

120

IV Perturbed KdV Equations

Note that the first sum is taken over n ≥ 1, not n 6 = 0, which accounts for the “missing” factor 1/2. The phase space hr• is endowed with the Poisson structure {F ,G} = i

X

σn

n6=0

∂ F ∂G , ∂vn ∂v−n

where σn = sgn(n) is the sign of n, and the equations of motion in the new coordinates are given by ∂ Hc v˙n = iσn , n 6= 0. ∂v−n This is most easily seen by observing that X ∂ F ∂vn X ∂F ∂F = = · γ −1 e−2πinx ∂v(x) ∂vn ∂v(x) ∂vn n n6=0

n6=0

and calculating {F ,G} on H0N . Since the transformed Poisson structure is nondegenerate, it also defines a symplectic structure 1X dvn ∧ dv−n ω= i n≥1

•

on hr , according to which the above equations of motion are the usual Hamiltonian equations with Hamiltonian Hc (v). The associated Hamiltonian vector field with Hamiltonian H is given by XH = i

X

σn

n6=0

∂H ∂ . ∂v−n ∂vn

The vector field of the quadratic Hamiltonian 3c takes values in hr•−3 for v in hr• , hence it is unbounded of order 3. Strictly speaking, it is not a genuine vector field. The vector field of the cubic Hamiltonian G is also unbounded, but only of order 1. More precisely, we have the following regularity property of X G . Lemma 14.1. The Hamiltonian vector field X G is real analytic as a map from hr• into hr•−1 for each r ≥ 23 . Moreover,
kX G kr −1 = O kvkr2 . We remark that here and in the following, r − 12 is thought of being an integer. But everything holds for arbitrary r ≥ 32 as well. P Proof. We have G(v) = k+l+m=0 γk γl γm vk vl vm , hence X ∂G = 3γn γk γl vk vl = 3γn gn , ∂v−n k+l=n

14 Birkhoff Normal Form

121

where gn =

X

γk γl vk vl =

k+l=n

X

γk vk γn−k vn−k ,

k

and where all indices are nonzero integers. This restriction may be dropped by understanding that v0 = 0 and γ0 = 0. Defining w = (wn )n = (γn vn )n and g = (gn ) we see that gn = (w ∗ w)n , hence g = w ∗ w. For v ∈ hr• we have w ∈ hr•−σ with σ = 12 . Moreover, the space hr•−σ is a Banach algebra for r − σ > 12 , see [76]. Hence we have kgkr −σ = kw ∗ wkr −σ ≤ c kwkr2−σ ≤ c kvkr2 , and consequently kX G kr −1 ≤ c kgkr −σ ≤ c kvkr2 . The analyticity of X G follows with Theorem A.5 from the analyticity of each of its components and its local boundedness as a map from hr• into hr•−1 . u t The next theorem is the main result of this section. Theorem 14.2. There exists a real analytic symplectic coordinate transformation • v = 8(w) defined in a neighbourhood of the origin in h3/2 which transforms each Hamiltonian Hc , c ∈ R, into its Birkhoff normal form up to order four. More precisely, Hc ◦ 8 = 3c − B + K c with B=3

X

|wn |4 ,


kX K c k1/2 = O kwk43/2 .

n≥1

Moreover, for each r ≥ 3/2, the restriction of 8 to some neighbourhood of the origin in hr• defines a similar coordinate transformation in hr• , so that
kX K c kr −1 = O kwkr4 . Note that B happens to be independent of c and is a sum of terms each of which depends only on |wn |2 . Thus the modes are uncoupled up to order four. Before giving the proof of the theorem we mention that the transformation 8 not only normalizes the KdV Hamiltonian Hc up to order four, but indeed puts each Hamiltonian in the KdV hierarchy into its Birkhoff normal form up to order four. This is explained in appendix G. As an example we consider the second KdV equation at the end of this section. Another consequence is that the term K c is actually independent of c. Moreover, the construction of the transformation 8 allows us to determine the Birkhoff coefficients up to order 6. This is also explained in appendix G, and the somewhat lengthy calculations can be found in appendix H.

122

IV Perturbed KdV Equations

We turn to the proof of Theorem 14.2. To simplify notation we drop the subscript c and thus consider the Hamiltonian H = 3 + G, with X 3= λn |vn |2 , λn = n˜ 3 + 6cn. ˜ n≥1

Here and later, we use the convenient short hand n˜ = 2πn for integers n. The coordinate transformation 8 is constructed in two steps: 8 = 9 ◦ 4, where 9 eliminates the third order term G and replaces it by higher order terms, and 4 normalizes the resulting fourth order term. Each of these transformations is obtained as the time-1-map of the flow of some real analytic Hamiltonian vector field on hr• , whose Hamiltonian has to be chosen properly. Consider the first step, and let 9 = X 1F = X tF t=1 . Assuming for the moment that X tF is defined for 0 ≤ t ≤ 1 in some neighbourhood of the origin in hr• we can use Taylor’s formula to expand around t = 0: H ◦ 9 = 3 ◦ X tF t=1 + G ◦ X tF t=1 Z 1 = 3 + {3,F} + (1 − t) {{3,F},F} ◦ X tF dt Z 1 0 {G,F} ◦ X tF dt. +G+ 0

If we can solve the equation {3,F} + G = 0 with a homogeneous Hamiltonian F of order three, then we obtain Z 1 H ◦9 =3+ t {G,F} ◦ X tF dt 0

1 1 = 3 + {G,F} − 2 2

1

Z 0

(t 2 − 1) {{G,F},F} ◦ X tF dt

(14.3)

by partial integration. Here, {G,F} is homogeneous of order four, and the integral only contains terms of order five or more. P To solve {3,F} + G = 0 we make the ansatz F = Fklm vk vl vm and note that X {3,F} = −i (λk + λl + λm )Fklm vk vl vm , k,l,m

where we extend the definition of λn = n˜ 3 + 6cn˜ to all n. Since G contains only monomials vk vl vm with k + l + m = 0 and k, l, m 6= 0, also F need only contain

14 Birkhoff Normal Form

123

monomials of this kind. Under this condition, λk + λl + λm = 8π 3 (k 3 + l 3 + m 3 ) is independent of c. Moreover, we make the following observation. Lemma 14.3. Suppose k, l, m are nonzero integers with k + l + m = 0. Then k 3 + l 3 + m 3 = 3klm 6= 0. Indeed, if k + l + m = 0, then m = −k − l, and k 3 + l 3 + m 3 = −3k 2l − 3kl 2 = −3kl(k + l) = 3klm 6= 0. Hence it is justified to define F by setting  G klm  iFklm = λk + λl + λm  0

for k + l + m = 0, otherwise.

Then, at least formally, we have {3,F} + G = 0. The nonzero coefficients of F are more explicitly γk γl γm 1 1 1 γk γl γm 1 = = , 3 3 3 3 3 2πk · 2πl · 2πm 3 γ˜k γ˜l γ˜m 8π k + l + m √ with γ˜n = σn γn = sgn(n) 2π |n|, not to be confused with n˜ for 2πn. It follows, in exactly the same manner as in the proof of Lemma 14.1, that X F defines a real analytic vector field on hr• of order −1 with
kX F kr +1 = O kvkr2 iFklm =

for each r ≥ 23 . A fortiori, X F is a real analytic vector field on all of hr• for every r ≥ 32 . It follows that if X G is of order 1, then also X {G,F} = [X G , X F ] is of order 1. Moreover, in a small neighbourhood of the origin in hr• , the flow X tF exists for 0 ≤ t ≤ 1 and defines a local diffeomorphism 9 = X 1F with fixed point 0 and regular Jacobian D9, which is also an isomorphism of hr•−1 . It follows from (14.3) that the transformed vector field 9 ∗ X H − 3 = D9 −1 X H ◦ 9 − 3 is again analytic and unbounded of order 1. Moreover, by construction, H ◦ 9 is normalized up to terms of order three. This completes the first step of the normalization procedure. In the second step we normalize the resulting fourth order term 12 {G,F} in (14.3). This term is easily calculated. Recall that G=

X

γk γl γm vk vl vm ,

F=

1 X vk vl vm . 3i γ˜k γ˜l γ˜m

(14.4)

124

IV Perturbed KdV Equations

So we have {G,F} = i

X

σj

j6=0

=3

X

σj

j6=0

∂G ∂ F ∂v j ∂v− j X k,l : k+l=− j

X

= −3

k+l+m+n=0 k+l6=0

X

γ j γk γl vk vl ·

γk γl vk vl vm vn . γ˜m γ˜n

m,n : m+n= j

vm vn γ˜− j γ˜m γ˜n

We decompose the last sum into its contribution to the Birkhoff normal form and the rest, to be transformed away in a moment. The former consists of all terms for which k + m = 0 or k + n = 0, whereas terms with k + l = 0 do not occur. If for example we have k + m = 0, then k + l + m + n = 0 leads to l + n = 0, and the corresponding term reduces to γk γl vk vl vm vn = σkl |vk |2 |vl |2 , γ˜m γ˜n where σkl is the sign of kl. The same contribution is obtained for k + n = 0 and l + m = 0. Hence the normal form part of {G,F} is  X  X X |vk |4 = −3 |vk |4 , −3 2 σkl |vk |2 |vl |2 + k,l:k6=l

k6=0

k6 =0

since the first sum on the left hand side is zero as its terms annihilate each other. Thus we obtain X X 3 γk γl 1 {G,F} = −3 |vk |4 − vk vl vm vn 2 2 γ˜m γ˜n k>0

k+l+m+n=0 k+l,k+m,k+n6 =0

= −B − Q,

(14.5)

where B is the term stated in the theorem. The complete Hamiltonian at this stage is Z 1 1 2 H ◦9 =3− B− Q− (t − 1) {{G,F},F} ◦ X tF dt. 2 0 It remains to eliminate Q by another coordinate transformation 4. In complete analogy to the first step we let X 4 = X tF t=1 , F= Fklmn vk vl vm vn . k,l,m,n

We need to solve the equation X {3,F} = −i (λk + λl + λm + λn )Fklmn vk vl vm vn = Q. k,l,m,n

14 Birkhoff Normal Form

125

To this end it suffices that F contains only those monomials which also appear in Q. But for k + l + m + n = 0 we have λk + λl + λm + λn = 8π 3 (k 3 + l 3 + m 3 + n 3 ). Moreover, the following holds. Lemma 14.4. Suppose k, l, m, n are nonzero integers with k + l + m + n = 0, but k + l 6 = 0, k + m 6 = 0 and k + n 6 = 0. Then k 3 + l 3 + m 3 + n 3 = 3(k + l)(k + m)(k + n) 6 = 0. Indeed, with n = −(k + l + m) we have k 3 + l 3 + m 3 + n 3 = −3(k 2l + kl 2 + k 2 m + km 2 + l 2 m + lm 2 + 2klm), and the expression in parentheses equals (k + l)(k + m)(l + m) by a straightforward computation. Thus we may define Fklmn =

3 i γk γl 2 λk + λl + λm + λn γ˜m γ˜n

for the relevant indices k, l, m, n, and Fklmn = 0 otherwise. At least formally, the transformation 4 = X 1F then eliminates the term Q. To establish the regularity of the vector field X F we write 3 i γk γl 3 3 3 3 3 16π k + l + m + n γ˜m γ˜n 3 i γk γl γm γn = 5 3 3 3 3 mn 64π k + l + m + n

Fklmn =

and observe that
(k 3 + l 3 + m 3 + n 3 )mn ≥ max k 2 , l 2 , m 2 , n 2 . Indeed, if k is the biggest integer in absolute value, then we write the left hand side as 3(k + l)(k + m)(k + n)mn and use |(k + m)m|, |(k + n)n| ≥ |k − 1| ≥

2 |k|. 3

If, however, m is the biggest integer in absolute value, then we write the left hand side as 3(m + k)(m + l)(m + n)mn and use |(m + n)n| ≥ |m − 1| ≥

2 |m|. 3

126

IV Perturbed KdV Equations

By symmetry in k, l and m, n, this covers all cases. It follows that for X ∂ F/∂v j = (F jklm + · · · + Fklm j )vk vl vm k+l+m=− j

we have the estimate ∂F 3 γj ∂v ≤ 4π 3 j 2 j

X

γk γl γm |vk | |vl | |vm | ≤

k+l+m=− j

gj γ j3

,

where g j stands for the entire sum. Thus, kX F kr +1 ≤ kgkr −1/2 , and g = (gn )n6=0 is the three-fold convolution of w = (γn |vn |)n6=0 . As in the proof of Lemma 14.1, we have kgkr −1/2 ≤ c kwkr3−1/2 ≤ c kvkr3 for r ≥ 32 . So finally, kX F kr +1 = O kvkr3




for v ∈ hr• and r ≥ 32 . This establishes the regularity of the vector field X F , and finishes the proof of Theorem 14.2. As we explain in appendix G we obtain the same result by first P calculating the normal form for Hc at c = 0 and then adding the quadratic term 6c n γn2 |vn |2 to the result, as the latter is invariant under 8. Hence, it is no accident that the divisors in Lemmas 14.3 and 14.4 are independent of c. We already mentioned that the transformation 8 of Theorem 14.2 not only normalizes Hc up to order four, but puts every Hamiltonian in the KdV hierarchy into its own Birkhoff normal form up to order four – see appendix G. This is due to the fact that all these Hamiltonians are in involution. As an example we consider the second KdV equation ∂t u = ∂x5 u − 10u∂x3 u − 20∂x u∂x2 u + 30u 2 ∂x u, whose Hamiltonian H 2 (u) =

Z S1

1 2 2 uxx


+ 5uu 2x + 52 u 4 dx

is defined on H 2 . With u = v + c, [v] = 0, we can write 5 H 2 (u) = Hc2 (v) + c4 2 with Hc2 (v) = H 2 (v) + 10cH 1 (v) + 30c2 H 0 (v), R R
where H 1 = S 1 21 vx2 + v 3 dx is the KdV Hamiltonian, and H 0 = S 1 21 v 2 dx is the Hamiltonian of translation.

15 Global Coordinates and Frequencies

127

Theorem 14.5. The transformation 8 of Theorem 14.2 also takes each Hamiltonian Hc2 , c ∈ R, into its Birkhoff normal form of order four: Hc2 ◦ 8 = 32c − Bc2 + K c2 , where, with n˜ = 2πn, X
32c = n˜ 5 + 10cn˜ 3 + 30c2 n˜ |wn |2 , n≥1

Bc2

=

X

X
˜ |wk |2 |wl |2 , 15n˜ 2 + 30c |wn |4 − 10 (3 − 2δkl ) |k˜ l|

n≥1

k,l≥1

and K c defines a real analytic vector field of order 3 satisfying


4

X K c r −3 = O kwkr for w ∈ hr• for each r ≥ 52 . For the second KdV Hamiltonian, the fourth order term Bc2 does depend on c. Its first sum is the contribution of the term 21 {G,F} for this R case, and the second term is the normal form part arising from the quartic term 52 S 1 v 4 dx. The calculations are given in appendix G.

15 Global Coordinates and Frequencies The Birkhoff normal form theorem in section 14 provides symplectic coordinates in a neighbourhood of the origin in hr• in which the KdV Hamiltonian is a classically integrable system up to order four. These coordinates are completely sufficient, if we were only interested in small solutions u and their behavior under Hamiltonian perturbations of the equation. To study large solutions as well, however, we employ the global Birkhoff coordinates constructed in chapter III. The results from Theorem 12.6 relevant in this context may be formulated as follows. Theorem 15.1. There exists a canonical transformation • 9 : h1/2 → H00

with the following properties. (i) 9 is one-to-one, P onto, bi-analytic and symplectic with respect to the symplectic structures −i n≥1 dwn ∧ dw−n and h∂x−1 · , · i, respectively. (ii) For each N ≥ 0, the restriction of 9 to h •N +1/2 , denoted by the same symbol, is a map 9 : h •N +1/2 → H0N , which is one-to-one, onto, and bi-analytic as well.

128

IV Perturbed KdV Equations

• (iii) The coordinates w on h3/2 are global Birkhoff coordinates for KdV:

Hc ◦ 9 = Hˆ c (|w|2 ),

|w|2 = (wn w−n )n≥1 .

The same holds for all higher KdV equations when considered on h •N +1/2 with an appropriate N . (iv) This transformation satisfies 9(0) = 0 and d0 9 = F , where F is a weighted inverse Fourier transform as defined in (14.2). Let us introduce the actions I = (In )n≥1 ,

In = wn w−n .

• If w ∈ h3/2 , then I is an element of the Banach space `13 consisting of all real sequences u = (u n )n≥1 with X |u|`1 = n 3 |u n | < ∞. 3

n≥1 • → `13 , w 7→ I, is real analytic and onto the closed positive cone The map A : h3/2

 P`13 = u ∈ `13 : u n ≥ 0 for all n ≥ 1 . Of course, this map is not one-to-one. Addendum to Theorem 15.1. There exists a real analytic Hamiltonian Hˇ c on the closed positive cone P`13 such that Hˆ c (|w|2 ) = Hˇ c (I). Proof. The function Hˇ c is pointwise well defined on P`13 , since Hˆ c is a function of the amplitudes |w|2 and thus constant on the fibers of A over P`13 . To prove analyticity consider first the origin in `13 . We may expand Hˆ c into its Taylor series in w, replace wn w−n by In everywhere for all n, and re-index the coefficients. Thus, we obtain a Taylor series expansion of Hˇ c around the origin. At any other point I ∈ P`13√ , we can do the same by expanding Hˆ c around w = (wn )n6=0 with wn = w−n = In . Thus, there exists a Taylor series expansion of Hˇ c around every point in P`13 . u t From now on we drop the accents and write again Hc for the Hamiltonian as a function of I. Theorem 15.1 states that 9 puts Hc into a global Birkhoff normal form on all of • h3/2 . So does the transformation 8 of Theorem 14.2 locally at the origin up to order four. By the last item of Theorem 15.1, the change between these two coordinate systems is of the form “identity + higher order terms”, hence by its uniqueness the normal form up to order four is not affected – compare Theorem G.1. Theorem 14.2 therefore provides us with the Taylor series expansion of Hc up to order two.

15 Global Coordinates and Frequencies

129

Corollary 15.2. The transformed KdV Hamiltonian Hc expanded at I = 0 is of the form X X Hc (I) = λn I n − 3 In2 + . . . , n≥1

n≥1

where λn = n˜ 3 + 6cn˜ with n˜ = 2πn, and the dots stand for higher order terms in I. Before continuing our investigation of Hc , we consider the second KdV Hamiltonian. By Theorem 15.1, 9 takes Hc2 into a function of |w|2 on `25/2 as well. As above, this gives rise to a Hamiltonian Hˇ c2 of the actions I, which is real analytic on the closed positive cone P`15 defined analogously to P`13 . Theorem 14.5 then provides the first two terms of its Taylor series expansion. Theorem 15.3. The transformation 9 of Theorem 15.1 transforms the second KdV Hamiltonian Hc2 into Hˆ c2 (|w|2 ) = Hˇ c2 (I) =

X

λ2n In −

n≥1

1 X Ckl Ik Il + . . . , 2 k,l≥1

where Hˇ c2 is real analytic on P`15 and ( λ2n

5

3

2

= n˜ + 10cn˜ + 30c n, ˜

Ckl =

˜ −60k˜ l, 2 ˜ 10k + 60c,

k 6= l . k =l

The dots stand for higher order terms in I. Given the integrable Hamiltonian Hc as a function of I we may now define its frequencies ω in the usual fashion: ω = (ωn )n≥1 ,

ωn =

∂ Hc . ∂ In

This definition does not require the use of angular variables. Since ω is just the differential of Hc , we obtain a real analytic map from P`13 into the dual space of `13 . That is, ω : P`13 → `∞ −3 is real analytic, where `∞ β is the Banach space of all real sequences λ = (λ1 , λ2 , . . . ) with |λ|β = sup n β |λn | < ∞. n≥1

This follows from the general Cauchy estimate for analytic maps between Banach spaces, see Lemma A.2. Moreover, according to Corollary 15.2 we have the expansion ωn = λn − 6In + O2 (I), or ω = λ − 6I + . . . .

(15.1)

130

IV Perturbed KdV Equations

Note that λ = (λn )n≥1 itself is an element of `∞ −3 , so the above target space for ω can not be chosen smaller. To obtain a sharper statement, we consider the I-dependent part ω˜ = ω − λ = −6I + . . . of the frequencies ω. By Corollary F.5 they satisfy the asymptotic estimate |ω˜ n | = |ωn − λn | = O(n) uniformly on bounded subsets of the phase space, hence uniformly on bounded subsets of P`13 . This estimate also extends to small complex neighbourhoods around each point in P`13 . So we have a map ω˜ : P`13 → `∞ −1 , which is componentwise real analytic and locally bounded on complex neighbourhoods. Since the norm of the target space is a weighted sup-norm, this implies that the whole map is real analytic – see Theorem A.3. Theorem 15.4. The frequency map ˜ ω˜ : I 7→ ω(I) = ω(I) − λ of the KdV Hamiltonian is real analytic as a map from P`13 to `∞ −1 . In the next section we are going to describe an abstract infinite dimensional KAM theorem. To be applicable to perturbations of the KdV equation, however, the frequencies ω need to satisfy certain nondegeneracy conditions as functions of I. In particular, since our focus is on finite gap solutions, their dependence on finitely many non-zero actions is important. These properties are now established. Let A ⊂ N be an arbitrary finite index set, and let Z = N − A be its complement. According to this decomposition of N, we write I = (I A , I Z ),

I A = (In )n∈A ,

I Z = (In )n∈Z ,

and similarly in other cases. We are going to restrict ourselves to the subspace  ` A = u ∈ `13 : u Z = 0 ' R A . Of course, ω and ω˜ are also real analytic maps from the positive cone P` A ⊂ ` A into ∞ `∞ −3 and `−1 , respectively. P P For integer sequences k ∈ Z∞ , let k · ω = n≥1 kn ωn and |k| = n≥1 |kn |. We need to know that for a certain set of k, the frequency combinations k · ω do not vanish identically when considered as functions of I A . Moreover, the finite dimensional frequency map I A 7 → ω A has to have an almost everywhere regular Jacobian   ∂ωk Q A = (Q kl )k,l∈A = . ∂ Il k,l∈A

15 Global Coordinates and Frequencies

131

The following proposition and its addendum provide the essential facts. Recall that the frequencies also depend on the real parameter c, which we do not indicate explicitly. Proposition 15.5. For every c and every finite index set A ⊂ N the following holds on P` A . (i) The map I A 7 → ω A is nondegenerate in the sense that det Q A 6 ≡ 0. (ii) k · ω 6 ≡ 0 for all k ∈ Z∞ with k A 6 = 0 and |k Z | ≤ 2. (iii) k · ω 6 ≡ 0 for all k ∈ Z∞ with k A = 0 and 1 ≤ |k Z | ≤ 2, provided that 
c∈ / E A = − 32 π 2 i 2 ± i j + j 2 : i, j ∈ Z . Remark 1. In fact, Krichever proved that the map I A 7→ ω A is a local diffeomorphism everywhere, by using the representation of the frequencies by periods of certain Abelian differentials – see [9]. In the case c = 0, the second statement is proven in [11] using Schottky uniformization. Remark 2. E A is a discrete subset of the negative real axis, and there are no exceptional values for c > − 23 π 2 . Anyway, the Addendum below will show, that the condition c ∈ / E A can be dropped. Proof of Proposition 15.5. Recall that we have ωn = λn − 6In + O2 (I). Thus, Q A = −6E A + O(I), where E A is the identity matrix of dimension |A|. This proves the first item. Also, k · ω = k · λ − 6k A · I A + O2 (I A ), since I Z = 0 on ` A . For k A 6 = 0, the right hand side can not vanish identically in I A , so also the second item is proven. For k A = 0, however, the right hand side is not explicitly under control as a function of I A . Therefore, we instead restrict the parameter c so that k · ω| I=0 = k · λ = k Z · λ Z 6 = 0. For instance, consider the case where k Z · λ Z = λi ± λ j for distinct i, j ∈ Z . With λn = n˜ 3 + 6cn˜ we obtain the condition 0 6 = ı˜3 ± ˜3 + 6c(˜ı ± ˜) = (˜ı 2 ∓ ı˜˜ + ˜2 )(˜ı ± ˜) + 6c(˜ı ± ˜), or −6c 6 = ı˜2 ± ı˜˜ + ˜2 = 4π 2 (i 2 ± i j + j 2 ). The other cases lead to conditions which are contained in this one. This proves also the third item. u t

132

IV Perturbed KdV Equations

The proof of the preceding proposition is fairly short and simple. With more effort we can show that the restriction c ∈ / E A can be dropped from part (iii). The following is proven in appendix I, based on a result by Kramer. Addendum to Proposition 15.5. For any real parameter c and any finite index set A ⊂ N one has k · ω 6 ≡ 0 for all k ∈ Z∞ with k A = 0 and 1 ≤ |k Z | ≤ 2. In the remainder of this section we discuss the frequencies of the second KdV Hamiltonian, ∂ Hc2 ω2 = (ωn2 )n≥1 , ωn2 = . ∂ In According to Theorem 15.3 this defines a real analytic map ω2 : P`15 → `∞ −5 with expansion ω2 = λ2 − C I + . . . ,

C = (Ci j )i, j≥1 .

Note that λ2 = (λ2n )n≥1 itself belongs to `∞ −5 . As before, we consider the I-dependent term ω˜ 2 = ω2 − λ2 . By Theorem F.5 we have the asymptotic estimate 2 ω − λ2 = O(n 3 ) n n uniformly on bounded subsets of P`15 , and also on small complex neighbourhoods around each point. Hence we have the following analogue to Theorem 15.4. Theorem 15.6. The frequency map ω˜ 2 : P`15 → `∞ −3 ,

I 7→ ω˜ 2 (I) = ω2 (I) − λ2

of the second KdV Hamiltonian is real analytic. In appendix J we prove the following nondegeneracy properties of the second KdV Hamiltonian. For A ⊂ N, let Q 2A be the Jacobian of the map I A 7→ ω2A . Again, recall that the frequencies and the Jacobian depend on the real parameter c. Proposition 15.7. For every finite index set A ⊂ N the following holds on P` A . (i) There exists an |A|-point set C 2A ⊂ R not containing 0 such that for c ∈ / C 2A , det Q 2A 6≡ 0. (ii) There exists an at most countable subset E A2 ⊂ R not containing 0 and accumulating at most at the points of C 2A such that for c ∈ / E A2 , k · ω2 6 ≡ 0 for all 0 6 = k ∈ Z∞ with 1 ≤ |k Z | ≤ 2.

16 The KAM Theorem

133

These statements are proven in appendix J by looking at the first two terms of the expansion of ω2 at I A = 0. It is reasonable to expect that the set of excluded values for c can be considerably reduced or even eliminated, by looking at more terms of this expansion, or by using more global arguments.

16 The KAM Theorem In this section we formulate an infinite dimensional KAM theorem that is applicable to small perturbations of KdV equations. We begin by explaining its set up in an informal way. In the previous section we saw that on some Hilbert space hr• of complex sequences w = (wn )n6=0 with w−n = w n the KdV Hamiltonians take the form H = H (|w|2 ),

|w|2 = (|wn |2 )n≥1 .

The same form results by truncating the Birkhoff normal forms of Theorems 14.2 and 14.5 to fourth order. In any of these cases, the equations of motion are w˙ n = i

∂H , ∂wn

n 6 = 0.

It is immediate that we have infinitely many nonnegative integrals of motion In = |wn |2 ≥ 0,

n ≥ 1,

all of which are functionally independent and in involution. So the equations of motion reduce to w˙ n = iωn (I)wn , where ωn (I) =

∂H (I), ∂ In

I = (In )n≥1 .

Q It follows that each motion takes place on the product of circles n≥1 { |wn |2 = In }, and the n-th coordinate circles around with fixed frequency ωn (I). Their combined motion is in general not quasi-periodic, but almost-periodic, that is, the uniform limit of trigonometric polynomials [63]. The dimension of the underlying invariant torus equals the number of nonvanishing actions In , and the whole phase space decomposes into such tori. So in this sense the Hamiltonian system is completely integrable. As up to now there is no genuine KAM theorem for the infinite dimensional tori in this system – and there might never be – we need to restrict our attention to quasiperiodic motions on finite dimensional tori. In the context of the KdV equation this corresponds to the evolution of finite gap solutions. Thus, we fix a finite index set A ⊂ N of cardinality |A| = m, let Z = N − A be its complement, and restrict ourselves to the finite dimensional space  ` A = w ∈ hr• : w Z = 0 .

134

IV Perturbed KdV Equations

Here we use the notation introduced in the previous section on page 130. This space is invariant and foliated into the invariant tori  T I A = w : |w A |2 = I A , w Z = 0 ⊂ hr• , I A ∈ PR A . That is, on T I A we have |wn |2 = In for n ∈ A, and wn = 0 otherwise. Their dimension is maximal, if In > 0 for all n ∈ A, which we will assume henceforth. That A. is, we assume that I A ∈ R+ To formulate a KAM theory for such invariant tori it is convenient to introduce angle-action coordinates (x A , y A ) in T A × R A locally around each torus T I A by writing p wn = In + yn e−ixn , n ∈ A, while keeping the coordinates wn for n ∈ Z . The integrable Hamiltonian becomes H = H (I A + y A , |w Z |2 ) X X = H (I A , 0) + ωn (I A )yn + ωn (I A ) |wn |2 + . . . , n∈A

where ωn (I A ) =

n∈Z

∂H (I A , 0), ∂ In

n ≥ 1,

and the dots stand for the integral remainder of Taylor’s formula of second order in y A and |w Z |2 . The phase space coordinates are now (x A , y A , w Z ), the I A are parameters, and the linearized equations of motion are x˙n = ωn (I A ),

w˙ n = iωn (I A )wn ,

y˙n = 0,

for n ∈ A and n ∈ Z , respectively. Thus, they reduce to constant coefficient form. Geometrically speaking, in the new coordinates (x A , y A , w Z ) we have an invariant torus Tm × {0} × {0}, on which the flow is given by m internal frequencies ωn , n ∈ A. They depend on I A and thus on the position of the torus in the given family. In the normal space, described by the coordinates w Z , we have an elliptic fixed point at the origin, which is characterized by the external frequencies ωn , n ∈ Z . Thus, for each I A , we have what we call an invariant, rotational, linearly stable m-torus. It turns out that in order to prove Theorems 13.1 and 13.2 it is sufficient to develop a KAM theory, which is concerned with the persistence of a single finite dimensional torus, such as T0 , in an infinite dimensional system, which depends on sufficiently many parameters. To formulate a general KAM theorem we therefore start now with the following set up. We consider small perturbations of a family of infinite dimensional integrable Hamiltonians N = N (x, y, u, v; ξ ) X 1X = ωn (ξ )yn + n (ξ )(u 2n + vn2 ). 2 1≤n≤m

n≥1

(16.1)

16 The KAM Theorem

135

The phase space is m m 2 2 Sm p = T × R × `p × `p

with coordinates (x, y, u, v), where Tm = Rm /2πZm denotes the usual m-torus, and n o X `2p = x ∈ `2 (N, R) : kxk2p = n 2 p |xn |2 < ∞ . n≥1

P

The symplectic structure is ian N depends on parameters

1≤n≤m

dxn ∧ d yn +

P

n≥1 du n

∧ dvn . The Hamilton-

ξ ∈ 5 ⊂ Rm , where 5 is a compact subset of Rm of positive Lebesgue measure. For example, 5 may be a compact Cantor set of positive measure. Note that this Hamiltonian N equals the Hamiltonian H above, after dropping the higher order terms and an irrelevant constant term, setting ξ = I A , (ω1 , . . . , ωm ) = ω A ,

(n )n≥1 = ω Z ,

and re-indexing the coordinates. Our aim is to prove the persistence of the invariant torus T0 = Tm × {0} × {0} × {0} of N at ξ together with its elliptic fixed point at (u, v) = (0, 0) under sufficiently small Hamiltonian perturbations H = N + P of N , for parameter values ξ in a large Cantor subset of 5. To this end we make the following three assumptions. Assumption A: Frequency Asymptotics. There exist two real numbers d > 1 and δ < d − 1 such that the following holds. First, the frequencies n are real valued functions of ξ of the form ¯n + ˜ n (ξ ), n (ξ ) =  ¯ n is independent of ξ and of the form  ¯ n = cn d + . . . , where the dots stand where  for an expansion in lower order terms in n. Second, the functions ξ 7→

˜ n (ξ )  , nδ

n ≥ 1,

are uniformly Lipschitz on 5, or equivalently, the map ˜ : 5 → `∞  −δ ,

˜ ) = ( ˜ n (ξ ))n≥1 ξ 7→ (ξ

is Lipschitz on 5. ¯ n is slightly For the proof of the abstract KAM theorem the assumption on  relaxed. The one given here, however, is more transparent and suffices when applied to the KdV Hamiltonians. We assume here that d > 1. The case d = 1 may also be handled, as it is done in [72, 109], but is somewhat more involved and not relevant for the application to the KdV equations. We therefore omit it for the sake of clarity.

136

IV Perturbed KdV Equations

Assumption B: Nondegeneracy. The map ξ 7 → ω(ξ ) between 5 and its image is a homeomorphism which is Lipschitz continuous in both directions. Moreover, for every k ∈ Zm and l ∈ Z∞ with 1 ≤ |l| ≤ 2 the resonance set Rkl = { ξ ∈ 5 : hk,ω(ξ )i + hl,(ξ )i = 0 } has Lebesgue measure zero. P For integer vectors such as l, we always understand that |l| = n |ln |. We note that the zero measure condition is satisfied, if each frequency is a real analytic function of ξ , and hk,ω(ξ )i + hl,(ξ )i 6 ≡ 0 on 5 for all relevant integer vectors k and l. Moreover, assumption A implies that the measure of Rkl is under control for almost all k and l, so that this assumption together with assumption A is relevant only for finitely many resonance sets. This will be made explicit in section 22. The third assumption is concerned with the perturbing Hamiltonian P and its Hamiltonian vector field, X P = (Py , −Px , Pv , −Pu )T . We use the notation i ξ X P for X P evaluated at ξ , and likewise in analogous cases. With Sp,C we denote the complexification of the phase space Sp = Spm . Assumption C: Regularity. There is a neighbourhood U p of T0 in Sp,C such that P is defined on U p × 5, and its Hamiltonian vector field defines a map X P : U p × 5 → Sq,C , where q satisfies p − q < d − 1. Moreover, i ξ X P is real analytic on U p for each ξ ∈ 5, and i w X P is uniformly Lipschitz on 5 for each w ∈ U p . The essential requirement is the following. For each ξ , the vector field i ξ X P , considered as a map from a subset of Sp to Sq , is of the order p −q. By the preceding assumption, this order must be strictly smaller than d − 1, where by assumption A, d is a lower bound for the order of the unperturbed vector field X N . Hence, using the language of partial differential equations, the perturbed system has to be semi-linear. In previous versions of the KAM theorem [72, 109], an additional requirement was p − q ≤ 0.

16 The KAM Theorem

137

That is, the perturbation was also required to be bounded as an operator. This is suitable for nonlinear Schr¨odinger and wave equations on a bounded interval, but not for the KdV equations considered here. This assumption was later removed by Kuksin in [74] so that the theory applies also to perturbed KdV equations. To state the KAM theorem we need to introduce some domains and norms. For s > 0 and r > 0 we introduce the complex T0 -neighbourhoods D(s, r ) = { |Im x| < s } × { |y| < r 2 } × { kuk p + kvk p < r } 2 2 ⊂ Cm × Cm × `p, C × `p,C = Sp,C . 2 2 Here, |z| = maxn |z n | for vectors in Cm , and `p, C is the complexification of `p . On Sq,C we introduce for W = (Wx , W y , Wu , Wv ) the weighted norm

kW kr,q = |Wx | +

1 1 1 W y + kWu kq + kWv kq . r r r2

For a map W : U × 5 → Sq,C , such as the Hamiltonian vector field X P , we then define the norms sup

kW kr,q;U ×5 = lip kW kr,q;U ×5

sup

(w,ξ )∈U ×5

= sup

kW (w, ξ )kr,q ,



1ξ ζ W sup r,q;U

ξ,ζ ∈5 ξ 6=ζ

|ξ − ζ |

,

where 1ξ ζ W = i ξ W − i ζ W , and sup

ki ξ W kr,q;U = sup kW (w, ξ )kr,q . w∈U

˜ : 5 → `∞ In a completely analogous manner, the Lipschitz semi-norm of the map  −δ is defined as 1ξ ζ  ˜ lip −δ ˜ ||−δ;5 = sup . ξ,ζ ∈5 |ξ − ζ | ξ 6 =ζ

˜ lip = ||lip , since  ¯ =− ˜ is independent of ξ . Note that || −δ;5 −δ;5 We introduce one more constant. By assumptions A and B, lip

lip

|ω|5 + ||−δ;5 ≤ M < ∞. Finally observe that if X P satisfies assumption C, then it does so with the T0 -neighbourhoods D(s, r ) for all s > 0, r > 0 sufficiently small. A complete proof of the following theorem will be given in chapter V.

138

IV Perturbed KdV Equations

Theorem 16.1. Suppose N is a family of Hamiltonians of the form (16.1) defined on a phase space Spm and depending on parameters in 5 so that assumptions A and B are satisfied. Then there exists a positive constant γ depending only on m, d, δ, the frequencies ω and  and the real number s > 0 such that for every perturbed Hamiltonian H = N + P that satisfies assumption C and the smallness condition sup

ε = kX P kr,q;D(s,r )×5 +

α lip kX P kr,q;D(s,r )×5 ≤ αγ M

for some r > 0 and 0 < α < 1, the following holds. There exist (i) a Cantor set 5α ⊂ 5 with meas(5 X 5α ) → 0 as α → 0, (ii) a Lipschitz family of real analytic torus embeddings 8 : Tm × 5α → Sp , (iii) a Lipschitz map ϕ : 5α → Rm , such that for each ξ ∈ 5α the map 8 restricted to Tm × {ξ } is a real analytic embedding of a rotational torus with frequencies ϕ(ξ ) for the system with Hamiltonian H = N + P at ξ . In other words, t 7 → 8(θ + tϕ(ξ ), ξ ),

t ∈ R,

is a real analytic, quasi-periodic solution for the Hamiltonian i ξ H for every θ ∈ Tm and ξ ∈ 5α . Moreover, each embedding is real analytic on D(s/2) = { |Im x| < s/2 }, and sup

cε α lip k8 − 80 kr, p;D(s/2)×5α ≤ , M α α lip sup |ϕ − ω|5α + |ϕ − ω|5α ≤ cε, M

k8 − 80 kr, p;D(s/2)×5α +

where 80 : Tm × 5 → T0 ,

(x, ξ ) 7 → (x, 0, 0, 0)

is the trivial embedding for each ξ , and c is a positive constant which depends on the same parameters as γ . Remark 1. The role of the parameter α is the following. In applications the size of the perturbation usually depends on a small parameter which we may also call ε. One then wants to choose α as a function of this parameter, for example α = ε/γ . This way one can control the size of 5 X 5α in terms of the size of the perturbation. See [76, 110] for examples. Remark 2. The complement of the Cantor set 5α can be written as 5 X 5α = 4α = 41α ∪ 42α , where 41α and 42α may be considered as the coarse and the fine structure of 4α , respectively. The latter depends on the perturbation P and satisfies meas(42α ) = O(α).

17 Proof of the Main Theorems

139

The former consists of finitely many resonance zones defined in terms of the unperturbed frequencies, and meas(41α ) → 0

as α → 0

by assumption B. However, it requires further assumptions on the frequencies to make the rate of convergence explicit.

17 Proof of the Main Theorems We now give the proofs of the two theorems in section 13 concerning perturbations of the first two KdV equations. Since they are treated in a completely analogous way, we focus on the proof of Theorem 13.1. Recall the set up of Theorem 13.1. Let A ⊂ N be a finite index set of cardinality m, let 0 ⊂ Rm + be a closed bounded set of positive Lebesgue measure, and T0 =

[

TJ

J ∈0

the union of all A-gap potentials with A-gap lengths in 0. Consider a perturbed KdV Hamiltonian H = Hc + εK , where Hc is the KdV Hamiltonian in (13.1), and K is real analytic on some complex neighbourhood V of T0 in H0,NC . Since all finite gap potentials are in H0,NC , the unperturbed Hamiltonian Hc is real analytic on V as well, while K satisfies

∂ K sup

≤1

∂u N ;V by assumption. As a first step we apply the global symplectic coordinate transformation of Theorem 15.1, 9 : h •N +1/2 → H0N , which introduces global Birkhoff coordinates for the KdV equation. We know by Theorems 11.9 and 11.10 that 9 maps the space h A diffeomorphically onto the manifold G A of A-gap potentials in such a way that every torus TI ⊂ h A is mapped diffeomorphically onto an isospectral torus T I ⊂ G A . Hence there exists a closed A such that bounded subset 5 ⊂ R+ 9(T5 ) = T0 . Since 9 is real analytic, there is also a complex neighbourhood U of T5 in the complexification of h •N +1/2 , which is mapped bi-analytically onto the neighbourhood V of T0 . If necessary, we choose V smaller. Hence we have the following diagram,

140

IV Perturbed KdV Equations

where each arrow represents a bi-analytic diffeomorphism given by the map 9: TI ⊂ T5 ⊂ U ⊂ h •N +1/2,C ↓ ↓ ↓ T J ⊂ T0 ⊂ V ⊂ H0,NC

.

Now we consider the transformed Hamiltonian H ◦ 9 = Hc ◦ 9 + εK ◦ 9 = Hˆ c + ε R ¯ = Hˆ c (|w|2 ) + ε R(w, w), which is real analytic on U ⊃ T5 . We first look at the integrable Hamiltonian, which can be written as Hˆ c (|w|2 ) = Hˇ c (I),

I ∈ `12N +1 ,

according to Theorem 15.1, where `12N +1 ⊂ `13 , as N ≥ 1. Using Taylor’s formula and the frequencies introduced in section 15 we can write Hˇ c (I 0 + I) = Hˇ c (I 0 ) +

X ∂ Hˇ c i≥1

∂ Ii

(I 0 ) Ii

X ∂ 2 Hˇ c Ii I j dt ∂ Ii ∂ I j 0 i, j≥1 X X = const + ωi (I 0 )Ii + Q i j (I 0 , I)Ii I j Z

+

1

(1 − t)

i≥1

i, j≥1

1

∂ωi (I 0 + t I) dt. ∂Ij

(17.1)

with Q i j (I 0 , I) =

Z

(1 − t)

0

(17.2)

Note that Q i j is symmetric in i and j. Using the asymptotics of Theorem 15.4 and Cauchy’s estimate we obtain X Q i j (I 0 , I)I j ≤ ci kIk`1 (17.3) 2N +1

j≥1

sufficiently uniformly in I 0 on some complex neighbourhood of T5 and |I|`1 2N +1 small. The perturbing Hamiltonian vector field in the original phase space is XK =

d ∂K . dx ∂u

17 Proof of the Main Theorems

141

It is defined on V and of order 1. Since 9 is a symplectic diffeomorphism of the two Hilbert scales (h •N +1/2 ) N ≥1 and (H0N ) N ≥1 , as explained in appendix K, the vector field of the transformed Hamiltonian R = K ◦ 9 is X R = 9 ∗ X K = D9 −1 X K ◦ 9, and it is also of order 1. Since we may choose the domain V so that the inverse of the Jacobian of 9 is uniformly bounded, we obtain

∂ K sup sup sup

kX R k N −1/2;U ≤ C kX K k N −1;V ≤ C ≤ C. (17.4) ∂u N ;V As a second step we introduce symplectic polar coordinates around the tori in the family T5 . To simplify notation we henceforth assume that A = { 1, . . . , m }. For each ξ = (ξ1 , . . . , ξm ) ∈ 5 we then introduce new coordinates by setting p p wn = ξn + yn e−ixn , w−n = ξn + yn eixn , 1 ≤ n ≤ m, and 1 wm+n = √ (u n − ivn ), 2

1 w−m−n = √ (u n + ivn ), 2

n ≥ 1.

This transformation is real analytic and symplectic on D(s, r ) = { |Im x| < s } × { |y| < r 2 } × { kuk p + kvk p < r } 2 2 ⊂ Cm × Cm × `p, C × `p,C = Sp,C

for all s > 0 and r > 0 sufficiently small, where p = N + 1/2. In the following we may fix such an s arbitrarily, while we keep the freedom to choose r smaller. Using the expansion of Hˇ c in (17.1) and setting I 0 = (ξ, 0) the integrable Hamiltonian in the new coordinates is, up to a constant depending only on ξ and dropping the accent, given by Hc = N + Q = N (y, u, v, ξ ) + Q(y, u, v, ξ ), where N=

X

ωn (ξ )yn +

1 2

X

n (ξ )(u 2n + vn2 ),

n≥1

1≤n≤m

n (ξ ) = ωm+n (ξ ), 2 2 ) for i ≥ m + 1, + vi−m and, with Ii = yi for 1 ≤ i ≤ m and Ii = 21 (u i−m

Q=

X i, j≥1

Q i j (ξ, I)Ii I j .

142

IV Perturbed KdV Equations

As the notation indicates, N will play the role of the integrable normal form depending on the parameters ξ , which is perturbed by Q and R. We check assumptions A, B and C of the KAM Theorem 16.1 for this normal form. Its external frequencies n may be written as ¯n + ˜ n (ξ ) n (ξ ) =  ¯ n = n (0) and  ˜ n (ξ ) = n (ξ ) − n (0). We have with  ¯ n = 8π 3 (m + n)3 + 12πc(m + n)  and a map ˜ = ( ˜ n )n≥1 : 5 → `∞  −1 , which by Theorem 15.4 is real analytic on some complex neighbourhood of 5. Hence that map is also Lipschitz by Cauchy’s estimate. So assumption A is satisfied with d = 3 and δ = 1. To verify assumption B we recall that by Proposition 15.5 we have   ∂ωi det 6≡ 0 ∂ξ j 1≤i, j≤m on 5. Since this determinant is a real analytic function, it is non-zero almost everywhere on 5. In particular, for any given η > 0 we may excise from 5 a relatively open subset 5η with meas(5η ) < η such that on 5 X 5η the above determinant is uniformly bounded away from zero. Moreover, we may cover 5 X 5η by finitely many closed subsets 5ι , so that on each such subset the map ξ 7→ ω(ξ ) is a bianalytic homeomorphism onto its image in Rm . Henceforth it suffices to consider each such parameter set 5ι one at a time. On such a set we have hk,ω(ξ )i + hl,(ξ )i 6 ≡ 0 for every k ∈ Zm and l ∈ Z∞ with 1 ≤ |l| ≤ 2 by Proposition 15.5 and its Addendum. Since each such expression is analytic in ξ , its zero set is a set of measure zero. Thus, assumption B is satisfied for each subset 5ι . As a consequence of assumption A and B we also have lip

lip

|ω|5 + ||−1;5 ≤ M < ∞,

−1 lip ω ≤ L < ∞. ω(5X5 ) η

It remains to check assumption C. There are two contributions to the perturbation P: P = Q + ε R. From the form of Q and the estimate of Q i j as given by equations (17.2) and (17.3) we obtain with Cauchy’s estimate

sup

X Q ≤ cr 2 . r, p−1;D(s,r )×5 ι

17 Proof of the Main Theorems

143

2 2 ) for i ≥ m + 1, Just observe that Ii = yi for 1 ≤ i ≤ m and Ii = 21 (u i−m + vi−m 2 and that |I|`12N +1 ≤ cr on the domain D(s, r ) × 5ι . For the second term in P we have to take into account the weight factors in the definition of the norm k · kr, p−1 to obtain from (17.4) the estimate sup

kX R kr, p−1;D(s,r )×5ι ≤

c . r2

The same bounds hold for the respective Lipschitz semi-norms, since both vector fields are real analytic in ξ , the above estimates extend to some complex neighbourhood of 5ι , and Cauchy’s estimate applies. So altogether we obtain  α ε def sup lip kX P kr, p−1;D(s,r )×5ι ≤ C r 2 + 2  = kX P kr, p−1;D(s,r )×5ι + M r for α ≤ M and all small r > 0. In particular, we have verified that X P : U p × 5ι → Sq,C , with U p = D(s, r ) ⊂ Sp,C and q = p − 1. Since 1 = p − q < d − 1 = 2, X P has the required regularity properties. To meet the smallness condition of the KAM theorem for P = Q + ε R choose now √ 2C √ r 2 = ε, α= ε, γ with ε so small that α < 1. Here, C is taken from the preceding estimate, and γ is taken from the KAM theorem. We then obtain √  ≤ 2C ε = γ α as required. The conclusions of Theorem 13.1 now follow immediately from the conclusions of the KAM theorem. We only comment on the measure theoretic statement. For each 5ι we have
meas 5ι X 5ι,α → 0 as ε → 0. We have only finitely many such patches 5ι , which cover the original parameter domain 5 up to a set of measure η. By first choosing η and then ε small enough we can arrange that   [ meas 5 X 5ι,α → 0 as ε → 0. ι

The proof of Theorem 13.1 is now complete. The proof of Theorem 13.2 is completely analogous. The only differences are that we have d = 5, δ = 3, and q = p − 3. Moreover, the perturbation R may depend on u x .

V The KAM Proof

18 Set Up and Summary of Main Results In the following we give a complete proof of the infinite dimensional KAM theorem used in chapter IV to study small Hamiltonian perturbations of KdV equations. To make this presentation independent of chapter IV we begin by recalling the set up. We consider small perturbations of a family of infinite dimensional integrable Hamiltonians, where each Hamiltonian admits an invariant n-torus, on which the flow is linear with fixed frequencies, while in the remaining coordinates there is an infinite dimensional elliptic equilibrium. The aim is to prove the persistence of this torus under such perturbations. Extending the set up of the previous discussion in view of a wider class of applications, the underlying phase spaces will also allow for exponentially decaying sequences. It turns out that this does not complicate the construction and the pertaining estimates at all, but is rather a matter of notation. More specifically the unperturbed family is given by the normal form Hamiltonians N = N (x, y, u, v; ξ ) X X (18.1) = ω j (ξ )y j + 1  j (ξ )(u 2 + v 2 ) j

2

j

j≥1

1≤ j≤n

on a phase space n 2 2 Sp,a = Tn × Rn × `p,a × `p,a

with coordinates (x, y, u, v), depending on parameters ξ ∈ 5 ⊂ Rn . 2 denotes the Hilbert Tn is the usual n-torus Rn /2πZn with 1 ≤ n < ∞, and `p,a space of all real sequences w = (w1 , w2 , . . . ) with X kwk2p,a = j 2 p e2a j |w j |2 < ∞. j≥1

146

V The KAM Proof

The parameter set 5 may be any compact subset of Rn of positive Lebesgue measure. For example, 5 may be a compact Cantor set of positive measure. We will assume that p ≥ 0 and a ≥ 0. In the following the parameters n and a are fixed. Therefore we drop them from the notation and for simplicity write n Sp = Sp,a ,

2 `p2 = `p,a ,

k · k p = k · k p,a .

P P The symplectic structure on Sp is 1≤ j≤n dx j ∧ d y j + j≥1 du j ∧ dv j . The equations of motion for N are therefore x˙ j = ω j (ξ ), y˙ j = 0,

u˙ j =  j (ξ )v j , v˙ j = − j (ξ )u j .

For each parameter ξ ∈ 5 there is an invariant torus T0 = T0n = Tn × {0} × {0} × {0}, on which the flow is rotational with internal frequencies ω(ξ ) = (ω1 (ξ ), . . . , ωn (ξ )). In the normal space described by the u, v-coordinates we have an infinite dimensional elliptic equilibrium at the origin, whose characteristic external frequencies are (ξ ) = (1 (ξ ), 2 (ξ ), . . . ). Hence, T0 is an invariant, rotational, linearly stable n-torus. Our aim is to prove the persistence of this torus under small perturbations N + P of the integrable Hamiltonian N for a large Cantor set of parameter values ξ . To this end we make the following three assumptions. Of these, assumption A* differs slightly from the corresponding assumption A in chapter IV in order to simplify minor technical points later on. See the remark following Theorem 18.1 below. Assumption A*: Frequency Asymptotics. There exist reals d > 1 and δ < d − 1 such that the following holds. First, i −  j ≥ m |i − j|(i d−1 + j d−1 ) (18.2) for all i 6 = j ≥ 0 uniformly on 5 with some constant m > 0. Here, 0 = 0. Second, the functions  j (ξ ) ξ 7→ jδ are uniformly Lipschitz on 5 for j ≥ 1. d Setting one of the indices to zero in (18.2) one has  j ≥ m j forδ j ≥ 1. On the other hand, the second assumption implies  j (ξ ) −  j (ζ ) ≤ c j for j ≥ 1 uniformly on 5. So the ξ -dependent part of the external frequencies is only of order δ, not d. We assume here that d > 1. The case d = 1 may also be handled, as it is done in [72, 109], but is somewhat more involved and not relevant for the application to the KdV equations. We therefore omit it.

18 Set Up and Summary of Main Results

147

Assumption B*: Nondegeneracy. The map ξ 7→ ω(ξ ) between 5 and its image ω(5) is a homeomorphism which is Lipschitz continuous together with its inverse. Moreover, for every k ∈ Zn and l ∈ Z∞ with 1 ≤ |l| ≤ 2 the resonance set  Rkl = ξ ∈ 5 : hk,ω(ξ )i + hl,(ξ )i = 0 P has Lebesgue measure zero. Here, |l| = j≥1 |l j |. Note that by assumption A*, the sets Rkl are empty for k = 0 and 1 ≤ |l| ≤ 2. The third assumption is concerned with the perturbing Hamiltonian P and its vector field, X P = (Py , −Px , Pv , −Pu )T . We use the notation i ξ X P for X P evaluated at ξ . With Sp,C we denote the complexification of the phase space Sp . Assumption C*: Regularity of Perturbation. There is a neighbourhood U p of T0 in Sp,C such that P is defined on U p × 5, and its Hamiltonian vector field defines a map X P : U p × 5 → Sq,C , where q ≥ 0 satisfies p − q < d − 1. Moreover, i ξ X P is real analytic on U p for each ξ ∈ 5, and i w X P is uniformly Lipschitz on 5 for each w ∈ U p . Without loss of generality we will also assume that δ ≥ 0 is chosen so that p − q ≤ δ < d − 1. To state the KAM theorem we need to introduce some domains and norms. For s > 0 and r > 0 we introduce the complex T0 -neighbourhoods D(s, r ) = { |Im x| < s } × { |y| < r 2 } × { kuk p + kvk p < r } 2 2 ⊂ Cn × Cn × `p, C × `p,C = Sp,C . 2 2 Here, |z| = max j |z j | for vectors in Cn , and `p, C is the complexification of `p . On Sq,C we introduce for W = (Wx , W y , Wu , Wv ) the weighted norm

kW kr,q = |Wx | +

1 1 1 W y + kWu kq + kWv kq . 2 r r r

For a map W : U × 5 → Sq,C , such as the Hamiltonian vector field X P , we then define the norms sup

kW kr,q;U ×5 = lip kW kr,q;U ×5

sup

(w,ξ )∈U ×5

= sup

ξ,ζ ∈5 ξ 6=ζ sup

kW (w, ξ )kr,q ,



1ξ ζ W sup r,q;U |ξ − ζ |

,

where 1ξ ζ W = i ξ W − i ζ W and ki ξ W kr,q;U = supw∈U kW (w, ξ )kr,q .

148

V The KAM Proof

In a completely analogous manner, the Lipschitz semi-norm of the external frequencies  is defined as 1ξ ζ  j −δ 1ξ ζ  j lip −δ ||−δ;5 = sup = sup sup . |ξ − ζ | ξ,ζ ∈5 |ξ − ζ | ξ,ζ ∈5 j≥1 ξ 6=ζ

ξ 6=ζ

Here and later, |λ|β = sup j≥1 j β |λ j | for sequences λ = (λ1 , λ2 , . . . ). Finally, we introduce two more constants. Assuming that assumptions A* and B* hold, there exist constants M and L with −1 lip lip lip ω |ω|5 + ||−δ;5 ≤ M < ∞, ≤ L < ∞. ω(5) The bound L will only be needed in section 22. Theorem 18.1 (KAM Theorem). Suppose N is a family of Hamiltonians of the form (18.1) defined on a phase space Sp and depending on parameters in 5 so that assumptions A* and B* are satisfied. Then there exists a positive constant γ depending only on n, d, δ, m, the frequencies ω and  and s > 0 such that for every perturbation H = N + P of N that satisfies assumption C* and the smallness condition α lip sup kX P kr,q;D(s,r )×5 ≤ αγ ε = kX P kr,q;D(s,r )×5 + M for some r > 0 and 0 < α < 1, the following holds. There exist (i) a Cantor set 5α ⊂ 5 with meas(5 X 5α ) → 0 as α → 0, (ii) a Lipschitz family of real analytic torus embeddings 8 : Tn × 5α → Sp , (iii) a Lipschitz map ϕ : 5α → Rn , such that for each ξ ∈ 5α the map 8 restricted to Tn × {ξ } is a real analytic embedding of a rotational torus with frequencies ϕ(ξ ) for the perturbed Hamiltonian H at ξ . In other words, t 7→ 8(θ + tϕ(ξ ), ξ ),

t ∈ R,

is a real analytic, quasi-periodic solution for the Hamiltonian i ξ H for every θ ∈ Tn and ξ ∈ 5α . Moreover, each embedding is real analytic on D(s/2) = {|Im x| < s/2}, and sup

α cε lip k8 − 80 kr, p;D(s/2)×5α ≤ , M α α sup lip |ϕ − ω|5α + |ϕ − ω|5α ≤ cε, M

k8 − 80 kr, p;D(s/2)×5α +

where 80 : Tn × 5 → T0 ,

(x, ξ ) 7→ (x, 0, 0, 0)

is the trivial embedding for each ξ , and c is a positive constant which depends on the same parameters as γ .

18 Set Up and Summary of Main Results

149

Remark 1. The smallness condition on γ is made more explicit through equations (19.8), (21.6) and (22.2). Remark 2. For a more explicit description of the Cantor set 5α we refer to section 22. Remark 3. The formulation of the KAM theorem here and in section 16 is identical, only the assumptions A and A* differ slightly. However, assuming assumptions A and B in section 16 one verifies that for any given η > 0 one can remove from 5 a subset 5η with meas(5η ) < η, such that on 5 X 5η assumption A* is satisfied with some constant m > 0. Therefore, the KAM theorem of section 16 follows from the theorem given here. We now give an outline of the proof of the theorem. We focus on the case q < p, that is, the case of unbounded perturbations. The case q ≥ p is much simpler and has been dealt with in [72, 109]. The proof of Theorem 18.1 employs the rapidly converging iteration scheme of Newton type to handle small divisor problems introduced by Kolmogorov, and involves an infinite sequence of coordinate transformations. At the ν-th step of the scheme, a Hamiltonian Hν = Nν + Pν is considered, which is a small perturbation of some normal form Nν . A transformation 8ν is set up so that Hν B 8ν = Nν+1 + Pν+1 with another normal form Nν+1 and a much smaller error term Pν+1 . For instance, kPν+1 k ≤ Cν kPν kκ for some κ > 1. This transformation is found by linearizing the above equation. Repetition of this process leads to a sequence of transformations 80 , 81 , . . ., whose infinite composition transforms the initial Hamiltonian H0 into a normal form N∞ up to a certain order. To describe the construction in more detail, let us drop the index ν. First we write H = N + P = N + R + (P − R), where R is obtained from P by truncating its Fourier and Taylor series expansion in a suitable way. The coordinate transformation 8 is written as the time-1-map of the flow X tF of a Hamiltonian vectorfield X F : 8 = X tF t=1 . Then 8 is symplectic. Moreover, we may expand H B 8 = H B X tF t=1 with respect to t at 0 using Taylor’s formula. Recall that d G B X tF = {G,F} B X tF . dt

150

V The KAM Proof

Thus we may write (N + R) B 8 = N B X tF t=1 + R B X tF t=1 Z 1 = N + {N ,F} + (1 − t) {{N ,F},F} B X tF dt 0

Z

1

+R+ 0

{R,F} B X tF dt

= N + R + {N ,F} +

Z 0

1

{R + (1 − t) {N ,F},F} B X tF dt.

The latter integral is of quadratic order in R and F and will be part of the new error term. The point is to find F such that N + R + {N ,F} = N+ is again a normal form, where ‘+’ is short for ‘ν + 1’. Equivalently, setting N+ = N + Nˆ , this amounts to solving the linear equation {F ,N } + Nˆ = R for F and Nˆ , when R is given. Suppose such a solution exists. Then (1 − t) {N ,F} + R = (1 − t) Nˆ + t R, and hence H B 8 = N+ + P+ = N+ + Q + (P − R) B 8 with 1

Z Q= 0

{(1 − t) Nˆ + t R,F} B X tF dt.

The term Q is of quadratic order in R, F and Nˆ . What makes this scheme more complicated than previous ones is the fact that the vector field X R is unbounded, whereas the vector field X F has to be bounded to generate a bona fide coordinate transformation. For most terms in F this presents no problem, because they are obtained from the corresponding terms in R by division with a large divisor. There is no such smoothing effect, however, for terms in R of the form X R j (x; ξ )(u 2j + v 2j ). j≥1

So we include them in Nˆ and hence in the new normal form N+ . Subsequently, however, we have to deal with generalized normal forms X N = hω(ξ ),yi + 21  j (x; ξ )(u 2j + v 2j ), j≥1

where now the coefficients  j in general depend on the angular variables x.

18 Set Up and Summary of Main Results

151

The linearized equation {F ,N } + Nˆ = R then leads to the following type of first order partial differential equation for functions on the torus Tn : −i∂ω u + λu + b(x)u = f,

x ∈ Tn ,

(18.3)

P where ∂ω = nj=1 ω j ∂x j . To obtain the required estimates for its solution we need to make the following three assumptions. Assumption U. The frequencies ω are diophantine in the sense that there are constants α > 0, τ > n and l > 0 such that |λ| ≥ αl and αl , |k|τ α |hk,ωi| ≥ τ , |k|

|hk,ωi + λ| ≥

for all 0 6 = k ∈ Zn , where |k| = |k1 | + · · · + |kn |. Assumption V. The function b is analytic on some strip D(s) = { |Im x| < s } around the torus Tn with [b] = 0 and def

kbks,τ =

X

|bˆk | |k|τ e|k|s ≤ γ α

k

for b =

P

k

bˆk eihk,xi with some γ > 0 and the same τ as before.

Assumption W. The function f is analytic on the same strip D(s) with def

k f ks = k f ks,0 < ∞. Lemma 18.2 (Kuksin [74]). Under assumptions U, V, W, equation (18.3) has a unique solution u that is analytic on D(s) and satisfies kuks−σ ≤ Moreover, if

ce2γ k f ks , αlσ τ

0 < σ ≤ s.

γ σ ≤ with |ω| = max |ωi |, then also λ 5 |ω| kuks−σ ≤

c k f ks , αlσ τ +n+3

0 < σ ≤ s.

The constant c can be chosen so that it only depends on τ in the first case, and only on τ and n in the second case. The second estimate is the difficult one. It is needed to obtain a uniform bound for kuks−σ for solutions of a family of equations of type (18.3) with no uniform bound on γ , but only a uniform bound on γ /λ.

152

V The KAM Proof

19 The Linearized Equation We consider the linearized equation mentioned above, {F ,N } + Nˆ = R. √ √ Using convenient complex notation z = (u − iv)/ 2 and z¯ = (u + iv)/ 2, where, however, u and v are complex, the generalized normal form reads N = hω(ξ ),yi + h(x; ξ ),z z¯ i. N is assumed to be regular on the domain D(s, r ) × 5 in the following sense: for each ξ ∈ 5, i ξ N is real analytic on D(s, r ), and for each w ∈ D(s, r ), i w N is Lipschitz on 5, as required in assumption C* of the KAM theorem. The right hand side R is also assumed to be regular on D(s, r ) × 5 and of the form X R= Rmn n¯ (x; ξ )y m z n z¯ n¯ (19.1) 2|m|+|n+n|≤2 ¯

in usual multi-index notation, where momentarily m, n and n¯ also denote multiindices. Hence, if we define deg(y m z n z¯ n¯ ) = 2 |m| + |n + n|, ¯

(19.2)

then R is a polynomial in y, z, z¯ of degree 2 with coefficients depending regularly on x and ξ . Moreover, the Hamiltonian vector field associated with R is assumed to define a regular map X R : Sp,C → Sq,C ,

p − q ≤ δ.

Note that X H = (∂ y H, −∂x H, i∂z¯ H, −i∂z H )T is the vector field of a Hamiltonian H in the complex notation. The mean value of a function u on Tn is defined as usual by Z 1 u(x) dx. [u] = (2π)n Tn The part of R in generalized normal form is defined as X X hRi = [R000 ] + R0n n¯ (x; ξ )z n z¯ n . [Rm00 ]y m + |m|=1

|n|=1

In the sequel we omit the term [R000 ], since it only depends on ξ and does not affect the dynamics. Note that in hRi the coefficients R0n n¯ are not averaged over Tn . Given N and R, we now seek a solution Nˆ and F of the linearized equation of ¯ + , ˜ where the same form as N and R, respectively. To this end, write  =  ¯ = [], 

˜ =  − [], 

19 The Linearized Equation

153

are the x-independent and x-dependent parts of the exterior frequencies , respec¯ and a smallness tively. We will impose small divisor and growth conditions on  ˜ condition on . To formulate these we define  X X  hki = max(1, |k|), jl j · j δ l j [l]δ = max 1, j≥1

j≥1

for k ∈ Zn and l ∈ Z∞ . For example, ( j 1+δ , l = ej, [l]δ = δ δ |i − j|(i + j ), l = ei − e j 6 = 0

.

Moreover, for a function u analytic on D(s) we set X uˆ k |k|τ e|k|s , kuks,τ = k∈Zn

as in assumption V above. Observe that by standard estimates for Fourier coefficients, c sup kuks,τ ≤ τ +n |u| D(s+σ ) (19.3) σ for σ > 0 with a constant c depending only on n. Lemma 19.1. Suppose that uniformly on 5, [l]δ , hkiτ ¯ )i| ≥ m [l]d−1 , |hl,(ξ

˜ j ≤ αγ0 , j −δ  s,τ

¯ )i| ≥ α |hk,ω(ξ )i + hl,(ξ

k 6= 0, |l| ≤ 2 0 < |l| ≤ 2,

(19.4)

j ≥ 1,

with constants τ ≥ n, d > 1, 0 < γ0 ≤ 1, m > 0, and a parameter 0 < α ≤ m. If γ0 is sufficiently small, then the linearized equation {F ,N } + Nˆ = R has a solution F, Nˆ , which is unique with the normalization hFi = 0, h Nˆ i = Nˆ , is regular on D(s, r ) × 5 in the above sense, and satisfies for 0 < σ < s the estimates sup

Bσ sup kX R kr,q;D(s,r )×5 , α   Bσ M lip sup kX k kX k ≤ R r,q;D(s,r )×5 + R r,q;D(s,r )×5 , α α

kX F kr, p;D(s−σ,r )×5 ≤ lip

kX F kr, p;D(s−σ,r )×5 and sup

sup

kX Nˆ kr,q;D(s−σ,r )×5 ≤ Bσ kX R kr,q;D(s,r )×5 ,   M lip lip sup kX Nˆ kr,q;D(s−σ,r )×5 ≤ Bσ kX R kr,q;D(s,r )×5 + kX R kr,q;D(s,r )×5 . α

154

V The KAM Proof lip

lip

Here, M = |ω|5 + ||−δ;D(s)×5 , and Bσ =

 c 1/β  a exp , b σ σ

1 δ = , β d −1−δ

(19.5)

with constants a, b, c depending only on n and τ . The crucial, somewhat hidden feature of the first two of these estimates is the ‘ p’ on their left hand sides and the ‘q’ on their right hand sides. That is, the solution X F is bounded in the stronger norm k · kr, p rather than k · kr,q . We also note that Bσ is increasing fast as σ tends to zero, but not too fast for a subsequent iteration of Newton type to work. This kind of estimates also occur if one uses the more general approximation functions of R¨ussmann [19, 116, 117] rather then the usual diophantine conditions – see for example [108]. Proof. Decompose R = R 0 + R 1 + R 2 , where R j comprises all terms in the expansion of R with |n + n| ¯ = j. Decompose similarly F, N and Nˆ , where neces1 1 ˆ sarily N = 0 and N = 0 by normalization. Comparing coefficients the linearized equation decomposes into  0 0 F ,N + Nˆ 0 = R 0 ,  1 F ,N = R1, (19.6)  2  0 2 2 2 F ,N + Nˆ = R − F ,N . We will see that with the chosen normalization and the diophantine conditions these equations determine Nˆ 0 , F 0 , F 1 and then Nˆ 2 , F 2 uniquely. The first equation is independent of z, z¯ and amounts to the classical, finitedimensional pde X ∂ω F 0 + Nˆ 0 = R 0 , ∂ω = ωi ∂xi . 1≤i≤n

This leads to Nˆ 0 = [R 0 ] and ∂ω F 0 = R 0 − [R 0 ] with [F 0 ] = 0. Their estimates are standard and of the same form – indeed much better – than the ones for F 1 , F 2 and Nˆ 2 obtained below. For later reference we record that

X

sup

F 0 r, p;D(s−2σ,r )

X

lip

F 0 r, p;D(s−4σ,r ) lip

Bσ sup kX R kr,q;D(s,r ) , α   Bσ M lip sup kX R kr,q;D(s,r ) + kX R kr,q;D(s,r ) , ≤ α α ≤

lip

where M = |ω|5 + ||−δ;D(s)×5 and Bσ as in (19.5). We drop the factor 5 from the domains during this proof for brevity, since it stays the same throughout. Note that X F 0 does not have any z, z¯ -component, so kX F 0 kr, p does not depend on p. Consider the second equation in (19.6). Writing R 1 = R 10 + R 01 = hR10 ,zi + hR01 ,¯z i

19 The Linearized Equation

and similarly F 1 it decomposes into  ij F ,N = R i j ,

155

i + j = 1,

and it suffices to study each equation individually. We have R10 = Rz z,¯z =0 and thus

1 sup

R10 sup ≤ kX R kr,q;D(s,r ) , q;D(s) r P where D(s) = { |Im x| < s }. Writing R 10 = hR10 ,zi = j≥1 R j (x; ξ )z j , and similarly F 10 , the equation {F 10 ,N } = R 10 further decomposes into ¯ j Fj +  ˜ j F j = iR j , i∂ω F j +  j F j = i∂ω F j + 

j ≥ 1.

˜j = ˜ j (x; ξ ). To Here we did not indicate the dependence on x and ξ . Recall that  obtain a solution of these equations with useful estimates we want to apply Kuksin’s Lemma of section 23. The assumptions of this lemma are now verified. By the diophantine condition in (19.4) we have uniformly on 5 α , hkiτ α j 1+δ hk,ωi +  ¯ j ≥ , hkiτ |hk,ωi| ≥

k 6= 0, k ∈ Zn , j ≥ 1.

¯ j and l j = j 1+δ in place So assumption U of Kuksin’s lemma is satisfied with λ =  ˜ = 0, and by the smallness condition in (19.4) we have of l. Moreover, []



 ˜ j ≤ αγ0 j δ , j ≥ 1, s,τ uniformly on 5. So assumption V of Kuksin’s lemma is satisfied with γ j = γ0 j δ ¯ j ≥ m j d for in place of γ . Finally, by the growth condition in (19.4), we have  j ≥ 1, and one verifies that  ¯ j ≥ |ω|sup γ 1+β , j ≥ 1, 5 j with β = (d − 1 − δ)/δ, if γ0 in the definition of γ j is sufficiently small. An explicit and stronger condition on γ0 is given in (19.8) below. Thus, Kuksin’s lemma and its corollaries apply, and so the unique solution F j satisfies the estimate sup Fj

D(s−2σ )

≤

Bσ sup R j D(s−σ ) , αl j

j ≥ 1.

Here and in the following, Bσ stands for a function of σ of the form (19.5) with various constants a, b, c, which only depend on n and τ , but not on j. Since l j = j 1+δ and p − q ≤ δ this and Lemma M.2 imply

10 sup

Bσ Bσ sup

F

R10 sup ≤ ≤ r kX R kr,q;D(s,r ) . p;D(s−2σ ) q;D(s) α α

156

V The KAM Proof

The same estimate holds for F 01 . Multiplying F 10 with z and F 01 with z¯ and using p ≥ 0 this gives 1 1 sup Bσ sup kX R kr,q;D(s,r ) , F D(s−2σ,r ) ≤ α r2 and finally with Cauchy’s estimate

Bσ sup

X 1 sup kX R kr,q;D(s,r ) . F r, p;D(s−3σ ) ≤ α To obtain Lipschitz estimates we study first the differences 1F j = i ξ F j − i ζ F j for ξ, ζ ∈ 5. We obtain i1R j = i∂ω 1F j +  j 1F j + i∂1ω F j + F j 1 j from (19), hence i∂ω 1F j +  j 1F j = i1R j − i∂1ω F j − F j 1 j , ¯ j + ˜ j . The right hand side is known, so 1F j uniquely solves the same with  j =  kind of equation as F j . So we obtain    sup sup  1 sup Bσ 1F j sup |1ω| 1R + F + 1 ≤ j D(s−σ ) j D(s−2σ ) j D(s) D(s−3σ ) αl j σ  sup sup  Bσ B 2 sup ≤ 1R j D(s−σ ) + σ2 R j D(s−σ ) |1ω| + 1 j D(s) . αl j α 2l j Multiplying by l j = j 1+δ and going to the vector norms for F 10 with the help of Lemma M.2 using p − q ≤ δ, we obtain

Bσ

1R10 sup

1F 10 sup ≤ q;D(s) p;D(s−3σ ) α  2

sup  B sup + σ2 R10 q;D(s) |1ω| + |1|−δ;D(s) . α Dividing by |ξ − ζ | 6 = 0 and taking the supremum over 5,  

10 lip M Bσ2

R10 lip

R10 sup

F ≤ + . q;D(s) q;D(s) p;D(s−3σ ) α α The same estimate applies to F 01 . So for the vector field of F 1 we finally get  

Bσ2 M lip sup

X 1 lip kX R kr,q;D(s,r ) + kX R kr,q;D(s,r ) . F r, p;D(s−4σ ) ≤ α α This concludes the discussion of F 1 . Consider now the third equation in (19.6). Writing R 2 = R 20 + R 11 + R 02 and similarly F 2 and N 2 this equation decomposes into  ij



  F ,N = R i j − R i j , Nˆ 11 = R 11 − F 0 ,N 2 , while Nˆ i j = 0 for i 6 = j.

19 The Linearized Equation

157

Consider the equation for F 11 , which is slightly more complicated than the ones for F 20 and F 02 . Writing R 11 = hR11 z,¯z i, we have R11 = Rz z¯ |z,¯z =0 . Thus R11 is the Jacobian of Rz with respect to z¯ at z¯ = 0. By Cauchy’s inequality we have an estimate in the induced operator norm:

11 sup 1 sup sup

R ≤ kRz kq;D(s,r ) ≤ kX R kr,q;D(s,r ) , q, p;D(s) r where k · kq, p denotes the operator norm induced by k · k p and k · kq in the source and target spaces, respectively. Now write more explicitly X R 11 = Ri j (x; ξ )z i z¯ j , i, j≥1

and similarly F 11 . The equation {F i j ,N } = R i j − hR i j i decomposes into i∂ω Fi j + (i −  j )Fi j = iRi j ,

i 6 = j,

and i∂ω F j j = 0. The normalization hF 11 i = 0 enforces F j j = 0, so it suffices to ¯ ij +  ˜ i j with  ¯ i j = [i j ] it consider the first equation. Letting i j = i −  j =  becomes ¯ i j Fi j +  ˜ i j Fi j = iRi j , i∂ω Fi j +  i 6= j, to which we apply Kuksin’s lemma. To this end we now verify assumptions U, V and W. Again, by the diophantine condition in (19.4) we have |hk,ωi| ≥

α , hkiτ

αl hk,ωi +  ¯ ij ≥ ij , hkiτ

(19.7)

for k 6 = 0 uniformly on 5 with li j = |i − j|(i δ + j δ ). So assumption U is satisfied ¯ i j and li j in place of l. Moreover, we have [ ˜ i j ] = 0, and by the smallness with λ =  condition in (19.4),





 ˜ i j ≤ k ˜ i ks,τ +  ˜ j ≤ αγ0 (i δ + j δ ). s,τ s,τ So V is satisfied with γi j = γ0 (i δ + j δ ) in place of γ . Finally, we have assumption  ¯ i j ≥ m |i − j|(i d−1 + j d−1 ) by the growth condition in (19.4), and one finds  ¯ i j ≥ |ω|sup γ 1+β , 5

ij

if for example γ0 ≤ cβ

m sup , |ω|5

β=

with some constant cβ depending only on β.

d −1−δ , δ

(19.8)

158

V The KAM Proof

So Kuksin’s lemma and its corollaries apply, and we obtain sup Fi j

D(s−2σ )

≤

Bσ sup Ri j D(s−σ ) , αli j

or sup Bσ 1 sup (i δ + j δ ) Fi j D(s−2σ ) ≤ Ri j D(s−σ ) , α |i − j|

i 6= j.

With Lemma M.3 this yields

11 sup

sup

F , F 11 q,q;D(s−2σ ) p, p;D(s−2σ )

Bσ Bσ sup

R11 sup kX R kr,q;D(s,r ) . ≤ ≤ q, p;D(s) α α

(19.9)

The same, and even better estimates hold for F 20 and F 02 . Multiplying with z, z¯ we then get 1 2 sup Bσ sup kX R kr,q;D(s,r ) , F D(s−2σ,r ) ≤ α r2 and finally with Cauchy’s estimate

X

sup

F 2 r, p;D(s−3σ,r )

≤

Bσ sup kX R kr,q;D(s,r ) , α

in complete analogy to the estimate for X F 1 above. The estimate for the Lipschitz semi-norm of X F 2 is obtained by the same arguments as the one for X F 1 , and the result is analogous. We therefore omit it. For the contribution to the normal form we obtain X ˆ ˆ j (x; ξ )z j z¯ j Nˆ 2 = Nˆ 11 = h(x; ξ ),z z¯ i =  j≥1



  ˆ j (x; ξ ) = R j j +  ˜ j ,F 0 = R j j + ∂x  ˜ j ,∂ y F 0 . We find that with  sup sup sup R j j ≤ j p−q kX R kr,q;D(s,r ) ≤ j δ kX R kr,q;D(s,r ) D(s−2σ ) and

 ∂ F 0 ∂x  ˜ j ,∂ y F 0 ≤ ∂x  j D(s−2σ ) y D(s−2σ )



sup ≤  j X 0 r, p;D(s−2σ,r ) sup γ0 Bσ kX R kr,q;D(s,r ) . s,τ

≤ j

δ

F

sup ˆ j ≤ Bσ kX R kr,q;D(s,r ) , using (19.3). Together with the estiHence, j −δ  s−4σ,τ mate for Nˆ 0 = [R 0 ] this implies sup

sup

kX Nˆ kr,q;D(s−4σ,r ) ≤ Bσ kX R kr,q;D(s,r ) .

19 The Linearized Equation

159

As to the Lipschitz estimate we have



 ˆ j = 1R j j + ∂x 1 ˜ j ,∂ y F 0 + ∂x  ˜ j ,∂ y 1F 0 . 1 This gives 1 ˆ j sup

D(s−4σ )

sup sup sup 1 ≤ 1R j j D(s−4σ ) + 1 j D(s−3σ ) ∂ y F 0 D(s−4σ ) σ sup 1 sup +  j D(s−3σ ) 1∂ y F 0 D(s−4σ ) σ Bσ sup sup sup δ kX R kr,q;D(s,r ) ≤ j k1X R kr,q;D(s,r ) + j δ |1|−δ;D(s) α

αγ0

1X 0 sup + jδ F r, p;D(s−4σ,r ) . σ

Dividing by j δ and |ξ − ζ | 6 = 0, taking the supremum over 5 and using the estimate of kX F 0 klip above this gives lip

lip

ˆ || −δ;5 ≤ (1 + γ0 Bσ ) kX R kr,q;D(s,r ) M sup + (Bσ + γ0 Bσ ) kX R kr,q;D(s,r ) α   M sup lip kX R kr,q;D(s,r ) . ≤ Bσ kX R kr,q;D(s,r ) + α From these estimates the ones for X Nˆ are readily derived. The final estimates of the lemma are obtained by replacing σ by σ/4 throughout the proof. This affects only the constants a, b, c in the expression of Bσ and completes the proof. u t For later reference the estimates of Lemma 19.1 may be condensed as follows. For λ ≥ 0, define sup lip kX krλ = kX kr + λ kX kr . The symbol ‘λ’ in kX krλ will always be used in this role and never has the meaning of exponentiation. Lemma 19.2. The estimates of Lemma 19.1 imply that Bσ λ kX R kr,q;D(s,r )×5 , α λ ≤ Bσ kX R kr,q;D(s,r )×5 ,

λ kX F kr, p;D(s−σ,r )×5 ≤ λ kX Nˆ kr,q;D(s−σ,r )×5

for 0 < σ < s and 0 ≤ λ ≤ α/M with another Bσ of the same form as in Lemma 19.1. λ The preceding lemma also gives us an estimate of kD X F kr, p, p;D(s−2σ,r/2)×5 with the help of Cauchy’s estimate. However, for the estimate (20.4) below we also need an estimate in terms of the operator norm k · kr,q,q .

160

V The KAM Proof

Lemma 19.3. Under the assumptions of Lemma 19.1, λ λ kD X F kr, p, p;D(s−σ,r )×5 , kD X F kr,q,q;D(s−σ,r )×5 ≤

Bσ λ kX R kr,q;D(s,r )×5 . α

λ Proof. It remains to consider the estimate of kD X F kr,q,q . We have



 Fyx 0 0 0  −Fx x −Fx y −Fx z −Fx z¯  . DXF =   iFz¯ x 0 iFz¯ z iFz¯ z¯  −iFzx 0 −iFzz −iFz z¯ The lower right 2 × 2-block is independent of z and z¯ and defines a bounded operator on `q2 by the estimate (19.9). Since Fz , Fz¯ are in `p2 and p ≥ q ≥ 0, the operators Fx z , Fx z¯ are bounded in `2p by Cauchy’s estimate, and hence bounded in `q2 as well. As to the estimates we note that

kD X F kr,q,q = Wr−1 D X F Wr q,q , where Wr = diag(1, r 2 , r, r ) according to the above block decomposition. In other words, the weighted norm kD X F kr,q,q is obtained by first multiplying the entries of D X F with the corresponding entries in the matrix 1 r2  r −2 1   r −1 r r −1 r 

r

r



r −1 r −1   1 1  1 1

and then taking the norm k · kq,q . In all the cases where an x-derivative is involved, such as Fx z , the proper estimate of each block is now obtained from the corresponding estimate of the vector field X F , such as Fz , Cauchy’s estimate with respect to x, and the fact that `p2 ⊂ `q2 . In the other cases we use the estimate (19.9). Finally, replacing again σ by σ/2 we obtain the claimed supremum estimates. The Lipschitz estimates are obtained in a similar fashion. u t

20 The KAM Step At the general ν-th step of the iteration scheme we are given a Hamiltonian Hν = Nν + Pν , where Nν = hων (ξ ),yi + hν (x; ξ ),z z¯ i

20 The KAM Step

161

is a generalized normal form, and Pν is a small perturbation of it. Both are assumed to be regular on D(sν , rν ) × 5ν in the sense described on page 152, where 5ν is some closed bounded subset of Rn . On 5ν we assume that [l]δ , hkiτ ¯ ν (ξ )i| ≥ m ν [l]d−1 , |hl,

˜ ν, j j −δ  ≤ αν γ0 , s ,τ

¯ ν (ξ )i| ≥ αν |hk,ων (ξ )i + hl,

ν

k 6= 0, |l| ≤ 2 0 < |l| ≤ 2,

(20.1)

j ≥ 1,

¯ν + ˜ ν with  ¯ ν = [ν ]. Moreover, we assume that where we let ν =  lip

lip

|ων |5ν + |ν |−δ;D(sν )×5ν ≤ Mν . For the rest of this section we drop the subscript ‘ν’ from the notation, and write the symbol ‘+’ for ‘ν + 1’ for simplicity. Thus, P = Pν , P+ = Pν+1 , and so on. Also, we write u l v, if u ≤ cv with some constant c ≥ 1, which only depends on n and τ . To perform one step of the iteration we assume that sup

kX P kr,q;D(s,r )×5 +

αη2 α lip kX P kr,q;D(s,r )×5 ≤ 0 M Bσ

with some 0 < η < 1/16 and 0 < σ < s, σ ≤ 1, where Bσ0 is a function of σ of the form (19.5) with certain sufficiently large constants a, b, c depending only on n and τ , which could be made explicit. Using the notation of Lemma 19.2 the smallness condition may also be written as λ kX P kr,q;D(s,r )×5 ≤

αη2 Bσ0

(20.2)

for 0 ≤ λ ≤ α/M. – From now on we also drop the factor ‘5’ from the notation of domains, since it stays the same throughout this section. Approximation of P We approximate P by its Taylor polynomial R of order 2 in y, z, z¯ in the sense of (19.2), which is of the form (19.1). This amounts to corresponding approximations of the partial derivatives Px , Py , Pz , Pz¯ , which constitute the vector field X P . Since P is real analytic, Rx , R y , Rz , Rz¯ and their remainders are given by certain Cauchy integrals, which we can estimate exactly as in a finite dimensional setting. We therefore obtain λ λ kX R kr,q;D(s,r ) l kX P kr,q;D(s,r ) , (20.3) λ kX P − X R kληr,q;D(s,4ηr ) l η kX P kr,q;D(s,r ), for 0 ≤ λ ≤ α/M and 0 < η < 1/16.

162

V The KAM Proof

Solution of the Linearized Equation The hypotheses (20.1) are of exactly the same form as those of Lemma 19.1 for the linearized equation. Hence, if γ0 is sufficiently small, Lemma 19.1 applies, and we can solve {F ,N } + Nˆ = R. With Lemmas 19.2 and 19.3 and estimates (20.3) we obtain λ λ kX Nˆ kr,q;D(s−σ,r ) ≤ Bσ kX P kr,q;D(s,r ) , (20.4) Bσ λ λ kX F kr, kX P kr,q;D(s,r p;D(s−σ,r ) ≤ ), α λ for 0 ≤ λ ≤ α/M. The bound in the second line also holds for kD X F kr, p, p;D(s−σ,r ) λ and for kD X F kr,q,q;D(s−σ,r ) . Together with the smallness assumption (20.2) we get λ λ 2 kX F kr, p;D(s−σ,r ) , kD X F kr,q,q;D(s−σ,r ) ≤ η

Bσ . Bσ0

(20.5)

Coordinate Transformation We now choose Bσ0 in such a way that the preceding estimate in particular gives sup

kX F kr, p;D(s−σ,r ) ≤ η2

Bσ η2 σ ≤ 0 c0 Bσ

with some suitable constant c0 ≥ 1. Then the flow X tF of the vector field X F exists on D(s − 2σ, r/2) for −1 ≤ t ≤ 1 and takes this domain into D(s − σ, r ). Similarly, it takes D(s − 3σ, r/4) into D(s − 2σ, r/2). Together with Lemma M.4 we obtain

t

λ

X − id λ l kX F kr, F p;D(s−σ,r ) , r, p;D(s−2σ,r/2)

λ

D X t − I λ l kD X F kr,q,q;D(s−σ,r F ), r,q,q;D(s−3σ,r/4)

(20.6)

for 0 ≤ λ ≤ α/M. The latter estimate also holds in the k · kr, p, p -norm, which we do not write down explicitly. In particular, we notice that X tF for −1 ≤ t ≤ 1 is a symplectic transformation, which transforms Hamiltonian vector fields of order p − q according to the transformation rule of appendix K, since D X tF is a bounded linear operator both on TSp,C and TSq,C . The New Hamiltonian Subjecting the Hamiltonian H = N + P defined on D(s, r ) to the symplectic trans formation 8 = X tF t=1 we obtain the new Hamiltonian H B 8 = N+ + P+ on D(s − σ, r/2), where N+ = N + Nˆ and Z 1 {R(t),F} B X tF dt, P+ = (P − R) B X 1F + 0

20 The KAM Step

163

with R(t) = (1 − t) Nˆ + t R. Hence, as X tF is symplectic, X P+ =

(X 1F )∗ (X P

− X R) +

Z 0

1

  (X tF )∗ X R(t) , X F dt.

We will show below that for any vector field Y ,

t ∗ λ

(X ) Y l kY kληr,q;D(s−2σ,4ηr ) F ηr,q;D(s−4σ,ηr )

(20.7)

for 0 ≤ t ≤ 1. We already estimated kX P − X R kληr,q in (20.3), so it remains to consider the commutator [X R(t) , X F ]. First, we have λ λ λ kX R(t) kr,q;D(s−σ,r ) ≤ kX Nˆ kr,q;D(s−σ,r ) + kX R kr,q;D(s−σ,r ) λ ≤ Bσ kX P kr,q;D(s,r )

by (20.3) and (20.4). Moreover, we have the pointwise estimate







X R(t) , X F ≤ D X R(t) · X F + D X F · X R(t) r,q r,q r,q



≤ D X R(t) r,q, p kX F kr, p + kD X F kr,q,q X R(t) r,q . By the product rule for Lipschitz-norms and Cauchy’s estimate we thus obtain

 

X R(t) , X F λ r,q;D(s−2σ,r/2)

λ λ l D X R(t) r,q, p;D(s−2σ,r/2) kX F kr, p;D(s−2σ,r/2)

λ λ + kD X F kr,q,q;D(s−2σ,r/2) X R(t) r,q;D(s−2σ,r/2) 2 Bσ  λ kX P kr,q;D(s,r l ) α for 0 ≤ λ ≤ α/M. Hence, also

 

X R(t) , X F λ

ηr,q;D(s−2σ,r/2)

  1

X R(t) , X F λ r,q;D(s−2σ,r/2) η2  2 Bσ λ l 2 kX P kr,q;D(s,r ) . αη ≤

Together with the estimate of X P − X R in (20.3) and with (20.7) we finally arrive at the estimate

X P λ +

λ l η kX P kr,q;D(s,r )+ ηr,q;D(s−4σ,ηr )

2 Bσ  λ kX k P r,q;D(s,r ) αη2

for 0 ≤ λ ≤ α/M. This is the bound for the new perturbation.

(20.8)

164

V The KAM Proof

Proof of Estimate (20.7) Fix −1 ≤ t ≤ 1 and set 8 = X tF . Consider the pull back 8∗ Y = D8−1 Y B 8. In view of the estimate for X tF − id, 8 maps the domain U = D(s − 4σ, ηr ) into the domain V = D(s − 3σ, 2ηr ). Hence,

∗ sup

sup

8 Y

D8−1 sup kY kηr,q;V , ≤ ηr,q;U ηr,q,q;V and by Lemma M.4,

D8−1 sup

ηr,q,q;V

≤1+

1

D X −t − I sup l1 F r,q,q;V η2 sup

sup

by (20.5) and (20.6). So we have k8∗ Y kηr,q;U l kY kηr,q;V . For the Lipschitz semi-norm we have to take into account that both 8 and Y depend on the parameters in 5. Therefore,

sup

sup

18∗ Y sup ≤ 1D8−1 ηr,q,q;U kY B 8kηr,q;U ηr,q;U

sup sup + D8−1 ηr,q,q;U k1(Y B 8)kηr,q;U

sup sup l 1(D8−1 − I ) ηr,q,q;U kY kηr,q;V sup

sup

sup

+ k1Y kηr,q;V + kDY kηr,q, p;V k18kηr, p;U

sup 1 sup l 2 1(D8−1 − I ) r,q,q;U kY kηr,q;V η 1 sup sup sup + k1Y kηr,q;V + kY kηr,q, p;W k18kηr, p;U σ with W = D(s − 2σ, 4ηr ) by Cauchy’s estimate. Dividing by |ξ − ζ |, taking the supremum over 5 and using the Lipschitz estimates for X tF and D X tF from (20.6) we obtain

∗ lip M sup lip

8 Y kY kηr,q;W . l kY kηr,q;V + ηr,q;U α From these estimates (20.7) follows. The New Generalized Normal Form This is N+ = N + Nˆ = hω+ (ξ ),yi + h+ (x; ξ ),z z¯ i ˆ For Nˆ , defined similarly in terms of ωˆ and , ˆ with ω+ = ω + ωˆ and + =  + . we have the estimate λ λ kX Nˆ kr,q;D(s−σ,r ) ≤ Bσ kX P kr,q;D(s,r ) sup

for 0 ≤ λ ≤ α/M. The weighted norm implies that we have |ω| ˆ ≤ kX Nˆ kr,q and

21 Iteration and Convergence

165

ˆ q ≤ r kX Nˆ ksup on D(s, r ), and consequently || ˆ q− p ≤ kX Nˆ ksup . The same kzk r,q r,q holds for the Lipschitz semi-norms. Since p − q ≤ δ we obtain λ ˆ λ−δ;D(s−σ ) ≤ Bσ kX P kr,q;D(s,r |ω| ˆ λ5 + || )

(20.9)

for 0 ≤ λ ≤ α/M. In order to control the assumptions of the KAM step for the iteration we notice that the last estimate also implies

sup ˆ j sup j −δ  ≤ Bσ kX P kr,q;D(s,r ) s−2σ,τ with a Bσ with different constants a, b, c using (19.3). Moreover, sup

sup

ˆ ˆ |hl,i| D(s−σ ) ≤ |l|δ ||−δ;D(s−σ ) λ ≤ [l]d−1 Bσ kX P kr,q;D(s,r )

≤ mˆ [l]d−1 , λ if Bσ kX P kr,q;D(s,r ˆ Finally, for k 6 = 0, ) ≤ m.

ˆ ≤ |k| |ω| ˆ −δ |hk,ωi ˆ + hl,i| ˆ + [l]d−1 || λ ≤ |k| [l]d−1 · Bσ kX P kr,q;D(s,r )

(20.10)

[l]d−1 ≤ αˆ , hkiτ provided λ Bσ kX P kr,q;D(s,r ) ≤

αˆ hkiτ +1

.

In the next section we will make sure that mˆ  m and αˆ  α, so that conditions (20.1) are essentially preserved.

21 Iteration and Convergence To iterate the KAM step infinitely often we now choose sequences for all its parameters. The guiding principle is to choose η so as to minimize the bound on the next perturbation, to keep α, m and M essentially constant, and to keep close track of the size ε of X P . First we make some heuristic considerations. We define Bσ0 , which is used in the smallness condition (20.2) of the KAM step, to be the largest among all functions Bσ encountered during the KAM step. Further, we define c0 to be twice the largest of all the constants c that are implicit in the ‘l’-notation. To make both terms in the bound for X Pν+1 of equal size, namely

X P

ν+1

λν

ην rν ,q

l ην εν +

Bσ0ν

αν ην2

εν2 ,

166

V The KAM Proof

we choose ην by ην3 =

Bν0 εν , αν

Bν0 = Bσ0ν .

One then obtains εν+1 =

c0 2

   κ−1 B0 Bν ην εν + ν 2 εν2 = c0 ην εν = ενκ , αν αν ην

where κ = 4/3 and Bν = c03 Bν0 = c03 Bσ0ν .

(21.1)

Note that Bν is different from Bσ . Iteration of the identity for εν+1 then gives κ−1 ν  ν−1 Y  Bµ  κ µ+1 κ εν = ε0 . αµ

(21.2)

µ=0

We will choose Bν and αν so that the product in this expression increases with ν. Further, choosing σν ∼ σ0 /ν 2 and αν ≥ α0 /2 the product will converge as ν → ∞. So the scheme will converge exponentially fast, if we roughly choose κ−1

ε0 <

 ∞  Y αµ κ µ+1 µ=0

Bµ

.

We now give the precise set up. For ν ≥ 0 let αν =

α0 (1 + 2−ν ), 2

and σν =

mν =

σ0 , 1 + ν2

m0 (1 + 2−ν ), 2

σ0 =

s0 , 20

Mν = M0 (2 − 2−ν )

sν+1 = sν − 4σν ,

so that s0 > s1 > s2 > · · · > s0 /2. Define Bν and εν by (21.1) and (21.2), and let ην3 =

Bν0 εν , αν

rν+1 = ην rν .

This defines the domains Dν = D(sν , rν ). Finally, we let λν =

αν , Mν

ν

K ν = K 0κ ,

1 − τ +1

K 0 = γ0

,

where κ = 4/3, and γ0 ≤ 1 is the small parameter of Lemma 19.1 appearing in (19.4) and (20.1) and required to satisfy (19.8).

21 Iteration and Convergence

167

Lemma 21.1 (Iterative Lemma). Suppose that ε0 ≤

∞ κ−1 α0 γ0 Y − κ µ+1 Bµ , 4

α0 ≤

µ=0

m0 . 2

(21.3)

Suppose Hν = Nν + Pν is regular on Dν × 5ν , where Nν is a generalized normal lip lip form with coefficients satisfying |ων |5ν + |ν |−δ;Dν ×5ν ≤ Mν and [l]δ , hkiτ ¯ ν (ξ )i| ≥ m ν [l]d−1 , |hl,

˜ ν, j j −δ  ≤ (α0 − αν )γ0 , s ,τ

¯ ν (ξ )i| ≥ αν |hk,ων (ξ )i + hl,

ν

on 5ν , and Pν satisfies



X P λν ν

rν ,q;Dν ×5ν

k 6= 0, |l| ≤ 2 0 < |l| ≤ 2,

(21.4)

j ≥ 1,

≤ εν .

Then there exists a Lipschitz family of real analytic symplectic coordinate transformations 8ν+1 : Dν+1 × 5ν → Dν and a closed subset [ ν+1 Rkl (αν+1 ) ⊂ 5ν , 5ν+1 = 5ν X |k|>K ν |l|≤2

where ν+1 (α) = Rkl



¯ ν+1 (ξ )i| < α ξ ∈ 5ν : |hk,ων+1 (ξ )i + hl,

 [l]δ , hkiτ

such that for Hν+1 = Hν B 8ν+1 = Nν+1 + Pν+1 the same assumptions as above are satisfied with ‘ν + 1’ in place of ‘ν’. In comparing assumptions (20.1) and (21.4) note that α0 − αν ≤ αν . Proof. In view of the definition of ην , namely ην3 = Bν0 εν /αν , the smallness condition of the KAM step, namely εν ≤ αν ην2 /Bν0 , is equivalent to εν ≤

αν . Bν0

To verify this inequality we argue as follows. As Bν and αν−1 are increasing with ν, κ−1

κ−1

ν

 Y  κ ∞  ∞  Y Bµ κ µ+1 Bν Bν κ µ−ν+1 = ≤ . αν αν αµ µ=ν µ=ν By the definition of εν in (21.2), the bound αν ≥ α0 /2 and the smallness condition on ε0 in (21.3),  Y  κ−1 κ ν ∞   γ κ ν Bµ κ µ+1 εν Bν0 0 −3 ≤ c0 ε0 ≤ c0−3 ≤ 1. αν αµ 2 µ=0

168

V The KAM Proof

So the smallness condition of the KAM step is satisfied for each ν ≥ 0. In particular, if c0 ≥ 16, then εν Bν0 γ0 ≤ ν+2 αν 2

(21.5)

in view of γ0 ≤ 1 and κ = 4/3, and also ην < 1/16. By the KAM step there exists a transformation 8ν+1 : Dν+1 × 5ν → Dν taking Hν into Hν+1 = Nν+1 + Pν+1 . By (20.8) and (21.2) the new perturbation Pν+1 then satisfies the estimate

X P

ν+1

  Bν0 2 c0 ≤ ην εν + ε rν+1 ,q;Dν+1 ×5ν 2 αν ην2 ν  0 κ−1  κ−1 Bν Bν = c0 ενκ = ενκ = εν+1 . αν αν

λν+1

Consider now the new normal form Nν+1 and its coefficients. By the estimates (20.9) and the ones following it and (21.5) we have, with λν = αµ /Mν , Mν Mν 0 Bν εν ≤ ν+2 , αν 2 αν γ0 0 ≤ Bν εν ≤ ν+2 ≤ (αν − αν+1 )γ0 , 2

lip ˆ ν |lip |ωˆ ν |5ν + | −δ;Dν+1 ×5ν ≤

ˆ ν, j j −δ  s

ν+1 ,τ

and, as α0 γ0 ≤ m 0 , sup

ˆ ν i| |hl, Dν+1 ×5ν [l]d−1

≤ Bν0 εν ≤

αν γ0 ≤ m ν − m ν+1 . 2ν+2

From this it follows that all assumptions in Lemma 21.1 except the diophantine conditions in (21.4) also hold for ‘ν + 1’ in place of ‘ν’. As to the diophantine conditions we use that ν

Bν0 εν ≤

αν γ0κ αν − αν+1 ≤ 2ν+2 K ντ +1

by the definition of K ν . Hence, by (20.10), ˆ ν i| ≤ (αν − αν+1 ) |hk,ωˆ ν i + hl,

|k| [l]d−1 K ντ +1

≤ (αν − αν+1 )

[l]d−1 hkiτ

¯ ν on for hki ≤ K ν on Dν+1 × 5ν . Using the diophantine conditions for ων and  ¯ ν+1 up 5ν we see that they also hold on 5ν for the new frequencies ων+1 and  to hki ≤ K ν . It remains to remove from 5ν the union of the open resonant zones ν+1 Rkl (αν+1 ) for hki > K ν and |l| ≤ 2 to obtain the parameter domain 5ν+1 on which all the diophantine conditions are satisfied. u t

21 Iteration and Convergence

169

With (20.4), (20.6) and (20.9) we also obtain the following estimates. We use that rν+1 = ην rν ≤ rν /4 and ην < 1/16. Lemma 21.2. For ν ≥ 0, k8ν+1 − idkrλνν, p;Dν+1 ×5ν , kD8ν+1 − I krλνν,q,q;Dν+1 ×5ν ≤ and

Bν0 εν , αν

ν |ων+1 − ων |λ5νν , |ν+1 − ν |λ−δ;D ≤ Bν0 εν . ν+1 ×5ν

The estimate of D8ν+1 − I holds also with ‘ p’ for ‘q’. We are now in a position to prove the KAM theorem. Suppose its assumptions are satisfied. To apply Lemma 21.1 with ν = 0, set N0 = N ,

P0 = P,

s0 = s,

r0 = r,

and similarly M0 = M, α0 = α, λ0 = λ = α/M and m 0 = m. Define γ in Theorem 18.1 by setting γ = γ0 γs ,

γs =

∞ κ−1 1 Y − κ µ+1 Bµ , 4

(21.6)

µ=0

where γ0 is the same parameter as before. Note that γs only depends on n, τ and s through the definition of the Bµ , while γ0 will depend on m, L M and other parameters as made explicit in equation (22.2), in addition to satisfying (19.8). The smallness condition (21.3) of Lemma 21.1 is then satisfied by the assumption of the KAM theorem:

λ def λ ε0 = X P0 r 0,q;D ×5 = kX P kr,q;D(s,r )×5 ≤ αγ = α0 γ0 γs . 0

0

0

The small divisor conditions are satisfied by setting [ 0 Rkl (α0 ), 50 = 5 X (k,l)6 =(0,0) |l|≤2

that is, by removing all resonance zones defined in terms of the unperturbed frequencies. The other two conditions in (21.4) follow immediately from assumption A* and ˜ 0 = 0.  Hence, Lemma 21.1 applies, and we obtain a decreasing sequence of domains Dν × 5ν and a sequence of transformations 8ν = 81 B · · · B 8ν : Dν × 5ν−1 → D0 , such that H B 8ν = Hν + Pν for ν ≥ 1. Moreover, the estimates of Lemma 21.2 hold.

170

V The KAM Proof

To prove the convergence of the 8ν we consider the operator norms kL W kr, p . W 6 =0 kW kr˜ , p

kLkr,˜r , p = sup

These norms satisfy kABkr,˜r , p ≤ kAkr,r, p kBkr˜ ,˜r , p for r ≥ r˜ as kW kr, p ≤ kW kr˜ , p . With the mean value theorem we then obtain

ν+1

sup sup sup

8 − 8ν r , p;D ≤ kD8ν kr0 ,rν , p;Dν k8ν+1 − idkrν , p;Dν+1 ν+1

0

and, by the chain rule, kD8ν kr0 ,rν , p;Dν ≤ sup

ν Y

D8µ sup

rµ ,rµ , p;Dµ

µ=1

≤

∞  Y 1+ µ=1



1 2µ+2

≤2

for all ν ≥ 1, using the estimates of Lemma 21.2 together with (21.5) and the fact that the rν are decreasing. Also,

ν+1

lip

8 − 8ν r , p;D 0

≤ kD8ν kr0 ,rν , p;Dν k8ν+1 − idkrν , p;Dν+1 lip

ν+1

sup

+ kD8ν kr0 ,rν , p;Dν k8ν+1 − idkrν , p;Dν+1 , lip

sup

and again by the chain rule and the preceding estimate of D8ν , kD8ν kr0 ,rν , p;Dν ≤ lip

ν X

D8µ lip µ=1 ν X

≤2

≤2

µ=1 ∞ X µ=1

It follows that

ν+1

λ0

8 − 8ν r , p;D 0

rµ ,rµ , p;Dµ

Y

D8ρ sup ρ6 =µ

rρ ,rρ , p;Dρ



D8µ − I lip

rµ ,rµ , p;Dµ

Mµ εµ Bµ0 M0 ≤ . αµ αµ α0

ν+1 ×5ν

l k8ν+1 − idkrλνν, p;Dν+1 ×5ν .

This shows that the 8ν converge uniformly on \ Dν × 5ν = D∗ × 5α , ν≥0

T where 5α = ν≥0 5ν and D∗ = D(s∗ ) × {0} × {0} × {0}, to a Lipschitz continuous family of real analytic torus embeddings 8 : Tn × 5α → Sp ,

22 The Excluded Set of Parameters

171

for which the estimates of Theorem 18.1 hold. Similarly, the frequencies ων converge uniformly on 5α to a Lipschitz continuous limit ω∗ , and the frequencies ν converge uniformly on D∗ × 5α to a regular limit ∗ , with estimates as in Theorem 18.1. The embedded tori are invariant under the perturbed Hamiltonian flow, and the flow on them is linear, because

X H B 8ν − D8ν · X N sup ν r0 ,q;Dν ×5α

sup sup ≤ kD8ν kr0 ,rν ,q;Dν ×5α (8ν )∗ X H − X Nν r ,q;D ×5 ν ν α

sup

l X Pν r ,q;D ×5 . ν

ν

α

Hence in the limit, X H B 8 = D8 · X N∗ on D∗ for each ξ ∈ 5α , where N∗ is the generalized normal form with coefficients ω∗ and ∗ . It remains to prove the claims about the set 5 X 5α . This is the subject of the next section.

22 The Excluded Set of Parameters During the iterative constructionTwe obtain a decreasing sequence of closed sets 50 ⊃ 51 ⊃ . . . such that 5α = ν≥0 5ν and 5 X 5α =

[

[

ν Rkl (αν ),

ν≥0 |k|>K ν−1 ,|l|≤2 κ ν /τ +1

where K −1 = 0, K ν = γ0 ν Rkl (αν )

 =

as defined in the iteration, and

¯ ν (ξ )i| < αν [l]δ ξ ∈ 5ν−1 : |hk,ων (ξ )i + hl, hkiτ



¯ ν are defined and Lipschitz continuous on 5ν−1 , and with 5−1 = 5. Here, ων and  0 =∅ ω0 = ω, 0 =  are the frequencies of the unperturbed system. Recall that R0l for 1 ≤ |l| ≤ 2 by assumption A*. Moreover, using a telescoping argument together with the estimates of Lemma 21.2 as well as (21.5) we have ¯ ν − |λν |ων − ω|λ5νν , | −δ;5ν ≤

α0 γ0 4

for all ν ≥ 0. We write 5 X 5α = 41α ∪ 42α , where [ 0 41α = Rkl (α0 ), 0<|k|≤K 0 ,|l|≤2

42α

=

[

[

ν≥0 |k|>max(K 0 ,K ν−1 ),|l|≤2

ν Rkl (αν ).

172

V The KAM Proof

The set 41α is defined in terms of the original frequencies and thus known a priori. Since |k| ≤ K 0 , it sort of describes the ‘coarse structure’ of 5 X 5α . The set 42α depends on the perturbation and is only known a posteriori. Since here |k| > K 0 , it describes the ‘fine structure’ of 5 X 5α . We first note that for each k there are at most finitely many nonempty resonance ν (α ). zones Rkl ν ν (α ) 6 = ∅, then Lemma 22.1. If Rkl ν

[l]d−1 ≤ θ |k|, sup

with θ = 4(1 + |ω|5 )/m and m = m 0 . ν (α ), then (21.4) implies that Proof. If there exists ξ ∈ Rkl ν

¯ ν (ξ )i| − αν [l]δ |hk,ων (ξ )i| ≥ |hl, hkiτ ≥ m ν [l]d−1 − αν [l]δ m ≥ [l]d−1 , 4 since [l]δ ≤ [l]d−1 for δ < d −1 and αν ≤ m ν /2, m ν ≥ m/2 by construction. Hence, m sup [l]d−1 ≤ |k| |ων (ξ )| ≤ |k|(1 + |ω|5 ). 4

t u

We now prove that the measure of the ‘coarse structure’ tends to zero as α tends to zero. Proposition 22.2. meas(41α ) → 0 as α → 0. Proof. By the preceding lemma for ν = 0, for each k there are only finitely many 0 (α ) is not empty. Moreover, |k| ≤ K . Hence, 41 is a finite union l for which Rkl ν 0 α 0 (α)) → 0 as of resonance zones. For each of its members we know that meas(Rkl α → 0 by assumption B* for l 6= 0, and by elementary volume estimates for l = 0. This proves the proposition. u t In the remainder of this section we estimate the measure of 42α . Proposition 22.3. If γ0 is sufficiently small and τ ≥ n + 1 + 2/(d − 1), then
meas 42α = O(α). The implicit constants are made explicit below. The two proposition together imply that meas(5 X 5α ) → 0 as α → 0. As a preparatory step we extend the frequencies ων and ν to Lipschitz maps defined on all of 5. Indeed, by Lemma M.5 each component of ων − ω and of ν −  has a Lipschitz continuous extension from 5ν to 5 which preserves minimum, maximum and Lipschitz semi-norm. Since we use the sup-norm for ω, and

22 The Excluded Set of Parameters

173

a weighted sup-norm for , doing this for each component we obtain extensions ˘ ν : 5 → `∞ of ν with |ω˘ ν − ω|λ5 = |ων − ω|λ5 and ω˘ ν : 5 → Rn of ων and  −δ ν λ λ ν ˘ ν − | | −δ;5 = |ν − |−δ;5ν . It follows that Rkl (αν ) is contained in ν R˘ kl (αν ) =



 ˘ ν (ξ )i < αν [l]δ . ξ ∈ 5 : hk,ω˘ ν (ξ )i + hl, hkiτ

Moreover, we need not distinguish between the different values of ν in ω˘ ν and ˘ ν . In the sequel we only use the fact that ω˘ ν and  ˘ ν are Lipschitz maps ω0 and 0  on 5 which satisfy the estimates 0 ω − ω λ , 0 −  λ 5

−δ;5

α0 γ0 α ≤ 4 4L M

≤

(22.1)

for 0 ≤ λ ≤ α/M, if we assume that γ0 ≤ 1/L M. Here we use for the first time the bound −1 lip ω ≤L<∞ ω(5) introduced in section 18 right before Theorem 18.1. Henceforth we consider functions ω0 , 0 which satisfy these estimates and may also depend on k and l, and define 0 Rkl (α)

 =



 [l]δ ξ ∈ 5 : k,ω0 (ξ ) + hl,0 (ξ )i < α τ hki



and 40α =

[

[

0 Rkl (α).

ν≥1 |k|>K ν−1 ,|l|≤2

It suffices to show that meas(40α ) = O(α), which will prove Proposition 22.3. Lemma 22.4. If |k| ≥ 8L M |l|δ and τ > n, then 0 meas (Rkl (α)) ≤ g

α , hkiτ

with g = 4θ L n M n−1 ρ n−1 , where ρ = diam 5 and θ is defined in Lemma 22.1. Proof. We introduce the unperturbed frequencies ζ = ω(ξ ) as parameters over the 0 ) in Z . Writing domain Z = ω(5) and consider the resonance zones R˙ kl = ω(Rkl 0 0 ˙ for the frequencies ω and  as functions of ζ , we then have by (22.1) ω˙ and  lip

|ω˙ − id| Z ≤ using λ = α/M and L M ≥ 1.

1 , 4

lip

˙ || −δ;Z ≤ 2L M,

174

V The KAM Proof

˙ )i. Choose a vector Now consider R˙ kl (α), and let φ(ζ ) = hk,ω(ζ ˙ )i + hl,(ζ n v ∈ {−1, 1} such that hk,vi = |k|, and write ζ = ζ (r ) = r v + w with r ∈ R and w ⊥ v. As a function of r , we then have, for t > s and ζ (t), ζ (s) ∈ Z , hk,ω(ζ ˙ )i|ts = hk,ζ i|ts + hk,ω(ζ ˙ ) − ζ i|ts 1 3 ≥ |k| |t − s| − |k| |t − s| = |k| |t − s| 4 4 and lip

˙ )i||ts ≤ 2 |l|δ || ˙ |hl,(ζ −δ |t − s| ≤ 4L M |l|δ |t − s| ≤

1 |k| |t − s|, 2

where for the last inequality we used the assumption of the lemma. Hence we have φ(r v + w)|ts ≥ 41 |k| |t − s| uniformly in w. It follows that for each w the point set {r : r v + w ∈ Z , |φ(r v + w)| < δ } is contained in the interval |r − r0 (w)| < 4δ/ |k|, where r0 can be chosen to depend miserably on w. Hence, with Fubini’s theorem and δ = α [l]δ /hkiτ , [l]δ meas(R˙ kl (α)) ≤ 4(diam Z )n−1 α . hkiτ +1 Going back to the original parameter domain 5 by the inverse frequency map ω−1 and noting that diam Z ≤ M diam 5, [l]δ ≤ [l]d−1 ≤ θ |k| by Lemma 22.1, and 0 ) ≤ L n meas(R ˙ kl ), the final estimate follows. u meas(Rkl t For the next lemma let σ = min(δ, d − 1 − δ) > 0 and δ/σ

K ∗ = 8L M · L ∗ ,

L ∗ = 16L Mθ.

Lemma 22.5. The measure estimate of Lemma 22.4 holds for any resonance zone 0 (α) with |k| ≥ K and |l| ≤ 2. Rkl ∗ Proof. Let |k| ≥ K ∗ . If l = 0, then Lemma 22.4 applies. So assume that l 6= 0. 0 (α) is empty, there is nothing to do. Otherwise, we also want to apply If Rkl Lemma 22.4 and thus verify its assumption. By Lemma 22.1 and our choice of σ , θ |k| ≥ [l]d−1 ≥

1 1 [l]d−1−δ |l|δ ≥ [l]σ |l|δ . 2 2

If [l]σ ≥ L ∗ = 16L Mθ, then division by θ gives |k| ≥ 8L M |l|δ , and Lemma 22.4 δ/σ applies. On the other hand, if [l]σ ≤ L ∗ , then |l|δ ≤ L ∗ , hence again δ/σ

8L M |l|δ ≤ 8L M · L ∗ So Lemma 22.4 applies also in this case.

t u

= K ∗ ≤ |k|.

22 The Excluded Set of Parameters

175

Proof of Proposition 22.3. Choose γ0 sufficiently small so that 1 − τ +1

K 0 = γ0

≥ K∗,

(22.2)

and thus K ν ≥ K 0 ≥ K ∗ for all ν ≥ 0. Then the estimate of Lemma 22.4 applies to all resonance zones in the union of 40α . For a fixed k, it suffices to consider l with [l]d−1 ≤ θ |k| according to Lemma 22.1. Taking into account that |l|d−1 ≤ 2 [l]d−1 we get  2 card l : |l| ≤ 2, [l]d−1 ≤ θ |k| l θ s |k|s , s= . d −1 Hence, by Lemmas 22.4 and 22.5, [  0 meas Rkl (α) l αgθ s l

If we choose τ ≥ n + 1 + s, then  meas

P

|k|>K ν

[ |k|>K ν ,|l|≤2

1 . |k|τ −s

|k|s−τ l K ν−1 and hence 

0 Rkl (α)

l αgθ s ·

1 . Kν

The sum of the latter inequality over all ν converges, and we obtain the estimate of Proposition 22.3. u t

VI Kuksin’s Lemma

23 Kuksin’s Lemma We consider the following first order partial differential equation coming up in the proof of the classical KAM theorem: −i∂ω u + λu + b(x)u = f,

x ∈ Tn ,

(23.1)

for functions on the torus Tn = Rn /2πZn , where ∂ω =

n X

ων ∂xν ,

ω = (ω1 , . . . , ωn ) ∈ Rn .

ν=1

We make the following assumptions, where |k| = |k1 | + · · · + |kn | for k ∈ Zn . Assumption U. The frequency vector ω is diophantine: there are constants α > 0, τ > n and l > 0 such that αl , |k|τ α |hk,ωi| ≥ τ , |k|

|hk,ωi + λ| ≥

for all 0 6 = k ∈ Zn . Also, |λ| ≥ αl. Assumption V. The function b is analytic on some complex strip D(s) = { x : |Im x| < s } ⊂ Cn R around Tn with mean value zero: [b] = Tn b(x) dx = 0. Moreover, def X ˆ kbks,τ = |bk | |k|τ e|k|s ≤ γ α k∈Zn

for b = k bˆk eihk,xi with some γ > 0 and the same τ as in assumption U. We also assume that s ≤ 1 for simplicity. P

178

VI Kuksin’s Lemma

Assumption W. The function f is analytic on the same complex strip D(s) and bounded in a weighted norm: def

k f ks =

X

| fˆk |e|k|s < ∞.

k

Lemma 23.1 (Kuksin [74, 73]). Under assumptions U, V and W, equation (23.1) has a unique solution u = L f that is analytic on D(s) and satisfies kuks−σ ≤

ce2γ k f ks , αlσ τ

Moreover, if

0 < σ ≤ s.

(23.2)

σ γ ≤ , λ 5 |ω|

with |ω| = max |ωi |, then also kuks−σ ≤

c αlσ τ +n+3

k f ks ,

0 < σ ≤ s.

(23.3)

The constant c can be chosen so that it only depends on τ in the first case, and only on τ and n in the second case. The second estimate – which is the difficult one – is not needed if there is a uniform bound on γ . But if a whole family of equations with no such bound is considered, then it gives a uniform bound for kuks−σ provided λ is substantially larger than γ . Corollary 23.2. If, in addition to the assumptions of Kuksin’s lemma, λ ≥ |ω|γ 1+β with some β > 0, then u = L f satisfies the estimate ce2(5/σ ) k f ks , αlσ τ +n+3 1/β

kuks−σ ≤

0 < σ ≤ s,

where c is a constant which depends only on n and τ . Proof. Indeed, with this assumption we have λ/γ ≥ |ω|γ β . Hence, the second estimate (23.3) applies, if γ β ≥ 5/σ . Otherwise, the first estimate (23.2) applies, with γ ≤ (5/σ )1/β . Combining these two estimates we obtain the estimate of the Corollary. u t For later reference we note that an analogous estimate holds also in terms of the sup-norm |u|s = supx∈D(s) |u(x)|. This follows from standard estimates for the Fourier coefficients of analytic functions, compare (19.3).

23 Kuksin’s Lemma

179

Corollary 23.3. Under the same assumptions as in Corollary 23.2, ce2(6/σ ) | f |s , ≤ αlσ τ +2n+3 1/β

|u|s−σ

0 < σ ≤ s,

where c is another constant which depends only on n and τ . The proof of the lemma has two parts. First, the solution u = L f is constructed by converting (23.1) via an integrating factor into an equation with constant coefficients. This also provides the first estimate (23.2). Second, for γ /λ sufficiently small, one considers a sequence of equations (23.1) where the frequency vector ω is approximated by rational ones and the variable coefficient b by trigonometric polynomials. Each of these equations has a unique periodic solution, which can be represented by an oscillatory integral with a complex valued phase function. A contour deformation is then constructed by a contraction argument, and using Cauchy’s theorem, these oscillatory integrals are estimated uniformly. Taking limits one obtains the second estimate (23.3). This scheme was first described by Kuksin [74]. For the convenience of the reader we present here our own version of it. Before giving the details we observe that we may divide equation (23.1) by α and thus replace ω, λ, b and f by α −1 ω, α −1 λ, α −1 b and α −1 f , respectively. In the assumptions U, V and W, the constant α is then replaced by 1. Henceforth, we can assume the normalization α = 1 for the rest of our considerations. The proof of the lemma now takes N steps, where N = 7. 1. Existence and Uniqueness To obtain an integrating factor B in (23.1), solve ∂ω B = b,

[B] = 0.

Then kBks =

X k6=0

X |bˆk | e|k|s |hk,ωi| k6=0 X ≤ |bˆk | |k|τ e|k|s

| Bˆ k |e|k|s =

k6=0

≤ kbks,τ ≤ γ . Hence, B is analytic in D(s). Now set u = e−iB v, f = e−iB g to obtain −i∂ω v + λv = g. This equation has a unique solution v by comparing Fourier coefficients: (hk,ωi + λ)vˆk = gˆ k ,

k ∈ Zn .

180

VI Kuksin’s Lemma

We get kvks−σ =

X k

≤

|gˆ k | e|k|(s−σ ) |hk,ωi + λ|

X1 k

l

|gˆ k |(1 + |k|τ )e|k|(s−σ )

1 ≤ kgks sup (1 + t τ )e−σ t l t≥0 1 c ≤ kgks τ l σ with some constant c depending only on τ . Going back we obtain the unique analytic solution u = e−iB v of (23.1) with the estimates kuks−σ ≤ ekBks kvks−σ ≤ eγ kvks−σ , kgks ≤ ekBks k f ks

≤ eγ k f ks ,

where we made use of the fact that the norm is multiplicative: kuvks ≤ kuks kvks . Hence, combining the inequalities above, we obtain kuks−σ ≤

e2γ c k f ks τ l σ

as stated in (23.2). 2. Approximation To obtain estimate (23.3) we now assume that σ γ ≤ , λ 5 |ω| and approximate ω by rational frequency vectors ων . The following is proven at the end of this section. Recall that α = 1. Lemma 23.4 (Approximation Lemma). There exist a sequence of frequency vectors 2π ων = · mν Tν with m ν ∈ Zn , Tν → ∞, and a sequence of constants K ν → ∞ such that for all ν, (a) (b) (c)

2π , Tν 1 |hk,ων i| ≥ , 2 |k|τ l |hk,ων + λi| ≥ , τ 2 |k| +n+2 |ω − ων | ≤

0 < |k| < K ν , |k| 6= 0.

23 Kuksin’s Lemma

181

Consequently, also γ σ ≤ λ 4 |ων |

(d) for all large ν. 3. The Periodic Problem

We now consider the approximate, periodic problem −i∂ων u + λu + bν (x)u = f, where ων =

2π · mν , Tν

bν =

X

bˆk eihk,xi .

0<|k|
Fixing ν we drop it from the notation in the following, which hopefully does not lead to confusion. We may solve again ∂ω B = b with [B] = 0. Using (b) of the Approximation Lemma we get, as before, kBks ≤ 2 kbks,τ ≤ 2γ , and with u = e−iB v, f = e−iB g, we obtain the equation −i∂ω v + λv = g. Its solution has the integral representation Z T i v(x) = ηT eiλt g(x + tω) dt, ηT = iλT . e −1 0 To obtain this identity, write w(t) = v(x + tω) and h(t) = g(x + tω). Both functions are periodic in t with period T . Substituting into the preceding equation we find d w(t) + iλw(t) = ih(t). dt With an integrating factor eiλt and a subsequent integration over [0, T ] this leads to the given integral representation. From this and the substitutions u = eiB v and f = eiB g results the representation of u as an oscillatory integral: Z T u(x) = ηT eiλt+iB(x+tω)−iB(x) f (x + tω) dt 0

= ηT

Z

T

eiλ(t+δ[B(x+tω)−B(x)]) f (x + tω) dt

0

with δ=

1 . λ

(23.4)

182

VI Kuksin’s Lemma

Our aim is to obtain a uniform estimate for x ∈ D(s − 2σ ). In the following we therefore fix x with |Im x| < s − 2σ and derive such an estimate for |u(x)|. 4. Stationary Phase Transformation To estimate the last integral we deform the path of integration into a curve 0 in C with the same endpoints. To this end we want to find a change of the integration variable t = φ(r ) with φ(0) = 0 and φ(T ) = T such that
φ(r ) + δ B(x + φ(r )ω) − B(x) = r. (23.5) We then obtain u(x) = ηT

T

Z

eiλr f (x + φ(r )ω)φ 0 (r ) dr,

0

which we will use to estimate u. To find φ we lift the problem to the torus Tn and write φ(r ) = r + ψ(r ω),

ψ : Tn → C.

Also, let r ω = ξ ∈ Tn . From (23.5) we obtain the equation
ψ(ξ ) + δ B(x + ξ + ψ(ξ )ω) − B(x) = 0

(23.6)

which we solve by a contraction argument. Consider the space A of functions ψ : D(σ ) → C with (i) (ii) (iii)

ψ analytic and 2π-periodic on D(σ ), ψ(0) = 0, |ψ|σ = sup |ψ(ξ )| ≤ 4γ δ. ξ ∈D(σ )

Define a map T on A by
T (ψ)(ξ ) = −δ B(x + ξ + ψ(ξ )ω) − B(x) . To show that T is well defined we have to verify that the argument of B is in the strip D(s). To this end we observe that γδ =

γ σ ≤ λ 4 |ω|

by (d) in the Approximation Lemma. Then |Im(ψ(ξ )ω)| ≤ |ψ|σ |ω| ≤ 4γ δ |ω| ≤ σ, hence |Im(x + ξ + ψ(ξ )ω)| < s −2σ +σ +σ = s. It follows that B(x +ξ +ψ(ξ )ω) and thus T (ψ) are well defined.

23 Kuksin’s Lemma

183

T maps A into A: (i) and (ii) clearly hold, and, as |B|σ ≤ kBks ≤ 2γ , |T (ψ)|σ ≤ 2δ |B|σ ≤ 4γ δ. Further, T is a contraction:
T (ψ)(ξ ) − T (χ)(ξ ) = δ B(x + ξ + χ ω) − B(x + ξ + ψω) Z 1 hd B(x + ξ(t)),(ψ − χ )ωi dt =δ 0

=δ

Z

1

b(x + ξ(t))(ψ − χ ) dt

0

with ξ(t) = ξ + ((1 − t)χ + tψ)ω. Hence, using assumption V and the fact that |Im(x + ξ(t))| < s for 0 ≤ t ≤ 1, |T (ψ) − T (χ )|σ ≤ δ kbks,τ |ψ − χ|σ ≤ γ δ |ψ − χ|σ ≤

1 |ψ − χ|σ , 2

since γ δ ≤ σ/4 |ω| ≤ 1/4 |ω|, and |ω| ≥ 1/2 for all approximating frequencies. Thus there exists a unique analytic solution ψ ∈ A of equation (23.6). Consequently, φ with φ(r ) = r + ψ(r ω) solves equation (23.5). In particular, φ − id has period T , since ω = 2πm/T . Moreover, differentiating the fixed point equation (23.6) with respect to r one gets ∂ω ψ = −δ hd B,ω + (∂ω ψ)ωi = −δb(1 + ∂ω ψ) and, again using assumption V and γ δ ≤ 1/2, |∂ω ψ|σ ≤ γ δ(1 + |∂ω ψ|σ ) ≤

1 (1 + |∂ω ψ|σ ), 2

hence |∂ω ψ|σ ≤ 1. 5. Transformation and Estimate We observe that φ maps the interval [0, T ] into a curve 0 in D(σ ) with the same endpoints as [0, T ]. By Cauchy’s theorem, the integral in (23.4) over [0, T ] is the same as over 0. Thus, substituting t = φ(r ) = r + ψ(r ω) in the integral (23.4) we obtain Z T Z T iλr 0 u(x) = ηT e f (x + φ(r )ω)φ (r ) dr = ηT eiλr h(x; ξ ) ξ =r ω dr 0

0

with h(x; ξ ) = f (x + ξ + ψ(ξ )ω)(1 + ∂ω ψ(ξ )) analytic on D(σ ), and |h(x; · )|σ ≤ 2 | f |s .

184

VI Kuksin’s Lemma

The above integral for u(x) is now easily evaluated. Since h is periodic in ξ , h(x; ξ ) =

X

hˆ k (x)eihk,ξ i ,

|hˆ k | ≤ |h|σ e−|k|σ .

k

Moreover Z 0

T

eiλr eihk,ξ i ξ =r ω dr =

Z

T

ei(λ+hk,ωi)r dr = −i

0

ηT−1 eiλT − 1 = . hk,ωi + λ hk,ωi + λ

Hence, u(x) =

X k

hˆ k (x) . hk,ωi + λ

With (c) of the Approximation Lemma, one obtains for every x ∈ D(s − 2σ ) the estimate |u(x)| ≤

1 1 c 2 X τ +n+2 −|k|σ |k| |h|σ + |h|σ ≤ | f |s · τ +n+3 , e l λ l σ k6=0

where c depends only on n and τ . We used that λ ≥ l by assumption U for α = 1. 6. Taking Limits Summarizing the preceding arguments we obtain the following result. For every frequency vector 2π · mν , m ν ∈ Zn , ων = Tν satisfying (b), (c) and (d) the equation −i∂ων u + λu + bν u = f has a unique analytic solution u ν satisfying |u ν |s−2σ ≤

c 1 | f |s · τ +n+3 , l σ

0 < σ ≤ s.

Letting ν → ∞ and taking (a) of the Approximation Lemma into account, we obtain approximating sequences ων → ω, bν → b. The corresponding u ν are uniformly bounded. Hence we can choose a convergent subsequence converging to some solution u of −i∂ω u + λu + bu = f with |u|s−2σ ≤

1 c | f |s · τ +n+3 . l σ

From this the second estimate (23.3) of the lemma follows. Incidentally, since the solution u is unique, actually the whole sequence u ν converges to u.

23 Kuksin’s Lemma

185

7. Proof of the Approximation Lemma For any t-interval I of the form I = [T, T + 1],

T ≥1=

2π |ω|

there is an integer vector m ∈ Zn such that |tω − 2πm| ≤ 2π for t ∈ I . Hence, |ω − ωt | ≤

2π , T

ωt =

2π · m, t

for each t ∈ I . This gives (a). To obtain (b) we estimate, for arbitrary t ∈ I and using assumption U, 1 1 2π ≥ − |k| |k|τ T 2 |k|τ

|hk,ωt i| ≥ |hk,ωi| − |hk,(ω − ωt )i| ≥ for 4π |k|τ +1 ≤ T , that is, for def

|k| ≤ K T =



T 4π

1/(τ +1)

.

This gives (b). For (c) we estimate similarly, again using assumption U, |hk,ωt i + λ| ≥ |hk,ωi + λ| − |hk,(ω − ωt )i| ≥

l l 2π ≥ − |k| τ |k| T 2 |k|τ

for t ∈ I and 4π |k|τ +1 ≤ lT . So it remains to consider k with 4π |k|τ +1 ≥ lT.

(23.7)

Assume that for some t ∈ I , |hk,ωt i + λ| ≤

l , 2

otherwise there is nothing to do. In particular, we have |hk,ωt i| ≥ l/2, since λ ≥ l. As long as this holds, we have for ϕ(t) := hk,ωt i + λ the estimate 0 1 ϕ (t) = |hk,ωt i| ≥ l ≥ l , t 2t 4T as t ≤ T + 1 ≤ 2T . Hence the measure of the subset of the t-interval I where |hk,ωt i + λ| ≤

l 2

|k|τ +n+2

can be estimated by 4T l 2T = τ +n+2 . τ +n+2 l 2 |k| |k|

(23.8)

186

VI Kuksin’s Lemma

Summing over all k ∈ Zn with (23.7) the total measure of the subset of t-values in I satisfying (23.8) can be estimated by c(n) ·

T K 0τ +2

,

where c(n) depends only on n, and K 0 is the lower bound of |k| in (23.7). That is, K 0 = c(l)T 1/(τ +1) . Hence the measure is bounded by c(n, l)T −1/(τ +1) <

2π = |I |, |ω|

if T is sufficiently large. Note that the dependence of the constant on l is irrelevant here, since we want to make T large anyhow, after l is given. Hence for each large T we can find t ∈ [T, T + 1] so that (23.8) does not hold. That is, for such t, (c) holds. This proves the approximation lemma.

VII Background Material

A Analyticity We discuss the notion of an analytic map between two complex Banach spaces and prove two frequently used characterizations of such maps. Much of this material is taken from appendix A in [112]. Let E and F be complex Banach spaces with norms | · | and k · k, respectively, and let U ⊂ E be open. A map f: U→F is analytic on U , if it is continuously differentiable on U . This is the straightforward generalization of the notion of an analytic function of one complex variable. It is convenient to introduce another notion of analyticity. The map f : U → F is weakly analytic on U , if for each u ∈ U , h ∈ E and L ∈ F ∗ , the function z 7→ L f (u + zh) is analytic in some neighbourhood of the origin in C in the usual sense. The radius of weak analyticity of f at u is the supremum of all r ≥ 0 such that the above function is defined and analytic in the disc |z| < 1 for all L ∈ F ∗ and h ∈ E with |h| < r . It is easy to see that the radius r of weak analyticity at u is equal to the distance ρ of u to the boundary of U . Clearly, r ≤ ρ by definition. On the other hand, if L and h are given with |h| < ρ, then the function z 7→ L f (u + zh) is well defined on the disc |z| < 1 and analytic in some neighbourhood of each point in it, since f is weakly analytic on all of U . Consequently, this function is analytic on |z| < 1, and so also r ≥ ρ. It follows that r = ρ. The notion of a weakly analytic map is weaker than that of an analytic map. For instance, every globally defined, but unbounded linear operator is weakly analytic, but not analytic. Remarkably, a weakly analytic map is analytic, if in addition it is locally bounded. Before we get to this result, we state two basic lemmata.

188

VII Background Material

Lemma A.1 (Cauchy’s Formula). Suppose f is weakly analytic and continuous on U . Then, for every u ∈ U and h ∈ E, Z f (u + ζ h) 1 f (u + zh) = dζ 2πi |ζ |=ρ ζ − z for |z| < ρ < r/ |h|, where r is the radius of weak analyticity of f at u. Proof. Fix u ∈ U , and let r > 0 be the radius of weak analyticity of f at u. Then the open ball of radius r around u is contained in U . For every h ∈ E, the integral Z f (u + ζ h) 1 |z| < ρ < r/ |h|, dζ, 2πi |ζ |=ρ ζ − z is well defined, since f is continuous and |ζ h| < r , |ζ − z| > 0 for |ζ | = ρ. Then, for every L ∈ F ∗ , Z Z f (u + ζ h) 1 L f (u + ζ h) 1 L dζ = dζ = L f (u + zh) 2πi ζ − z 2πi ζ −z |ζ |=ρ |ζ |=ρ by the usual Cauchy formula. Since this holds for all L, the statement follows. u t Lemma A.2 (Cauchy’s Estimate). Let f be an analytic map from the open ball of radius r around u in E into F, such that k f k ≤ M on this ball. Then kdu f k = max h6=0

kdu f (h)k M ≤ . |h| r

Proof. Let h 6 = 0 in E. Then φ(z) = f (u + zh) is an analytic map from the complex disc |z| < r/ |h| in C into F that is uniformly bounded by M. Hence kd0 φk = kdu f (h)k ≤

M · |h| r

by the usual Cauchy inequality. The above statement follows, since h 6 = 0 was arbitrary. u t The statement of the lemma is particularly transparent, when f is a complex valued function. Then du f is an element in the dual space E ∗ to E, and the induced operator norm is the norm | · | E ∗ dual to | · | E . So, for instance, if f is bounded in absolute value by M on the balls |u|∞ , |u|2 , |u|1 < r, in standard spaces, then |d0 f |1 , |d0 f |2 , |d0 f |∞ ≤

M , r

respectively, in both finite and infinite dimensional settings.

A Analyticity

189

Next we have a simple result about the analyticity of maps into sequence spaces with the sup-norm. Let U ⊂ E be open, and `∞ (F) the space of all sequences in a complex Banach space F endowed with the sup-norm | · |∞ . Theorem A.3. Suppose f : U → `∞ (F),

u 7 → f (u) = ( f n (u))n≥1 ,

is locally bounded, and each coordinate function f n is analytic on U with values in a complex Banach space F. Then f is analytic as a map from U into `∞ (F). Proof. We verify directly that f is continuously differentiable at each point u ∈ U , with du f = (du f n )n≥1 . Indeed, | f (u + h) − f (u) − du f (h)|∞ = sup | f n (u + h) − f n (u) − du f n (h)| F . n≥1

Since f is bounded in a fixed ball around u, each coordinate function f n is bounded by the same constant M on this ball. Cauchy’s estimate applied to f n then gives | f n (u + h) − f n (u) − du f n (h)| F ≤ M |h|2E for all small |h| E , uniformly in n. Hence, | f (u + h) − f (u) − du f (h)|∞ ≤ M |h|2E . The continuity of du f follows by the same token. u t We now turn to the basic characterization of analytic maps between complex Banach spaces. An infinitely often differentiable function f is said to be represented by its Taylor series near a point u, if f (u + h) =

X 1 d n f (h, . . . , h), n! u n≥0

for all sufficiently small h, with the series converging absolutely and uniformly. Theorem A.4. Let f : U → F be a map from an open subset U of a complex Banach space E into a complex Banach space F. Then the following three statements are equivalent. (1) f is weakly analytic and locally bounded on U . (2) f is analytic on U . (3) f is infinitely often differentiable on U , and is represented by its Taylor series in a neighbourhood of each point in U . Proof. (1) ⇒ (2) Suppose f is weakly analytic and locally bounded. We first show that f is continuous.

190

VII Background Material

Fix u ∈ U and choose r > 0 so small that sup k f (u + h)k = M < ∞. |h|≤r

As L f (u + zh) is continuous, we obtain, by the usual Cauchy formula, Z z L f (u + ζ h) L f (u + zh) − L f (u) = dζ 2πi |ζ |=1 (ζ − z)ζ for |z| < 1 and |h| < r , and for any L ∈ F ∗ . Hence, for |z| < 12 , L f (u + zh) − L f (u) ≤ 2M kLk, z where kLk denotes the operator norm of L. This estimate holds for all L ∈ F ∗ uniformly for |z| < 21 and |h| < r . Consequently,

f (u + zh) − f (u)

≤ 2M

z for |z| < 12 and |h| < r . From this, the continuity of f follows. Now, f being weakly analytic and continuous, Cauchy’s formula applies, and Z f (u + ζ h) 1 f (u + zh) = dζ 2πi |ζ |=1 ζ − z for |z| < 1 and |h| < r . It follows that f has a directional derivative in every direction h, namely Z 1 f (u + ζ h) f (u + zh) − f (u) δu (h) = lim = dζ. z→0 z 2πi |ζ |=1 ζ2 In fact, this limit is uniform in |v − u| < r/2 and |h| < r/2, since



Z

f (v + zh) − f (v)

z f (v + ζ h)

− δv (h) = dζ

≤ 2M |z| z 2πi |ζ |=1 ζ 2 (ζ − z) for |z| < 12 . It follows from this that f is continuously differentiable, hence analytic on U . (2) ⇒ (3) Suppose f is analytic on U . As before, fix u ∈ U and r > 0 such that sup|h|≤r k f (u + h)k = M < ∞. For h ∈ E and n ≥ 0, define Z n! f (u + ζ h) Pn (h) = dζ, 2πi |ζ |=ρ ζ n+1 where ρ > 0 is chosen sufficiently small. The integral is independent of ρ as long as ρ ≤ r/ |h|, since f is analytic. For instance, P0 (h) = f (u) and P1 (h) = du f (h). We show that Pn (h) is the n-th directional derivative of f in the direction h.

A Analyticity

191

First of all, Cauchy’s formula and the expansion m

X 1 1 1 = + m+1 n+1 ζ −1 ζ ζ (ζ − 1) n=0

give f (u + h) −

Z m X f (u + ζ h) 1 1 Pn (h) = dζ m+1 n! 2πi |ζ |=ρ ζ (ζ − 1) n=0

for |h| < r . Choosing ρ = r/ |h| for h 6= 0, the norm of the right hand side is bounded by  m |h| |h| M |h|m+1 . M = m r r − |h| r (r − |h|) Consequently, ∞ X 1 f (u + h) = Pn (h) n! n=0

for |h| < r . Moreover, the sum converges uniformly in every ball |h| ≤ ρ < r . We now show that each Pn is a homogeneous polynomial of degree n in h. That is, there exists a bounded symmetric n-linear map An such that Pn = Aˆ n , the polynomial associated to An by evaluating it on the diagonal. Consider the F-valued map An defined by An (h 1 , . . . , h n )   Z Z 1 n f (u + ζ1 h 1 + · · · + ζn h n ) = ... dζ1 . . . dζn , 2πi ζ12 . . . ζn2 |ζ1 |=ε |ζn |=ε where ε > 0 is sufficiently small, say ε < min1≤i≤n r/ |h i |. For every L ∈ F ∗ , the map (z 1 , . . . , z n ) 7 → L f (u + z 1 h 1 + · · · + z n h n ) is analytic in a neighbourhood of the origin in Cn . Hence, by the usual Cauchy formula for n complex variables, d d L An (h 1 , . . . , h n ) = ... L f (u + z 1 h 1 + · · · + z n h n ) . dz 1 dz n z 1 ,...,z n =0 It follows that An is linear and symmetric in all arguments. An is also bounded by a straightforward estimate. Finally, using Cauchy’s formula again,  n d L An (h, . . . , h) = L f (u + zh) = L Pn (h) dz z=0 for all L. Therefore, An (h, . . . , h) = Aˆ n (h) = Pn (h), as we wanted to show. Thus, on the ball of radius r around u, the map f is represented by a power series, which converges uniformly on every smaller ball around u. It is a basic fact that such a map is infinitely often differentiable. In particular, dun f = An for all n ≥ 0. (3) ⇒ (1) This is trivial. u t

192

VII Background Material

A special case of the preceding theorem arises for maps into a Hilbert space. Theorem A.5. Let f : U → H be a map from an open subset U of a complex Banach space into a Hilbert space with orthonormal basis (en )n≥1 . Then f is analytic on U if and only if f is locally bounded, and each “coordinate function” f n = h f ,en i : U → C is analytic on U . Moreover, the derivative of f is given by X d f (h) = d f n (h)en . n≥1

Proof. Let L ∈ H ∗ . By the Riesz representation theorem, P there is a unique element ` in H such that Lφ = hφ,`i for all φ in H . Write ` = n≥1 λn en , and set `m =

m X

λn en ,

m ≥ 1.

n=1

Then L is the operator norm limit of the functionals L m defined by L m φ = hφ,`m i. That is, as m → ∞, sup k(L − L m )(φ)k → 0. kφk≤1

Now, given x in U , choose r > 0 so that f is bounded on the ball of radius r around x. Fix h in the complex Banach space containing U with khk < r . On |z| < 1, the functions z 7 → L m f (x + zh) =

m X

λn f n (x + zh),

m≥1

n=1

are analytic by hypotheses and tend uniformly to the function z 7→ L f (x +zh), since f is bounded. Hence that function is also analytic on |z| < 1. This shows that f is weakly analytic and locally bounded. By Theorem A.4, the function f is analytic. Conversely, if f is analytic, then f is locally bounded, and each coordinate function f n is analytic. Finally, if f is analytic, then dx f (h) exists and is an element of H , hence can be expanded with respect to the orthonormal basis (en )n≥1 . Its n-th coefficient is hdx f (h),en i = dx h f ,en i(h) = dx f n (h) by the chain rule, since h · ,en i is a linear function. Thus, X dx f (h) = dx f n (h)en n≥1

as was to be proven. u t

A Analyticity

193

The next theorem may be considered a generalization of Theorem A.4. We say that a subset V ⊂ U of an open set U in a complex Banach space is an analytic subvariety, if locally it can be represented as the zero set of an analytic function taking values in Cn for some n ≥ 1. Theorem A.6. Let V1 , . . . , Vm be analytic subvarieties of an open subset U in a complex Banach space E. If f is a complex-valued function on U , which is (i) analytic on U X (V1 ∪ · · · ∪ Vm ), (ii) continuous on U , and, (iii) when restricted to each of the Vi , weakly analytic on Vi , then f is analytic on U . Proof. We are going to show that not only the restriction of f , but f itself is weakly analytic in every point in V1 ∪· · ·∪ Vm . Since f is also locally bounded by continuity, f is then analytic on U by Theorem A.4. Let D be a one-dimensional complex disc around an arbitrary point in U . Locally, we can write Vi = { q ∈ U : ϕi (q) = 0 }, 1 ≤ i ≤ m, with analytic, vector-valued functions ϕi . When restricted to D, each function ϕi either vanishes identically or has only a finite number of zeroes in D, possibly after shrinking D a bit. If at least one ϕi vanishes identically, then D is contained in some subvariety Vi , and f is analytic on D ⊂ Vi by assumption (iii). Otherwise, none of the functions ϕi vanishes identically on D, and consequently D ∩ (V1 ∪ · · · ∪ Vm ) is a finite set. Outside this set in D, f is analytic by hypotheses (i), and on all of D, f is continuous by hypotheses (ii). It follows that these singularities are removable, and that f is analytic on all of D. Since the disc D was arbitrary, it follows that f is weakly analytic. Hence f is analytic by Theorem A.4. u t Finally, we introduce the notion of a real analytic map. Let E, F be real Banach spaces, let E C , FC be their complexifications, and let U ⊂ E be open. A map f: U→F is real analytic on U , if for each point in U there is a neighbourhood V ⊂ E C and an analytic map g : V → FC , such that f =g

on U ∩ V.

It follows that a real analytic map can be extended into a Taylor series with real coefficients in a ball at each point. The converse is also true.

194

VII Background Material

B Spectra In this appendix we collect some basic facts about the spectra of Schr¨odinger operators on a finite interval. The main purpose is to fix notions and notations. Only in a few cases do we provide proofs, otherwise we refer for example to the references [80, 82, 84] and [112]. Fundamental Solution We consider the differential equation −y 00 + qy = λy

(B.1)

on the compact interval [0, 1] depending on a potential q ∈ L 2C = L 2C ([0, 1]) and a complex parameter λ ∈ C. By definition, a function y is a solution of this equation, if it is continuously differentiable, y 0 is absolutely continuous, and the equation holds almost everywhere for y 00 . One fundamental solution is given by the particular solutions y1 and y2 satisfying the initial conditions y1 (0, λ, q) = 1, y10 (0, λ, q) = 0,

y2 (0, λ, q) = 0, y20 (0, λ, q) = 1.

Any other solution of (B.1) is a linear combination of y1 and y2 with coefficients determined by its initial values. The associated Floquet matrix is the 2 × 2-matrix     m1 m2 y1 y2 F(λ, q) = (λ, q) = (1, λ, q), m 01 m 02 y10 y20 where we introduce the convenient notation m i = yi |x=1 ,

m i0 = yi0 x=1

for the values of this fundamental solution at x = 1. The Floquet matrix describes the shift of initial data at x = 0 to initial data at x = 1, since     y(1) y(0) = F(λ, q) y 0 (1) y 0 (0) for any solution y. Its determinant is 1 in view of the Wronskian identity def

W (y1 , y2 ) = y1 y20 − y10 y2 ≡ 1. Its trace, def

1 = tr F = m 1 + m 02 , is called the discriminant of q and is fundamental in discussing its periodic spectrum defined below. We note the following asymptotic behavior of the m-functions [112, chapter 1].

B Spectra

195

Proposition B.1. √



 m 1 (λ) = cos λ + O , |λ|1/2 √   sin λ cλ m 2 (λ) = √ +O , |λ| λ cλ

and √ √ m 01 (λ) = − λ sin λ + O(cλ ),   √ cλ m 02 (λ) = cos λ + O , |λ|1/2

locally uniformly on C × L 2C with cλ = e Im

√ λ

.

From this proposition one immediately obtains a first asymptotic estimate for the 1-function as well. But we also need the following refined estimate, which can be found in [84, Section 1.4] – see also Theorem C.3. 2 , Proposition B.2. For q ∈ H0, C

√   √ kqk2 sin λ cλ + O 1(λ) = 2 cos λ + 4 λ3/2 |λ|2

2 with c = e Im locally uniformly on C × H0, λ C

√ λ

.

As functions of λ and q, m 1 and m 2 as well as their x-derivatives are compact on C × L 2C , that is, continuous with respect to the weak topology. Moreover, they are also real analytic functions of λ and q, and their respective derivatives are denoted m˙ i and dm i . The latter has a representation in terms of a unique gradient, denoted ∂m i =

∂m i , ∂q(x)

such that dm i (v) =

Z

1

∂m i (x)v(x) dx

0

for all v ∈ L 2C . The same notation is used for all other gradients with respect to q. It is a general phenomenon in this context that such gradients are represented by products of solutions of the underlying equation. Proposition B.3. For i = 1, 2, ∂m i = (m 2 y1 − m 1 y2 )yi , ∂m i0 = (m 02 y1 − m 01 y2 )yi .

196

VII Background Material

Consequently, ∂1 = m 2 y12 +(m 02 −m 1 )y1 y2 −m 01 y22 . The latter gradient also admits the representation ∂1 = m 2 (λ, Tt q),

Tt q = q( · + t),

where q is understood to be periodically extended beyond [0, 1]. Proof. We only prove the last assertion. First of all, y2 (x, λ, Tt q) = y2 (x + t, λ, q)y1 (t, λ, q) − y1 (x + t, λ, q)y2 (t, λ, q), since both sides are solutions of the equation −y 00 + Tt qy = λy with the same initial data at x = 0. Hence, y2 (1, λ, Tt q) = y2 (1 + t)y1 (t) − y1 (1 + t)y2 (t)   = y2 (1)y1 (t) + y20 (1))y2 (t) y1 (t)   − y1 (1)y1 (t) + y10 (1)y2 (t) y2 (t) = m 2 y12 (t) + (m 02 − m 1 )y1 (t)y2 (t) − m 01 y22 (t) ∂1 = , ∂q(t) which proves the claim. u t Dirichlet Spectrum From now on we consider potentials q in the real space L 2 = L 2R ([0, 1]). But by analytic continuation, the following results extend to potentials in a sufficiently small complex neighbourhood of L 2 in L 2C . The spectrum of the differential operator −d 2 /dx 2 + q with Dirichlet boundary conditions is called the Dirichlet spectrum of q. It consists of those complex numbers λ, for which the equation −y 00 + qy = λy admits a nontrivial solution vanishing at both endpoints of [0, 1]. Clearly, λ is a Dirichlet eigenvalue of q if and only if y2 (1, λ, q) = 0. So the Dirichlet spectrum of q is precisely the zero set of the entire function m 2 . It turns out that the Dirichlet eigenvalues form an unbounded sequence of real numbers µ1 (q) < µ2 (q) < µ3 (q) < . . . , where each eigenvalue is a simple root of m 2 , as m˙ 2 does not vanish there in view of the next proposition below. Hence its algebraic multiplicity is one and thus equal to its geometric multiplicity, which is the dimension of the associated eigenspace. With µn we can thus associate a unique normalized Dirichlet eigenfunction gn by requiring that kgn k = 1 and gn0 (0) > 0. Clearly, y2 gn = , ky2 k µn and for the norm of y2 we have the following result.

B Spectra

197

Proposition B.4. At any Dirichlet eigenvalue, m 1 m 02 = 1 and m˙ 2 = m˙ 2 m 02 > 0. m1

ky2 k2 =

Proof. Since m 2 = 0 at a Dirichlet eigenvalue, we have m 1 m 02 = 1 by the Wronskian identity. In view of m 2 (λ + ε, q) = m 2 (λ, q − ε) and Proposition B.3, Z m˙ 2 = −

1

∂m 2 dx =

0

Z 0

which gives the second identity.

1


m 1 y22 − m 2 y1 y2 dx = m 1 ky2 k2 ,

t u

Next we recall the asymptotic behavior of Dirichlet eigenvalues and eigenfunctions R 1 and the product formula for m 2 from [112, Theorems 2.4 and 2.5]. Let [q] = 0 q(x) dx denote the mean value of q. Proposition B.5. For q ∈ L 2 , one has µn = n 2 π 2 + [q] + `2 (n) and √ gn = 2 sin πnx + O(1/n), √ gn0 = 2πn cos πnx + O(1). These estimates hold uniformly on bounded subsets of [0, 1] × L 2 . Proposition B.6. For q ∈ L 2 , m 2 (λ) =

Y µn − λ . n2 π2

n≥1

Considered as a function of q in L 2 , each Dirichlet eigenvalue is compact and real analytic. Its gradient ∂µn is obtained by differentiating m 2 (µn (q), q) = 0 with respect to q and using Propositions B.3 and B.4 – see [112, chapter 2]. Proposition B.7. Each Dirichlet eigenvalue µn is a compact, real analytic function on L 2 with gradient ∂m 2 ∂µn = − = gn2 . m˙ 2

µn

Periodic Spectrum Identify a function q ∈ L 2C with its periodic extension beyond [0, 1]. The spectrum of the operator −d 2 /dx 2 + q with periodic boundary conditions on the interval [0, 2] is called the periodic spectrum of q. It consists of those complex numbers λ, for which the equation −y 00 + qy = λy admits a nontrivial solution with period 2. As q is extended with period 1, a solution has period 2 if and only if it is either periodic or anti-periodic over [0, 1]. Therefore, λ is a periodic eigenvalue of q iff the Floquet matrix F of q either has an eigenvalue 1 or an eigenvalue −1, which in turn

198

VII Background Material

1 2 λ1 λ0

λ2 λ3

λ4

−2 Figure 6 A generic 1-function

is equivalent to its discriminant 1 being 2 or −2, respectively. Consequently, the periodic spectrum of q is precisely the union of the zero sets of the entire functions 1 − 2 and 1 + 2, or equivalently, the zero set of the entire function 12 − 4. To locate this zero set in the real case q ∈ L 2 , note that m 1 m 02 = 1 at any Dirichlet eigenvalue, and sgn m 02 (µn ) = (−1)n , thus ( 1 ≥ 2, n even, 1(µn ) = 0 + m 02 (µn ) m 2 (µn ) ≤ −2, n odd. ˙ has exactly n roots below (n + 1 )2 π 2 for all large n by a Counting Moreover, 1 2 Lemma completely analogous to [112, Lemma 2.2]. Taking into account its asymptotic behavior for λ → ±∞, the function 1 therefore must have a shape as depicted in figure 6. The periodic spectrum of a real potential q therefore consists of an unbounded sequence of periodic eigenvalues λ0 (q) < λ1 (q) ≤ λ2 (q) < λ3 (q) ≤ λ4 (q) < . . . , such that λ2n−1 ≤ µn ≤ λ2n for any n ≥ 1. Equality may occur in this sequence in every place with a ‘≤’-sign, and this case occurs precisely if all solutions of the corresponding differential equation are 2-periodic. Thus, also in the periodic case the geometric and algebraic multiplicities of an eigenvalue λm coincide, and could be either one or two. If λm is simple, then one can associate with it a unique normalized periodic eigenfunction f m by requiring that k f m k = 1 and f m (0) > 0, or f m (0) = 0 and f m0 (0) > 0. The results of Propositions B.8–B.13 are formulated for real potentials. By analytic continuation they extend to potentials in a sufficiently small complex neighbourhood of L 2 in L 2C , which can be chosen independently of n. Proposition B.8. For q ∈ L 2 , λ2n−1 , λ2n = n 2 π 2 + [q] + `2 (n), Consequently, γn = λ2n − λ2n−1 = `2 (n). These estimates hold uniformly on bounded subsets of L 2 .

B Spectra

199

Proof. These estimates follow by noting that, for example, λ2n (q) = µn (Tt q),

Tt q = q( · + t)

for some 0 ≤ t ≤ 1, and applying Proposition B.5.

t u

We also note the following improved estimate for potentials in H0N , which is used in section 11. See [61, 84], and in particular [51] for an elementary proof. Proposition B.9. For q ∈ H0N with N ≥ 1, X


n 2N |γn |2 + |µn − τn |2 = O(1)

n≥1

locally uniformly on H0N , as well as a small complex neighbourhood of it. The next product formulas complements the product formula for m 2 . Proposition B.10. For q ∈ L 2 , 12 (λ) − 4 = 4(λ0 − λ)

Y (λ2n − λ)(λ2n−1 − λ) . n4 π4

n≥1

At any Dirichlet eigenvalue µn , 12 (µn ) − 4 = (m 1 + m 02 )2 − 4m 1 m 02 = (m 1 − m 02 )2 by the Wronskian Identity. Hence, one can define a unique, real analytic root of this function by q ∗ 12 (µn ) − 4 = m 1 (µn ) − m 02 (µn ), compare equation (6.2). Like the Dirichlet eigenvalues, each periodic eigenvalue is a compact function of q in L 2 , and the proof is completely analogous to the one of [112, Theorem 2.3]. Unlike the former, however, the latter is a real analytic function of the potential only when it is simple. Proposition B.11. Each periodic eigenvalue λn is a compact function on L 2 , which is real analytic on the open domain where it is simple, with gradient ∂1 ∂λn = − = f n2 . ˙ λ 1 n Moreover, γn2 = (λ2n − λ2n−1 )2 and τn = (λ2n + λ2n−1 )/2 are real analytic on all of L 2 .

200

VII Background Material

Proof. We prove the statements concerning τn . By the product representation of 12 (λ) − 4 above and the residue theorem, Z ˙ 1 1(λ)1(λ) τn = λ 2 dλ, 2πi 0n 1 (λ) − 4 with some circuit 0n encircling counterclockwise precisely λ2n and λ2n−1 . Since 0n can be kept fixed locally, this shows that τn is a real analytic function of q. u t Proposition B.12.   1 τn = µn + hcos 2πnx ,qi + O n locally uniformly on L 2 . Proof. Write 1

1 Z τn − µn = (τn − µn )(tq) = 0

h(∂τn − ∂µn )(tq),qi dt.

0

We only need to consider the gradients where the n-th gap√is open. In this case, the two √ normalized periodic eigenfunctions can be written as 2 sin(x − xn ) + O(1/n) and 2 cos(x − xn ) + O(1/n). With Propositions B.5, B.7 and B.11 we get     1 1 ∂τn = 1 + O , ∂µn = 1 − cos 2πnx + O , n n which gives the result. u t ˙ By the Counting Lemma mentioned above, 1 ˙ has a unique root Consider now 1. ˙ in each interval [λ2n−1 , λ2n ], denoted λn , and no other roots. These roots determine ˙ completely. the function 1 Proposition B.13. For q ∈ L 2 , ˙ 1(λ) =−

Y λ˙ n − λ n2 π2

n≥1

and λ˙ n − τn = O



γn2 log n n



locally uniformly on L 2 for all large n. Also, λ˙ n − τn = O(γn2 ) for any fixed n ≥ 1. Remark. In fact, we can prove the stronger estimate λ˙ n − τn = O(γn2 /n) using Proposition D.9 and the remark preceding it.

B Spectra

201

Proof. We only prove the estimate for λ˙ n . Fix q in L 2 . By Proposition B.10, 12 (λ) − 4 = with χn (λ) = 4

(λ2n − λ)(λ − λ2n−1 ) χn (λ) n2 π2

(B.2)

λ − λ0 Y (λ2m − λ)(λ2m−1 − λ) . n2 π2 m 4 π4 m6 =n

By the asymptotics of the periodic eigenvalues and Lemma L.1 we have   log n χn (λ) = 1 + O n in a neighbourhood of size O(1) around the interval [λ2n−1 , λ2n ], uniformly in a neighbourhood of q. Consequently,   log n χ˙ n (λ) = O n by Cauchy’s estimate on a neighbourhood of similar size. Now, λ˙ n may be characterized as the unique zero of (12 )˙ near τn . Differentiating equation (B.2) with respect to λ and multiplying with n 2 π 2 , we may thus characterize λ˙ n as the unique solution of 0 = 2(λ − τn )χn + (λ2n − λ)(λ − λ2n−1 )χ˙ n near τn . With 4(λ2n − λ)(λ − λ2n−1 ) = γn2 − 4(λ − τn )2 , this is equivalent to λ˙ n being the unique root of 8(λ − τn )χn − 4(λ − τn )2 χ˙ n + γn2 χ˙ n = 0

(B.3)

near τn . As q is real, this immediately gives the claimed estimate. The following argument also applies to q in a complex neighbourhood of L 2 . Consider the last term in the last equation, gn = γn2 χ˙ n , as a small perturbation of the other terms, f n = 8(λ − τn )χn − 4(λ − τn )2 χ˙ n . Clearly, f n (τn ) = 0. Moreover, in a neighbourhood Un of size O(1) around τn , the λ-derivative of f n is f˙n = 8χn − 4(λ − τn )2 χ¨ n ∼ 8 for large n, while gn = γn2 O(log n/n), locally uniformly in q. It thus follows from the inverse function theorem that for all large n, Un contains a unique solution λ˙ n of equation (B.3), and that λ˙ n − τn = O(gn ). This proves the result for large n. The same argument applies for any individual n when γn → 0, since then gn is much smaller than f˙n , and the implicit function theorem applies as well. So also the last statement is proven. u t

202

VII Background Material

Isospectral Sets  Let L 20 = q ∈ L 2 : [q] = 0 , and consider the set  Iso(q) = p ∈ L 20 : spec( p) = spec(q) of all potentials in L 20 with the same periodic spectrum as q. To give a topological description of this set we introduce the following notation. For a real interval [a, b], define ( { (a, 0), (b, 0) } ∪ (a, b) × { −1, 1 }, a < b, [[a, b]] = { (a, 0) }, a = b. Endow this subset of R2 with the coarsest topology which makes the projections onto the two factors continuous. In this way, [[a, b]] is homeomorphic to a circle with center (a + b)/2 and radius (a − b)/2, with (a, 0) and (b, 0) being the points of intersection of this circle with the interval [a, b]. For the following result see [89]. Theorem B.14. For any q ∈ L 20 , the map Y µ × σ : Iso(q) → [[λ2n−1 , λ2n ]] n≥1


p 7→ µn ( p), σn ( p) n≥1 , where σn ( p) = sign

q ∗ 12 (µn ( p), q) − 4)

is a homeomorphism, when the right hand side is endowed with the product topology. Thus, the set Iso(q) is homeomorphic to a torus, whose dimension equals the number of non-collapsed spectral gaps [λ2n−1 , λ2n ]. In particular, every isospectral set is compact. The proof of Theorem B.14 uses the following coordinates, briefly mentioned in section 6, see [43, 89]. For q ∈ L 20 and n ≥ 1 let µˆ n (q) = µn (q) − n 2 π 2 ,

κn (q) = log (−1)n m 02 (µn (q)).

It is not difficult to verify that (κn )n≥1 belongs to `21 , while, by Proposition B.5, (µˆ n )n≥1 belongs to `2 .

a

b

∼ =

a

Figure 7 The set [[a, b]]

b

B Spectra

203

Theorem B.15. The map µ × κ : L 20 → `2 × `21
q 7 → µˆ n (q), κn (q) n≥1 is a real analytic embedding. For a complete proof of this theorem see [112]. Proof of Theorem B.14. Fix q ∈ L 20 and its associated ‘circles’ [[λ2n−1 , λ2n ]] for n ≥ 1. The periodic spectrum is the same for each p in Iso(q), so also their 1functions are the same. Hence, for any n ≥ 1 and p ∈ Iso(q), λ2n−1 ≤ µn ( p) ≤ λ2n , q ∗ 12 (µn ( p)) − 4 6 = 0 iff λ2n−1 < µn ( p) < λ2n . Therefore, the map µ × σ is well defined on Iso(q) and takes values in the target space as stated. To show that this map is onto, pick (µ¯ n , σ¯ n ) ∈ [[λ2n−1 , λ2n ]] arbitrarily for each n ≥ 1. Then define   q (−1)n + 2 1(µ¯ n , q) − σ¯ n 1 (µ¯ n , q) − 4 . κ¯ n = log 2 By the asymptotic behavior of µ¯ n and 1 one verifies that (κ¯ n )n≥1 belongs to `21 . So by Theorem B.15 there exists a unique potential p ∈ L 20 with (µ × κ)( p) = (µ¯ n , κ¯ n )n≥1 . We claim that p ∈ Iso(q) with (µ × σ )( p) = (µ¯ n , σ¯ n )n≥1 . To prove this, write m n = m 02 (µ¯ n ) = (−1)n eκ¯ n for n ≥ 1. We have m 1 (µn ) = 1/m 02 (µn ) at any Dirichlet eigenvalue, hence 1 + mn , mn q 1 ∗ 12 (µ¯ n , p) − 4 = − mn , mn 1(µ¯ n , p) =

for n ≥ 1. On the other hand, by the definition of κ¯ n we have   q 1 + 2 mn = 1(µ¯ n , q) − σ¯ n 1 (µ¯ n , q) − 4 . 2

(B.4)

204

VII Background Material

From this one directly calculates 1 + m n = 1(µ¯ n , q), mn q 1 + − m n = σ¯ n 12 (µ¯ n , q) − 4. mn Comparing this with equation (B.4) one sees that the 1-functions of p and q agree at the points µ¯ n , n ≥ 1. Since the 1-function is uniquely determined by its values at µ¯ n due to its interpolation property, we conclude that p and q have the same 1function and hence the same periodic spectrum, so p belongs to Iso(q). Furthermore, σn ( p) = σ¯ n for all n ≥ 1. So (µ × σ )( p) = (µ¯ n , σ¯ n )n≥1 . Q Thus, µ × σ maps Iso(q) onto n≥1 [[λ2n−1 , λ2n ]]. By the same token, this map is also one-to-one, since µ × κ is an isomorphism. It is continuous with respect to the product topology, since all data are continuous functions of q. To prove the continuity of the inverse map, we show that Iso(q) is compact. By the asymptotics of the 1-function given in Proposition B.2 the L 2 -norm of q is fixed, once the periodic spectrum is fixed. Thus, given any sequence in Iso(q), we can extract a weakly convergent subsequence, since it is bounded. This subsequence converges even strongly, since also the L 2 -norms converge. Clearly, the limit function also belongs to Iso(q) by the continuity of the periodic eigenvalues. Thus, Iso(q) is compact. u t Density of Finite Gap Potentials The following result was first proven by Marˇchenko and Ostrowski using inverse spectral theory – see [84] and later [49]. The proof given here is more elementary and was initiated by [27]. Theorem B.16. Finite gap potentials are dense in L 20 . To prove this theorem we introduce for potentials in L 20 the complex quantities αn = τn − µn + i 2πnκn ,

n ≥ 1.

Lemma B.17. There exists a complex neighbourhood W of L 20 so that each αn is a complex analytic function on W with asymptotics   1 2πinx αn = he ,qi + O n locally uniformly on W .

B Spectra

205

Proof. There exists a complex neighbourhood W of L 20 so that µn , κn and τn are real analytic functions on W for all n ≥ 1. So each αn is a complex analytic function on W as well. Moreover, τn − µn = hcos 2πnx ,qi + O(1/n), 2πnκn = hsin 2πnx ,qi + O(1/n), locally uniformly on W by Lemma B.12 and [112, p. 59]. Hence, αn = hcos 2πnx ,qi + i hsin 2πnx ,qi + O(1/n), which proves the asymptotics.

t u

Lemma B.18. For q in L 20 and any n ≥ 1, γn (q) = 0 iff αn (q) = 0. Proof. Fix q and n. If γn = 0, then µn = τn , and the n-th Dirichlet eigenfunction gn is also a periodic or anti-periodic eigenfunction. But then 0 y (1, µn ) = 1, 2 whence also κn = 0, and thus αn = 0. Conversely, if αn = 0, then κn = 0 implies that gn is a periodic or anti-periodic eigenfunction, hence µn is also a periodic eigenvalue. Since in addition µn = τn , the corresponding gap must be collapsed. u t Consider now the map A : L 20 → `2C ,

q 7→ (αn (q))n≥1 .

By Lemma B.17 and Theorem A.5, this map is complex analytic. By Lemma B.18, q in L 20 is a finite gap potential, if and only if all but finitely many coordinates of A(q) vanish. To prove Theorem B.16, however, we instead consider the map G = A B 8 : `2C → `2C , where 8 : `2C → L 20 ,

(ξn )n≥1 7→ Re

X

ξn e2πinx

n≥1

is the inverse of the restriction of the discrete Fourier transform to L 20 . Since 8 is a linear isomorphism, it suffices to prove the following statement. Proposition B.19. For ξ in a dense subset of `2C , all but finitely many coordinates of G(ξ ) vanish.

206

VII Background Material

Proof. In view of Lemma B.17, G is a real analytic map, when considered as a map (Re ξ, Im ξ ) 7 → (Re G(ξ ), Im G(ξ )). It is of the form I + K , where K maps `2C into the smaller spaces o n X n 2β |ξn |2 < ∞ , `2β,C = ξ ∈ `2C :

0<β<

n≥1

1 . 2

It follows with Cauchy’s inequality that on some ball around any given point in `2C , the Jacobian d K is uniformly bounded as a linear map `2C → `2β,C . Consequently, kTN d K k ≤

1 2

on the same ball for all sufficiently large N in the operator norm on `2C , where TN denotes the projection onto all except the first N coordinates in `2C . Now fix ξ o in `2C , and let ε > 0 be so small that the preceding estimate holds on the 4ε-ball B around ξ o for all sufficiently large N . We may then fix N so large that also

TN G(ξ o ) < ε. Writing ξ = ξ N + ζ N with ζ N = TN ξ we then have TN G(ξ ) = TN G(ξ N + ζ N ) = ζ N + TN K (ξ N + ζ N ) with

dζ TN K ≤ 1 N 2 uniformly on B. The map ζ N 7→ ζ N + TN K (ξ No + ζ N ) is thus a local diffeomorphism, and by the inverse function theorem the image of the ball kζ N k < 4ε under this map covers a ball of radius 2ε around TN G(ξ o ). Consequently, in view of kTN G(ξ o )k < ε, there exists ξ˜ = ξ No + ζ˜ N with



ξ˜ − ξ o = ζ˜ N − ζ o < 4ε N such that TN G(ξ˜ ) = 0. Since ε > 0 can be chosen arbitrarily small, this proves the claim. u t Remark 1. The proof incidentally shows that for a finite gap potential any finite number of Fourier coefficients can be prescribed arbitrarily. Remark 2. The preceding proof can be extended to show that finite gap potentials are also dense in any space H0N with N ≥ 1. Essentially, this requires to appropriately improve the asymptotic estimates of Lemma B.17.

C KdV Hierarchy

207

C KdV Hierarchy There is no generally established notion of the KdV hierarchy, and the definitions found in the literature are typically connected with specific ways of constructing it – see [95] and subsequently [31, 35, 41, 78, 83, 87, 104], among others. In this book we define the KdV hierarchy as a sequence of Hamiltonian equations d ∂ Hn , dx ∂u

ut = with Hamiltonians Hn =

Z S1

n ≥ 0,

pn (u, u x , . . . ) dx.

(C.1)

Each pn is a polynomial in u and its derivatives up to order n, such that these Hamiltonians are in the KdV algebra, the Poisson algebra of all Hamiltonians in involution with the KdV actions. One way to obtain such a sequence is as follows. Consider the Floquet matrix     m1 m2 y1 y2 F(λ) = (1, λ) = (1, λ) m 01 m 02 y10 y20 associated with the equation −y 00 + qy = λy, and its discriminant 1(λ) = tr F(λ). Since det F(λ) = 1 and 1(λ) > 2 for λ < λ0 (q), where λ0 (q) denotes the zero-th periodic eigenvalue of q, the two eigenvalues of F(λ) are real, positive, and distinct for λ → −∞. Moreover, exactly one eigenvalue, denoted w(λ), is greater than 1. For this eigenvalue one computes p 1(λ) + 12 (λ) − 4 1(λ) log w(λ) = log = arcosh . (C.2) 2 2 It turns out that for λ → −∞, this quantity admits an expansion of the form log w(λ) ∼

X √ −λ − n≥0

1 4n+1

Gn √ 2n+3 , −λ

whose coefficients G n are of the form (C.1). As they only depend on 1 and thus on the periodic spectrum of q, these functions belong to the KdV algebra. Indeed, they define a KdV hierarchy, and H n = (−1)n G n is referred to as the n-th KdV Hamiltonian. Calculation First note that w(λ) for λ < λ0 (q) is the Floquet multiplier of any expanding solution of −y 00 + qy = λy (C.3) over the period 1. As these solutions have no roots, we begin by writing them in a special form – see [84]. From now on we assume q to be a smooth, real-valued, 1-periodic function on the real line with mean-value zero.

208

VII Background Material

Lemma C.1. Let  Z g = exp µx +

x

 σ (r, µ) dr .

0

Then g is a solution of (C.3) with λ = −µ2 and µ sufficiently large, iff σ satisfies the Riccati equation σ 0 + 2µσ + σ 2 = q. (C.4) In particular, g is a Floquet solution of (C.3) iff σ is 1-periodic. Its Floquet multiplier is then   Z 1 w(λ) = exp µ + σ (r, µ) dr . 0

Proof. Writing g as above, we have g 0 = (µ + σ )g,

g 00 = (µ + σ )2 g + σ 0 g.

Hence, with λ = −µ2 , −g 00 + (q − λ)g = (−2µσ − σ 2 − σ 0 + q)g, from which the first claim follows. To be a Floquet solution the data of g at 0 and 1 must be proportional, or g 0 (1) g 0 (0) = , g(0) g(1) since g never vanishes. With g 0 = (µ + σ )g this amounts to µ + σ (0) = µ + σ (1), which gives the second claim. The multiplier of g is then   Z 1 g(1) = g(1) = exp µ + σ (r, µ) dr . t u g(0) 0 The solution σ of the Riccati equation (C.4) depends on µ and q, and we show that it admits an expansion at µ = ∞. To keep things simple, we consider formal expansions. Lemma C.2. Let q be smooth and 1-periodic. Making for µ → ∞ the formal ansatz X sn σ ∼ , (−2µ)n n≥0

σ is a formal solution of (C.4) iff s0 = 0, s1 = −q and X sn+1 = sn0 + sn−m sm , n ≥ 1. 0≤m≤n

Moreover, all sn are 1-periodic, too.

C KdV Hierarchy

209

Proof. Inserting the ansatz into (C.4) and writing µ˜ = −2µ we obtain q = σ 0 + 2µσ + σ 2 X s 0 − µs ˜ n X X sn−m sm n = + µ˜ n µ˜ n n≥0 n≥0 0≤m≤n  X X 1  X sn+1 0 = s + s s − µs ˜ 0. n−m m − µ˜ n n µ˜ n n≥0

0≤m≤n

n≥0

Comparing terms of order −1, 0, 1, . . . in 1/µ˜ gives the first result. As the sn are polynomial expressions in q and its derivatives, they are 1-periodic. u t By a straightforward calculation, s1 = −q, s3 = −q 00 + q 2 , s5 = −q (4) + 6qq 00 + 5q 0 − 2q 3 , 2

s7 = −q (6) + 10qq (4) + 28q 0 q 000 2

2

+ 19q 00 − 30q 2 q 00 − 50qq 0 + 5q 4 , while the even coefficients s2 , s4 , . . . are all exact with 1-periodic primitives. Letting Z Sn =

1

sn (x) dx,

0

the terms S2 , S4 , . . . thus vanish, while 1

Z

q dx = 0

S1 = − 0 1

Z

q 2 dx,

S3 = 0

Z S5 = − 0 1

Z S7 = 0

1


qx2 + 2q 3 dx,


qx2x + 10qqx2 + 5q 4 dx,

by partial integration. In this book the Sn define the KdV hierarchy of Hamiltonians through Hn =

(−1)n S2n+3 . 2

210

VII Background Material

In particular, Z 1 1 2 q dx, 2 0 Z
1 1 2 H1 = qx + 2q 3 dx, 2 0 Z
1 1 2 2 H = qx x + 10qqx2 + 5q 4 dx. 2 0

H0 =

Theorem C.3. For a smooth, 1-periodic potential q with mean value zero, arcosh

X (−1)n 1(λ) √ Hn ∼ −λ − , 2 4n+1 √−λ 2n+3 n≥0

for λ → −∞, where each H n is an integral over [0, 1] of a polynomial expression in q and its derivatives up to order n given by H n = (−1)n S2n+3 /2. Proof. Combining equation (C.2) with the results of the preceding two lemmas we obtain, with λ = −µ2 , arcosh

X Sn 1(λ) = log w(λ) = µ + . 2 (−2µ)n n≥0

Re-indexing the series and setting H n = (−1)n S2n+3 /2 gives the result. u t

VIII Psi-Functions and Frequencies

D Construction of the Psi-Functions In this appendix we prove the following theorem stated in section 8. In the form presented it is due to [6], but the proof given here is much simpler, and the normalizing constants are explicitly computed. See also [90] for prior results. – For notations we refer to sections 6 and 7. Theorem D.1. There exists a complex neighbourhood W of L 20 such that for each q in W there exist entire functions ψn , n ≥ 1, satisfying Z 1 ψn (λ) p dλ = δmn 2π 0m c 12 (λ) − 4 for all m ≥ 1. These functions depend analytically on λ and q and admit a product representation 2 Y σmn − λ ψn (λ) = , πn m 2 π2 m6 =n

whose complex coefficients

σmn

depend real analytically on q and satisfy 2 n σ − τm ≤ C |γm | m m

for all m, locally uniformly on W and uniformly in n. We prove this theorem with the help of the implicit function theorem. To this end we reformulate the statement in terms of a functional equation. In the following, it is convenient to denote σmn as σ¯ mn , and to use the former symbol for general `2 -sequences. Moreover, σ¯ m = m 2 π 2 + σm throughout this appendix.

212

VIII Psi-Functions and Frequencies

For σ = (σm )m≥1 in `2 and n ≥ 1 define an entire function φn (σ ) by φn (σ, λ) =

Y σ¯ m − λ . m 2 π2

m6 =n

For q in L 20 and m ≥ 1 define a linear functional Am (q) on the space of entire functions by Z 1 φ(λ) p dλ. Am (q)φ = c 2π 0m 12 (λ, q) − 4 Locally, one can choose the contours 0m to be independent of q, and one can choose them arbitrarily close to the real interval G m (q) = [λ2m−1 (q), λ2m (q)], so that Am is actually well defined on the space of real analytic functions on the real line. For each n ≥ 1 we then consider on `2 × L 20 the functional equation F n (σ, q) = 0, where F n = (Fmn )m≥1 with ( Fmn (σ, q)

=

Anm (q)φn (σ ), σ¯ n − τn (q),

m 6= n, m = n,

(D.1)

and, for m 6 = n, n Anm = wm Am ,

n wm = 2πm

n2 − m 2 . n2

In fact, each function Fmn is defined and real analytic on some complex neighbourhood U of `2 × L 20 , which is independent of n and m. We show that under some mild provisions there exists a unique solution σ n (q) of F n (σ, q) = 0, which is real analytic in q and extends to some complex neighbourhood of L 20 independently of n. We then verify that σ¯ mn = τm + O(γm2 /m), and that this solution satisfies An (q)φn (σ n (q)) =

πn . 2

Thus the functions ψn = will have the required properties.

2 φn (σ n ) πn

D Construction of the Psi-Functions

213

Real Solutions Before constructing real solutions we first establish the proper setting of the functionals F n . Lemma D.2. For each n ≥ 1, equation (D.1) defines a map F n : `2 × L 20 → `2 (σ, q) 7→ F n (σ, q), which is real analytic and extends analytically to the complex neighbourhood U of `2 × L 20 introduced above. Moreover, this neighbourhood U can be chosen so that all F n are locally uniformly bounded on it. Proof. Fix n, and consider Fmn for m 6= n. By the definition of φn and the product formula for 12 − 4 in Proposition B.10, σ¯ m − λ φn (σ, λ) p = p ζmn (λ) c s 2 (λ2m − λ)(λ − λ2m−1 ) 1 (λ) − 4

(D.2)

for λ near 0m with ζmn (λ) =

(−1)m+1 n 2 π 2 Y σ¯ l − λ p . √ 2 + λ − λ0 σ¯ n − λ l6=m + (λ2l − λ)(λ2l−1 − λ)

(D.3)

The absolute value of the infinite product is 1 + O(log m/m) by Lemma L.2 uniformly on bounded subsets of `2 × L 20 , since λ = m 2 π 2 + O(1) near 0m . On the other hand, locally uniformly around any point in `2 × L 20 we may choose δ > 0 and the contours 0m in such a way that inf min |σ¯ n − λ| ≥ δ,

m6=n λ∈0m

max |σ¯ m − λ| = O(ρm ),

λ∈0m

where def

ρm = |σ¯ m − τm | + γm +

1 . m

n , we then get Taking into account the definition of the weights wm   log m n n wm ζm = 1 + O m

near 0m , and furthermore Z 1 σ¯ m − λ p w n ζ n dλ = O(ρm ) 2π 0m s (λ2m − λ)(λ − λ2m−1 ) m m by Lemma M.1. These estimates hold uniformly in n.

(D.4)

(D.5)

214

VIII Psi-Functions and Frequencies

Taking into account the definition of Anm we altogether have Fmn (σ ) = O(ρm ),

m 6= n,

locally uniformly on `2 × L 20 . It follows that F n maps `2 × L 20 into `2 . Exactly the same arguments can be used to show that F n maps U into `2C and that the same estimates hold. Again, the last bound depends on σ and q in a locally uniform fashion, but not on n. Therefore, F n is locally bounded on the complex neighbourhood U uniformly in n. We already noticed that each function Fmn is real analytic on U . Analyticity of the entire map F n then follows with Theorem A.5. u t `2

Next we consider the Jacobian of F n with respect to σ . At any given point in × L 20 this Jacobian is a bounded linear operator Q n : `2 → `2 ,

which is represented by an infinite matrix (Q nmr ) with elements Q nmr =

∂ Fmn ∂ ∂φn = An φn = Anm , ∂σr ∂σr m ∂σr

m, r 6 = n,

while Q nmn = Q nnm = δmn . But first we make a simple observation, which is used several times below. Lemma D.3. If φ is real analytic on the real line, and Am φ = 0 for some m ≥ 1, then φ has a root in [λ2m−1 (q), λ2m (q)]. Proof. By assumption, Am φ =

1 2π

Z 0m

φ(λ) p dλ = 0 c 12 (λ) − 4

with a contour 0m around G m sufficiently close to the real axis. If γm > 0, then we may shrink the contour to the interval [λ2m−1 , λ2m ] to obtain 1 π

Z

λ2m

λ2m−1

φ(λ) p dλ = 0, 12 (λ) − 4

which is possible only when φ changes sign in this interval. If γm = 0, we may extract the factor (λ − τm )2 from the product representation of 12 (λ) − 4 and note that the contour integral above turns into a Cauchy integral around τm , which then gives φ(τm ) = 0. u t This simple lemma is the motivation why we look for entire functions ψn of the form Y σn − λ m ψn (λ) = cn m 2 π2 m6=n

D Construction of the Psi-Functions

215

in the first place. It also shows that we have to look for the zeroes σmn in the interval G m (q) = [λ2m−1 (q), λ2m (q)]. It therefore makes sense to restrict ourselves to the open domain V ⊂ `2 × L 20 characterized by λ2k−2 + λ2k−1 λ2k + λ2k+1 < σ¯ k < , 2 2

k ≥ 1.

As a consequence, any solution (σ, q) in V leads to a monotone sequence σ¯ mn , which in turn makes σ unique. Lemma D.4. On V ⊂ `2 × L 20 , the diagonal elements Q nmm never vanish and satisfy Q nmm



log m =1+O m



Q nmr

ρm =O 2 |m − r 2 |



for m 6 = n, while 

for m 6 = r and m, r 6 = n, with ρm defined in (D.4). These estimates hold uniformly in n. Proof. By the definition of φn , for r 6 = n, 1 Y σ¯ l − λ ∂φn φn = 2 2 = . ∂σr σ¯ r − λ r π l 2 π2 l6=n,r

Hence, for λ near 0m and m, r 6 = n, ∂φn 1 φn 1 p p = c c ∂σr 12 (λ) − 4 σ¯ r − λ 12 (λ) − 4 σ¯ m − λ ζmn p = s σ¯ r − λ (λ2m − λ)(λ − λ2m−1 ) with ζmn as in (D.3). Taking into account the weights in the definition of Anm we get Q nmr =

1 2π

Z 0m

n ζn σ¯ m − λ wm m p dλ. σ¯ r − λ s (λ2m − λ)(λ − λ2m−1 )

For the diagonal element Q nmm =

1 2π

Z 0m

n ζn wm m p dλ, s (λ2m − λ)(λ − λ2m−1 )

the claimed estimate now follows immediately with (D.5) and Lemma M.1. Moreover, Q nmm does not vanish by Lemma D.3, since ζmn has no root in [λ2m−1 , λ2m ].

216

VIII Psi-Functions and Frequencies

To estimate the off-diagonal terms Q nmr , note that   σ¯ m − λ ρm =O σ¯ r − λ |m 2 − r 2 | for λ near 0m with ρm given by (D.4), and apply again (D.5) and Lemma M.1. n ζ n , and the latter has been As Q nmr depends on n only through the product wm m estimated in (D.5) uniformly in n, these estimates hold uniformly in n and locally uniformly on V . u t Lemma D.5. At any point in V the Jacobian Q n of F n with respect to σ is of the form Q n = Dn + K n , where D n : `2 → `2 is an isomorphism in diagonal form and K n : `2 → `2 is compact. Proof. Set D n = diag(Q nmm ), the diagonal of Q n . By the preceding lemma, we have 0 6 = Q nmm ∼ 1, so D n : `2 → `2 has a bounded inverse. Moreover, K n = Q n − D n is a bounded linear operator on `2 with vanishing diagonal and elements   ρm n K mr = Q nmr = O , m 6 = r, |m 2 − r 2 | again by the preceding lemma. Clearly, X K n 2 < ∞, mr m,r

so K n is Hilbert-Schmidt, hence compact.

t u

Lemma D.6. At any given point in V ⊂ `2 × L 20 , each Jacobian Q n for n ≥ 1 is one-to-one and hence a linear isomorphism `2 → `2 . Proof. Fix n ≥ 1. To show that Q n is one-to-one, suppose that Q n h = 0 for some h ∈ `2 . Then clearly h n = 0, since Q nnr = δnr for r ≥ 1 by definition. For m 6= n, we get X ∂ Fn X X hr ∂φn m 0= hr = Anm h r = Anm φn . ∂σr ∂σr σ¯ r − λ r r r 6=n

Thus, using straightforward estimates, def

ψ(λ) =

X r 6=n

X h r Y σ¯ l − λ hr φn = σ¯ r − λ r 2 π2 l 2 π2 r 6=n

l6 =r,n

D Construction of the Psi-Functions

217

defines a function that is entire and satisfies Am ψ = 0 for all m 6= n, hence has a root ξm in each interval [λ2m−1 , λ2m ] with m 6 = n by Lemma D.3. Consequently, letting σ¯ n = ξn = τn , def

ψ∗ (λ) =

X hr σ¯ n − λ ψ= φ∗ , 2 2 σ¯ r − λ n π

φ∗ =

r 6=n

Y σ¯ l − λ , l 2 π2 l≥1

is also an entire function with roots ξm , m ≥ 1. Evaluating ψ∗ on the circles |λ| = Rk = (k + 12 )2 π 2 we find that √ X    hr 1 sin λ ψ∗ (λ) = φ∗ (λ) = O √ σ¯ r − λ k λ r 6=n

by Lemma L.3. On the other hand, by the same lemma we also have √    log k sin λ def Y ξl − λ χ∗ (λ) = 1+O = √ k l 2 π2 λ l≥1

uniformly on the same circles |λ| = Rk . The quotient ψ∗ /χ∗ is thus an entire function with   ψ∗ (λ) =O 1 . sup k |λ|=Rk χ∗ (λ) By Liouville’s theorem this is only possible for ψ∗ = 0, hence h = 0 by evaluating ψ∗ at the points σ¯ m , m 6 = n. This shows that Q n is one-to-one. By the preceding lemma and the Fredholm Alternative, Q n is thus an isomorphism. u t Lemmas D.2 and D.6 allow us to apply the implicit function theorem to any particular solution of F n (σ, q) = 0 in the domain V . The upshot is the following result. Proposition D.7. For any n ≥ 1 there exists a unique real analytic map σ n : L 20 → `2 with graph in V such that F n (σ n (q), q) = 0 everywhere. Indeed, σ¯ mn (q) ∈ G m (q) for each m ≥ 1 at every point q. Remark. To be precise, uniqueness holds within the class of all such analytic maps with graph in V . Proof. First we claim that for any solution of F n (σ, q) = 0 in V one has σ¯ mn (q) ∈ G m (q),

m ≥ 1.

(D.6)

For m = n this is obvious by definition. For any m 6 = n, the fact that Am φn (σ ) = 0 and Lemma D.3 imply that φn has some root ξm in G m . But φn has exactly the roots

218

VIII Psi-Functions and Frequencies

σ¯ 1 < σ¯ 2 < . . . with σ¯ m ∼ m 2 π 2 and m 6 = n, and no other roots. Consequently, σ¯ m = ξm ∈ G m for all m 6 = n, which proves the claim. Now, by Lemma D.6 and the implicit function theorem, any particular solution of F n (σ, q) = 0 in V can be uniquely extended locally such that σ is given as a real analytic function of q. This local solution can be extended by the continuation method along any path from q to any given point in L 20 , since ∂ F n /∂σ is a linear isomorphism everywhere on V and the compactness property (D.6) must hold for any continuous extension. Since L 20 is simply connected, any particular solution of F n (σ, q) = 0 in V thus extends uniquely and globally to a real analytic map σ n : L 20 → `2 with graph in V satisfying F n (σ n (q), q) = 0 everywhere. At q = 0 one solution is given by σ (0) = 0, as one verifies using Cauchy’s formula. Since G m (0) = { m 2 π 2 } for all m ≥ 1, this solution is also unique. Hence there is one and only one such analytic map. u t Complex Extension Proposition D.8. All real analytic maps σ n : L 20 → `2 of Proposition D.7 extend to a common complex neighbourhood of L 20 . Proof. To verify that the solutions σ n of Proposition D.7 all extend to a complex neighbourhood of `2 × L 20 independent of n we first show that at every real point q, the inverses of the Jacobians Q n (σ n (q), q) =

∂ Fn n (σ (q), q) ∂σ

are bounded uniformly in n. Consider the Jacobian Q n = (Q nmr ) at a point in V . We have, for m, r 6 = n, Z ∂φn dλ wn p Q nmr = m c 2π 0m ∂σr 12 (λ) − 4 with Q nmm = 1 + O



 log m , m

Q nmr = O



 ρm , |m 2 − r 2 |

m 6= r

by Lemma D.4 locally uniformly on V . In this identity for Q nmr we can pass to the limit n → ∞ to obtain Z Y σ¯ l − λ ∂φ∗ dλ p Q nmr → Q ∗mr = m , φ∗ = , c l 2 π2 0m ∂σr 12 (λ) − 4 l≥1 where φ∗ is the limit of φn =

Y σ¯ l − λ n2 π2 = φ∗ . σ¯ n − λ l 2 π2

l6=n

D Construction of the Psi-Functions

219

By Lemma D.4, the Q ∗mr satisfy the same asymptotic estimates as Q nmr and define a bounded operator Q ∗ on `2 . Moreover, the same estimates imply that Q n → Q ∗ in the `2 -operator norm locally uniformly on V . The diagonal elements Q ∗mm do not vanish by Lemma D.3, since ∂φ∗ /∂σm has no root in G m . Hence, by the same arguments used in the proofs of Lemmas D.5 and D.6, Q ∗ is boundedly invertible on `2 at every point in V . As the set 5(q) =

Y

G m (q) − m 2 π 2




m≥1

is compact in `2 , Q ∗ is indeed uniformly boundedly invertible for σ in 5(q) for any fixed q. By continuity, then also Q n (σ, q) is uniformly boundedly invertible for all large n for σ in 5(q), and hence for all n. By Lemma D.2 the maps F n are analytic and locally uniformly bounded on a common complex neighbourhood of V uniformly in n. Using Cauchy’s estimate, the variation δ Q of Q n with respect to σ and q can thus be kept as small as needed by restricting oneself to a sufficiently small complex neighbourhood of 5(q) × { q }. Using the standard estimate



(Q + δ Q)−1 ≤ 2 Q −1

if

−1 2 kδ Qk ≤ Q −1

for a perturbation δ Q of Q = Q n , this gives us a similar uniform bound on the inverses of the Jacobians on this complex neighbourhood. The result then follows by the implicit function theorem. u t Remark. Continuing the preceding proof one can actually show that σ n (q) → λ˙ (q) = (λ˙ m (q))m≥1

as n → ∞,

˙ This follows from the fact that where λ˙ m denote the roots of 1. ˙ = Am 1

1 2π

Z 0m

˙ 1(λ) p dλ = 0 12 (λ) − 4

for all m ≥ 1 and a uniqueness argument. Asymptotics Proposition D.9. The components of σ n = (σmn ) satisfy 2 n σ¯ − τm ≤ C γm m m

for all m, locally uniformly in a complex neighbourhood of L 20 and uniformly in n.

220

VIII Psi-Functions and Frequencies

Proof. Fix σ n = (σmn ), and drop the superscript n for this proof. There is nothing to prove for σ¯ n = τn , so consider σ¯ m with m 6= n. In view of their construction by the implicit function theorem, we have a first crude estimate σ¯ m = τm + O(1),

(D.7)

which we refine now. Since σ n solves F n (σ, q) = 0, we have Z 1 φn (σ, λ) p 0= dλ c 2π 0m 12 (λ) − 4 Z 1 σ¯ m − λ p = ζm (λ) dλ, 2π 0m s (λ2m − λ)(λ − λ2m−1 ) def

using (D.2). Writing ζm (λ) = ξm + (ζm (λ) − ξm ) with ξm = ζm (τm ) 6 = 0 and noting that, by a simple computation, Z σ¯ m − λ 1 p dλ = σ¯ m − τm , 2π 0m s (λ2m − λ)(λ − λ2m−1 ) we obtain (σ¯ m − τm )ξm =

1 2π

Z 0m

(λ − σ¯ m )(ζm (λ) − ξm ) p dλ. s (λ2m − λ)(λ − λ2m−1 )

(D.8)

If γm = 0, then the right hand side vanishes, and there is nothing to do. So assume that γm 6 = 0. Choosing the contour 0m close of order γm to G m , |ζm (λ) − ξm | = |ζm (λ) − ζm (τm )| ≤ Mm γm along 0m , where Mm denotes the supremum of ζm0 over the convex hull of 0m . In view of (D.3), (D.7) and Lemma L.2 we have for all large m the asymptotic estimate wm ζm (λ) = O(1) in a neighbourhood of size m around 0m . Hence, wm Mm = O(1/m), wm ξm = 1 + O(log m/m) by Cauchy’s estimate and (D.5), respectively. Moreover, for any m ≥ 1, the eigenvalues λ2m (q), λ2m−1 (q) as well as σ¯ m (q) are contained in an isolating neighbourhood Um locally uniformly in q, whence also |wm ξm | ≥ c > 0 for all m, n locally uniformly in q. From all this and (D.8) we thus obtain, in view of Lemma M.1, |σ¯ m − τm | |ξm | = sup |λ − σ¯ m | O(Mm γm ), 0m

D Construction of the Psi-Functions

and subsequently |σ¯ m − τm | = sup |λ − σ¯ m | O 0m

γ  m

m

221

.

Together with (D.7) this gives |σ¯ m − τm | = O(γm /m). But this in turn implies sup |λ − σ¯ m | ≤ |σ¯ m − τm | + sup |λ − τm | = O(γm ), 0m

0m

which finally gives the claimed estimate. The same arguments show that these estimates hold on some complex neighbourhood of L 20 . Since only the asymptotic behavior of the periodic eigenvalues of q and the initial estimate (D.7) enter, but not n, they hold locally uniformly on this neighbourhood and uniformly in n. u t Normalization Consider now the not yet normalized entire functions ϕ˜n = φn (σ n ). Clearly, we have Am ϕ˜n = 0 for any m 6 = n, and An ϕ˜n 6= 0 by Lemma D.3. Indeed, writing ϕ˜n n2 π2 p = p c s (λ2n − λ)(λ − λ2n−1 ) 12 (λ) − 4 (−1)n+1 Y σ¯ l − λ p · +√ + 2 λ − λ0 l6=n (λ2l − λ)(λ2l−1 − λ)

(D.9)

for λ near 0n in analogy to (D.2), Lemmas L.2 and M.1 lead to    log n πn An ϕ˜n = 1+O . 2 n Thus, we may set cn φn (σ n ) πn with a constant cn = 2 + O(log n/n) depending real analytically on q. This weaker result is completely sufficient to develop the theory of Birkhoff coordinates in chapter III. The proof of Theorem D.1 is complete once we show the following result. ψn =

Proposition D.10. For any n ≥ 1, An ϕ˜n =

πn . 2

We prove this, maybe surprising, fact in the next appendix with the help of Riemann bilinear relations. It is connected with the rigidity of periodic spectra, illustrated for example by the fact that the positive gap lengths alone together with λ0 already determine the entire spectrum of a potential.

222

VIII Psi-Functions and Frequencies

A Sampling Formula Finally we establish a sampling formula for ψn which is used in section 10. For more general formulas of this type see [90]. Proposition D.11. For each n ≥ 1, X ψn (µk ) m 2 (λ) = ψn (λ) m˙ 2 (µk ) λ − µk k≥1

everywhere on L 20 , where m 2 (λ) = y2 (1, λ) and the µk are the Dirichlet eigenvalues of q. Proof. Fix n ≥ 1. By the product expansions for ψn in the theorem above and m 2 in appendix B one has the crude estimate ψn (µk ) = O(1) for k 6 = n m˙ 2 (µk ) locally uniformly on L 20 , by using Lemma L.2 and the asymptotic estimates of the µk and σmn . Hence the sum converges to a real analytic function of λ and q. By the continuity of both sides on L 20 it then suffices to verify the identity on the dense subset of finite gap potentials. So fix such a finite gap potential. Then λ2k−1 = µk = σkn = τk = λ2k and ψn (µk ) = 0 for all k > K , with K sufficiently large. It remains to show that X ψn (µk ) m 2 (λ) = ψn (λ). m˙ 2 (µk ) λ − µk

1≤k≤K

By the product representations for ψn and m 2 , ψn (λ) =

2 πn

Y 1≤m≤K m6 =n

σmn − λ · P(λ) m 2 π2

and, for 1 ≤ k ≤ K , m 2 (λ) 1 =− 2 2 λ − µk k π with P(λ) =

Y 1≤m≤K m6 =k

µm − λ · P(λ), m 2 π2

Y τm − λ . m 2 π2

m>K

Factoring out P(λ), both sides of the sampling formula reduce to polynomials in λ of degree at most K − 1. One also checks with l’Hospital’s rule, that both sides agree at the points µ1 , . . . , µ K , while P does not vanish there. Consequently, both sides must be equal. u t

E A Trace Formula

223

E A Trace Formula In this appendix we prove the following trace formula which is used in the proof of Lemma 11.6 as well as Proposition D.10. Theorem E.1. For any q ∈ L 20 , X n≥1

2πn In =

1 2

Z

1

q(x)2 dx.

0

Note that the right hand side is the zero-th Hamiltonian in the KdV hierarchy which corresponds to translation. Hence the theorem says that in the angle-action coordinates introduced in chapter III the zero-th KdV Hamiltonian is just X H0 = 2πn In , n≥1

and its frequencies are just ∂ H0 = 2πn, ∂ In

n ≥ 1.

This result will be extended to higher KdV Hamiltonians in the next appendix. Riemann Bilinear Relations The proof of the trace formula is based on the Riemann bilinear relations for meromorphic differentials on a Riemann surface. Specifically, we consider the Riemann surface  6(q) = (λ, z) : z 2 = 12 (λ, q) − 4 ⊂ C2 associated with a potential q. In the case of an N -gap potential this is a hyperelliptic surface of genus N , which may be viewed as two copies of the complex plane slit open along the N +1 intervals (−∞, λ0 ), (λ1 , λ2 ), . . . , (λ2N −1 , λ2N ) and then glued together crosswise along the slits. On 6(q) consider the canonical basis (ak ) and (bk ) of cycles known as a- and bcycles, as indicated in Figure 8. Each of the a-cycles is homotopic to the corresponding a 0 -cycle in Figure 9, while the b-cycles are the same. The cycles in Figure 9 only intersect in λ0 , as required by the 2nd reciprocity law – see [52, chapter XX], for example. The configuration given p first, however, is more convenient for computations. By convention, the root 12 (λ) − 4 on the sheet containing the a-cycles is given by the c-root p c i 12 (λ) − 4 > 0 for λ ∈ (λ0 , λ1 ), as defined in section 6 on page 62. We will use the Riemann bilinear relations on 6(q) in the form of the 2nd reciprocity law.

224

VIII Psi-Functions and Frequencies

a1

a2

λ0 b1

b2

Figure 8 a- and b-cycles for N = 2

Proposition E.2. Let q be an N -gap potential. If η and χ are two meromorphic differentials on 6(q) with a single pole at infinity with vanishing residuum, then N

1 X 2πi k=1

Z

η

ak

Z

χ− bk

Z ak

χ

Z bk

  Z η = Resλ=∞ χ

λ

λ0

 η ,

(E.1)

where the integral is taken above the negative real axis on that sheet of 6(q) containing the a-cycles. We also need an asymptotic representation of the 1-function. Note that the following is a weaker form of Theorem C.3. Proposition E.3. For q ∈ L 20 and λ < λ0 (q), arcosh where H 0 =

  1 H0 1 1(λ) √ = −λ − √ 3 + O 2 , 4 −λ 2 λ

R 1 1 2 2 0 q(x) dx

is the zero-th Hamiltonian in the KdV hierarchy.

a10

a20

λ0 b1

b2

Figure 9 a 0 - and b-cycles with basepoint λ0 for N = 2

E A Trace Formula

225

Proof. By Proposition B.2, √  Im √λ  √ sin λ e 1(λ) = 2 cos λ + 2Q 3/2 + O λ λ2 √ λ = iµ and  µ 1(λ) Q e = cosh µ − 3 sinh µ + O . 2 µ µ4

with 4Q = H 0 . For λ = −µ2 with µ > 0, this gives

Expanding arcosh at cosh µ and noting that arcosh0 (cosh µ) =

1 , sinh µ

arcosh00 (cosh µ) = −

cosh µ sinh3 µ

,

we obtain 1(λ) = arcosh(cosh µ) 2   µ  Q 1 e + − 3 sinh µ + O sinh µ µ µ4   µ 2 1 cosh µ˜ Q e − − sinh µ + O 2 sinh3 µ˜ µ3 µ4   Q 1 =µ− 3 +O , µ µ4
√ where µ˜ = µ + O 1/µ3 . With µ = −λ the result follows. u t arcosh

Proof of Theorem E.1 It suffices to prove the trace formula for q in the dense union of N -gap potentials, N arbitrary, as both sides of the formula are continuous in q. We apply the Riemann bilinear relations to the differentials ˙ 1(λ) dλ, η=p 12 (λ) − 4

˙ λ1(λ) χ=p dλ. 12 (λ) − 4

Clearly, they are meromorphic on 6(q) with a single pole at infinity. To determine its residue, note that η = d arcosh 1(λ)/2, hence   1 2 η = − 2 − 3Qz + . . . dz (E.2) λ=−z −2 z by the preceding proposition with 4Q = H 0 . Thus, the residues of η and χ at infinity vanish, and we can apply the Riemann bilinear relations.

226

VIII Psi-Functions and Frequencies

R R We have ak η = 0, since η is exact near ak , and ak χ = π Ik by the definition of the actions Ik in section 7. Shrinking bk to the real axis in the usual way, Z

λ1

˙ 1(λ) p dλ c λ0 12 (λ) − 4 Z λ2k−1 ˙ 1(λ) p + ··· + 2 dλ, c λ2k−2 12 (λ) − 4

Z

η=2

bk

since the integrals along the slits cancel each other. With Z

λ2l+1

λ2l

Z ˙ ˙ 1(λ) (−1)l+1 λ2l+1 1(λ) p p dλ = dλ c + i λ2l 12 (λ) − 4 4 − 12 (λ) 1(λ) λ2l+1 = (−1)l i arcsin 2 λ2l

= −πi, we obtain obtain

R bk

η = −2πik. Hence, for the left hand side of the bilinear relation we N

1 X 2πi

Z

k=1

ak

η

Z

χ−

Z

bk

χ

ak

Z

 X N η = πk Ik .

bk

k=1

On the other hand, with λ = −z −2 and (E.2), Z λ χ η = (z −4 + 3Q + . . .) (z −1 − Qz 3 + . . .) dz λ0

= (· · · + 2Qz −1 + . . .) dz, hence  Z Resλ=∞ χ

λ λ0

 η = 2Q.

This proves the theorem. Proof of Proposition D.10 We end this section by proving Proposition D.10 stated at the end of appendix D. It again suffices to consider N -gap potentials with N sufficiently large. We apply the Riemann bilinear relations to the meromorphic differentials ˙ 1(λ) η=p dλ, 12 (λ) − 4

ϕ˜n (λ) χ=p dλ. 12 (λ) − 4

Note that χ has no poles at τk for k > N , since also ϕ˜n vanishes there by Proposition D.7.

F Frequencies

227

R R As before, akRη = 0 and bk η = −2πik for 1 ≤ k ≤ N . By construction of the function ϕ˜n , also ak χ = 0 for k 6= n. The left hand side of the bilinear relation thus amounts to Z Z Z  Z N Z 1 X η χ− χ η =n χ = 2πn An ϕ˜n , 2πi ak bk ak bk an k=1

with An as on page 212. On the other hand, a straightforward calculation using a representation analogous to (D.9) with λ = −z −2 gives
dz χ = n 2 π 2 z 3 1 + O(z 2 ) 3 z
2 2 2 = n π 1 + O(z ) dz. Together with

Rλ

λ0

(E.3)

η = z −1 − Qz 3 + . . . for λ = −z −2 by (E.2) we obtain  Z Resλ=∞ χ

λ

λ0

 η = n2 π2.

Combining this with the left hand side of the bilinear relation above we arrive at 2πn An ϕ˜n = n 2 π 2 . This proves the proposition. F Frequencies By the results of chapter III, and in particular Theorem 11.13, every Hamiltonian H m in the KdV hierarchy, when expressed in terms of the Birkhoff coordinates in a suitable subspace of h1/2 , is a function of the actions In = (xn2 + yn2 )/2 alone and thus classically integrable. Hence their associated frequencies ωnm are ωnm =

∂ Hm , ∂ In

n ≥ 1.

For example, the zero-th KdV Hamiltonian is H 0 = ωn0 =

∂ H0 = 2πn, ∂ In

R 1 1 2 2 0 q(x) dx,

and

n ≥ 1,

by Theorem E.1. To obtain an asymptotic estimate of the frequencies of higher KdV Hamiltonians H m , we use a procedure first developed in [50]. To start we recall that in Theorem C.3 we obtained the Hamiltonians in the KdV hierarchy as the coefficients in the formal expansion of the 1-function at infinity. More precisely, we have the following result, which extends Theorem E.3.

228

VIII Psi-Functions and Frequencies

Theorem F.1. For a smooth, 1-periodic potential q with mean value zero, arcosh

X (−1)m Hm 1(λ) √ , ∼ −λ − √ 2 4m+1 −λ 2m+3 m≥0

for λ → −∞. Expressing a potential q in terms of the Birkhoff coordinates (x, y) of Theorem 6.1, we may view the 1-function as an analytic function of λ and (x, y). As 1 is a spectral invariant, it is indeed an analytic function of λ and the actions I alone. Thus it makes sense to consider its gradient with respect to In for each n ≥ 1, and to introduce the one-forms ∂1/∂ In dλ. ηn = p 12 (λ) − 4 These are holomorphic one-forms on 6(q) except possibly at infinity. Proposition F.2. For an N -gap potential q, each ηn with 1 ≤ n ≤ N is a holomorphic one-form on 6(q) with ηn ∼ −

X m≥0

1

ωnm

4m+1 √−λ 2m+3

dλ

for λ → −∞. Proof. On the proper sheet of 6(q) we can write   1(λ) ∂ ηn = arcosh dλ ∂ In 2 n. u and apply Theorem F.1 and the definition of the frequencies ωm t

It turns out that these one-forms ηn can be identified with the one-forms ψn χn = p dλ, 2 1 (λ) − 4 with ψn as in Theorem 8.1 or D.1. Proposition F.3. At an N -gap potential, χn = −2ηn for 1 ≤ n ≤ N . Proof. Consider an N -gap potential q and its Riemann surface 6(q). By the preceding proposition, ηn with 1 ≤ n ≤ N is a holomorphic one-form on 6(q). Further, χn is holomorphic at infinity in view of the expansion (E.3), hence on all of 6(q) by its definition. To prove the claim, it therefore suffices to compare the a-periods of ηn and χn . On one hand, by the construction of the ψ-functions, Z χn = 2πδmn . am

F Frequencies

229

On the other hand, recall from the proof of Theorem 7.1 on page 64 that for a circuit 0m sufficiently close to [λ2m−1 , λ2m ], the principal branch of the logarithm   p c φ(λ) = log (−1)n 1(λ) − 12 (λ) − 4 is well defined along 0m and depends analytically on In . Further, ∂ ∂1/∂ In φ(λ) = − p . c ∂ In 12 (λ) − 4 Therefore we get Z Z ηn = am

0m

∂1/∂ In ∂ p dλ = − c 2 ∂ In 1 (λ) − 4

Z 0m

φ(λ) dλ = −πδmn

by equation (7.2). Hence Z

χn = −2

am

Z

ηn

am

for all m ≥ 1, and the result follows. u t Comparing the expansions of χn and ηn in λ = −z −2 we now obtain identities for the frequencies of the first and second KdV Hamiltonian. Theorem F.4. The frequencies of the first and second KdV Hamiltonian, defined for potentials in the appropriate Sobolev spaces, are ωn1 = 8πn (τn − rn ), ωn2 = 32πn (τn2 − rn τn + sn ), with rn = −

λ0 X n + (σm − τm ) 2 m≥1

and sn =

X
3 2 λ0 X n λ0 − (σm − τm ) + τm2 − τm σmn + γm2 /8 8 2 m≥1

m≥1

1X n + (σm − τm ) (σrn − τr ). 2 m6 =r

Remark 1. One verifies that the coefficients rn and sn are locally uniformly 1/2 bounded on L 20 and H0 , respectively. P Remark 2. One can show that λ0 /2 = m≥1 (λ˙ m − τm ) and hence also X rn = (σmn − λ˙ m ). m≥1

230

VIII Psi-Functions and Frequencies

Proof of Theorem F.4. The frequencies ωn1 and ωn2 are analytic on the appropriate Sobolev spaces, since the corresponding KdV Hamiltonians are. By Propositions B.11, D.7 and D.9, the functions τn , γn2 and σmn are all real analytic on L 20 and satisfy σmn − τm = O(γm2 /m),

τn = n 2 π 2 + O(1),

γn = `2 (n),

locally uniformly on L 20 . By continuity, it therefore suffices to prove the identities on a dense subset of finite gap potentials. Choose an N -gap potential q. Then, for λ near −∞, Y πn ψn 1 σm − λ p p = , √ + + 2 τ − λ λ − λ (λ − λ)(λ2m−1 − λ) n 0 1 (λ) − 4 2m 1≤m≤N

+

where σm = σmn , and in particular σn = τn . Hence, on the canonical sheet near −∞ above the real axis, χn

λ=−z −2

=

2πn 5(z 2 ) dz p −2 + τn + z λ0 + z −2 z 3

=

2πn 5(z 2 ) p dz 1 + τn z 2 + 1 + λ 0 z 2

with 5(z 2 ) =

Y 1≤m≤N

1 + σm z 2 p . + (1 + λ2m−1 z 2 )(1 + λ2m z 2 )

Using the identity (1 + λ2m−1 z 2 ) (1 + λ2m z 2 ) = 1 + 2τm z 2 + (τm2 − γm2 /4)z 4 , one obtains 1 + σm z 2 p = + (1 + λ2m−1 z 2 )(1 + λ2m z 2 ) 1 + (σm − τm )z 2 + (τm2 − τm σm + γm2 /8)z 4 + O(z 6 ). p Combining this with the standard expansion of 1/ + 1 + λ0 z 2 one finds that 5(z 2 ) p = 1 + r˜n z 2 + s˜n z 4 + O(z 6 ) + 2 1 + λ0 z with r˜n = −

X λ0 + (σmn − τm ) 2 1≤m≤N

and an expression for s˜n , which is analogous to sn , but with all summation indices restricted to { 1, 2, . . . , N }.

F Frequencies

231

As γm = 0 and τm = σmn for all m > N , these coefficients agree at an N -gap potential with the coefficients rn and sn given above. Together with 1 = 1 − τn z 2 + τn2 z 4 ∓ . . . , 1 + τn z 2 we thus have χn

λ=−z −2


= 2πn 1 − (τn − rn )z 2 + (τn2 − rn τn + sn )z 4 ∓ . . . dz.

On the other hand, by Proposition F.2, ηn

λ=−z −2

=

1 X ωnm 2m z dz. 2 4m m≥0

In view of Proposition F.3 we thus obtain, for an N -gap potential, ωn1 = 8πn (τn − rn ),
ωn2 = 32πn τn2 − rn τn + sn , for 1 ≤ n ≤ N . u t Corollary F.5. The frequencies of the first and second KdV Hamiltonian satisfy the asymptotic identities ωn1 = (2πn)3 + O(n), ωn2 = (2πn)5 + O(n 3 ), locally uniformly on the appropriate Sobolev spaces. Proof. Just observe that τn = n 2 π 2 + O(1) locally uniformly on L 20 , and use the identities stated in the preceding theorem together with the result that on the appropriate Sobolev spaces the rn and sn are bounded in n locally uniformly. u t Remark. In fact, Theorem F.4 allows to derive more precise asymptotic estimate than the ones presented in the preceding corollary. But the latter are sufficient for our purposes, and were obtained by different methods in [7], see also [42].

IX Birkhoff Normal Forms

G Two Results on Birkhoff Normal Forms Generalized Normal Forms Consider a Hamiltonian on the space hr• introduced in section 14 of the form H = H2 + H3 + . . . , where the Hk are homogeneous of degree k in v ∈ hr• . Assume that the quadratic term is already in normal form: X H2 = λn |vn |2 n≥1

with certain frequencies λ1 , λ2 , . . . . Assume further that these frequencies are nonresonant up to order m ≥ 3: X X |kn | ≤ m, k n λn 6 = 0 whenever 1≤ (G.1) n≥1

n≥1

where k1 , k2 , . . . are arbitrary integers. Then there exists a symplectic transformation v = 8(w) = w + . . . , where the dots stand for higher order terms, which takes H into its Birkhoff normal form up to order m: H B 8 = H2 + N4 + · · · + Nm + . . . , where the Nk , 4 ≤ k ≤ m, are homogeneous terms of order k, which are actually functions of |wn |2 , n ≥ 1, and where the trailing dots stand for arbitrary terms of order strictly greater than m [26, 98]. In particular, the normalized part of the Hamiltonian contains no monomials of odd order. Moreover, the coefficients of the terms up to order m are uniquely determined by H , in the sense that for any normalizing transformation 8 of the form 8 = id + . . . one obtains H B 8 = H2 + N4 + · · · + Nm + . . . .

234

IX Birkhoff Normal Forms

The normalization process may be taken to any order beyond m and may include resonant terms corresponding to resonant frequencies, leading to a so called resonant Birkhoff normal form. Its coefficients, however, are no longer uniquely determined, except for coefficients of the terms of order m + 1 and m + 2. To formulate this result, denote by Pk the space of all homogeneous functions of degree k. The nullspace Nk of the operator {H2 , · } in Pk together with its range Rk in Pk provide an invariant splitting Pk = Nk ⊕ Rk . Any term in an expansion H = H2 + H3 + . . . can be uniquely written as Hk = Nk + Rk ,

Nk ∈ Nk , Rk ∈ Rk ,

and such a Hamiltonian H is said to be in generalized normal form up to order m, if Rk = 0 for 3 ≤ k ≤ m. Theorem G.1. Suppose H2 is nonresonant up to order m ≥ 3, and H is in Birkhoff normal form up to order m: H = H2 + N4 + · · · + Nm + Hm+1 + Hm+2 + . . . . Then the generalized Birkhoff normal form is uniquely determined up to order m + 2 with respect to transformations 8 of the form 8 = id + . . . , and is obtained by projecting Hm+1 and Hm+2 onto Nm+1 and Nm+2 , respectively. Proof. The last statement is evident by considering the normalization process: as m + 1 ≥ 4, removing from Hm+1 all nonresonant terms will generate only terms of order ≥ m + 3, hence it will not affect the normal form terms of Hm+2 . So it remains to prove uniqueness. Suppose the Hamiltonian is transformed into the normal form H = H2 + N4 + · · · + Nm + Nm+1 + Nm+2 + . . . , where Nm+1 , Nm+2 are resonant normal form terms in Nm+1 and Nm+2 , respectively. Consider a symplectic transformation 8 = id + . . . so that 0 0 H B 8 = H2 + N40 + · · · + Nm0 + Nm+1 + Nm+2 + ...

is in similar normal form. 8 can be written as ∗ 8 = F3∗ B F4∗ B · · · B Fm+2 B 8m+3 ,

where Fk∗ is the time-1-map of the flow of a homogeneous Hamiltonian Fk of degree k, and where 8m+3 is of the form “identity + terms of order m + 3 or more”. ∗ is already Suppose that for some k with 3 ≤ k ≤ m the effect of F3∗ B · · · B Fk−1 known to be nil, where the composition is understood to be the identity for k = 3: ∗ H B F3∗ B · · · B Fk−1 = H2 + N4 + · · · + Nm + Nm+1 + Nm+2 + . . . .

G Two Results on Birkhoff Normal Forms

235

Then ∗ H B F3∗ B · · · B Fk−1 B Fk∗ = H2 + N4 + · · · + Nm + Nm+1 + Nm+2 + . . .

+ {H2 ,Fk } + {N4 ,Fk } + · · · +

1 2

{{H2 ,Fk },Fk } + . . . ,

which by assumption on Fk∗ is in normal form up to order m + 2. Thus, {H2 ,Fk } + Nk ∈ Nk . In view of Nk ∈ Nk and the splitting Pk = Nk ⊕ Rk we conclude that {H2 ,Fk } = 0, hence Fk ∈ Nk . As H2 is nonresonant up to order m, all elements in N2 ⊕ · · · ⊕ Nm commute. Hence, {N j ,Fk } = 0,

4 ≤ j ≤ m.

In particular, as N3 = 0 by the assumption m ≥ 3, we have F3 = 0. We conclude that Fk∗ does not affect the normal form terms of H up to order m + 2 included, whereas there might be a contribution of order m + 3 from {Nm+1 ,F4 }. By induction, this holds for 3 ≤ k ≤ m. ∗ . Since {H ,F 0 Next, consider the effect of Fm+1 2 m+1 } + Nm+1 = Nm+1 is in normal form by assumption, one obtains {H2 ,Fm+1 } ∈ Nm+1 and thus {H2 ,Fm+1 } = 0,

0 Nm+1 = Nm+1 .

This transformation therefore affects H B 8 only through {H4 ,Fm+1 } at the earliest, 0 that is, at terms of order m + 3. So we also have Nm+2 = Nm+2 . ∗ The same argument applies to Fm+2 . Finally, the transformation 8m+3 has no effect on terms up to order m + 2. u t Simultaneous Normalization We now consider a second, similar Hamiltonian G = G 2 + G 3 + . . . , which we assume to be in involution with H : {G, H } = 0. The finite dimensional version of this situation was investigated carefully by Ito [56]. Here we only need a simplified version for infinite dimensional systems. Theorem G.2. Suppose H satisfies the nonresonance condition (G.1), and 8 = ˆ is a real analytic symplectic transformation around the origin in hr• that id + 8 takes the Hamiltonian H into its Birkhoff normal form up to order m. Then 8 also normalizes every other Hamiltonian G up to order m, if G is in involution with H and starts with quadratic terms.

236

IX Birkhoff Normal Forms

Proof. The proof proceeds by induction. Suppose we know already that the transformed Hamiltonian G B 8 = G 2 + G˜ 3 + · · · + G˜ k−1 + G˜ k + . . . is in normal form up to order k − 1 with 2 ≤ k ≤ m. For k = 2 this is true, since G is assumed to start with quadratic terms. Thus we can start the induction. Since 8 is symplectic, H B 8 and G B 8 are in involution: {H B 8,G B 8} = {H ,G} B 8 = 0. Collecting terms of order k and using analogous notation for H B 8, we obtain X {H2 , G˜ k } + { H˜ l , G˜ k+2−l } = 0, 3≤l≤k

where the latter sum is absent for k = 2. For k > 2, this sum vanishes, since all the functions involved are known to be in normal form and hence in involution. Hence, {H2 , G˜ k } = 0. Since H2 is assumed to be nonresonant up to order m, G˜ k is in normal form. u t For our purposes we need a slight generalization of Theorem G.2 to the effect that the nonresonance condition (G.1) need only be required for those terms which are actually present in H and G. Addendum to Theorem G.2. The preceding theorem remains valid, if H and G belong to some Poisson subalgebra of Hamiltonians on hr• , and the nonresonance conditions (G.1) hold for all those integer sequences (kn )n≥1 which determine monomiQ als n≥1 vnkn appearing in the expansion of some element in this subalgebra. Proof. Let us consider the case at hand. The Hamiltonians of the KdV hierarchy are integrals of polynomials in u and its derivatives. When expressed in the v-coordinates via the ansatz (14.2), def X v(x) = F (v) = γn vn e2πinx n6=0

√ with γn = 2π |n|, we obtain a Hamiltonian H = H2 + H3 +. . . with homogeneous terms of the form X H j1 ... jk v j1 · · · v jk , (G.2) Hk = j1 +···+ jk =0

where j1 , . . . , jk are arbitrary nonzero integers. Such Hamiltonians form a Poisson subalgebra, since the condition j1 + · · · + jk = 0 is preserved under Poisson brackets. For this subalgebra it suffices to pose a weaker

G Two Results on Birkhoff Normal Forms

237

form of the nonresonance condition (G.1), namely X k n λn 6 = 0 n≥1

when 1≤

X

|kn | ≤ m

and

X

n≥1

The reason is that for Fk =

P

j1 +···+ jk =0

X

{Fk , H2 } = i

nkn = 0.

n≥1

F j1 ... jk v j1 · · · v jk , we have

(λ j1 + · · · + λ jk )F j1 ... jk v j1 · · · v jk ,

j1 +···+ jk =0

where we understand that λn = −λ−n for n < 0. If λ j1 + · · · + λ jk 6= 0 for any term with j1 + · · · + jk = 0 not already in normal form, then one can normalize H by a symplectic transformation as described in section 14. But this condition is equivalent with X X kn λn 6 = 0, nkn = 0, n≥1

n≥1

by setting kn = card{ jl : jl = n}. Moreover, if {G k , H2 } = 0 for a G k with the same representation as Fk above, then one concludes that G k must be in normal form. This establishes the Addendum in our special case. The general case is handled in the same way. u t As a first application of Theorem G.2 we consider the zero-th Hamiltonian in the KdV hierarchy, Z X 1 v 2 dx = γn2 |vn |2 , H 0 (v) = 2 S1 n≥1

for v = F (v). This Hamiltonian is quadratic and in involution with the KdV Hamiltonian, so the transformation 8 of Theorem 14.2 normalizes it up to order four. Recall that 8 = 9 B 4 = F3∗ B F4∗ , where F3 , F4 are homogeneous of order 3 and 4, respectively. As H 0 B F3∗ = H 0 + {H 0 ,F3 } +

1 {{H 0 ,F3 },F3 } + . . . , 2

and as F3∗ normalizes the KdV Hamiltonian up to order three, we conclude from Theorem G.2 that {H 0 ,F3 } = 0. This implies that H 0 B F3∗ = H 0 . Arguing similarly for F4∗ we obtain the following result.

238

IX Birkhoff Normal Forms

Corollary G.3. The zero-th Hamiltonian H 0 in the KdV hierarchy is invariant under the normalizing transformation 8. That is, H 0 B 8 = H 0 . The same argument suggest an alternate way to determine the Birkhoff normal form of the KdV Hamiltonian Hc = H 1 + 6cH 0 . It suffices to determine 8 for the Hamiltonian with c = 0, that is for H0 = H 1 , so that H0 B 8 = H 1 B 8 is in normal form. For general c 6= 0, Hc B 8 = (H 1 + 6cH 0 ) B 8 = H 1 B 8 + 6cH 0 is then also in normal form. In particular, we conclude that the higher order terms do not depend on c. As a second application of Theorem G.2 and its Addendum we determine the Birkhoff normal form of the second Hamiltonian in the KdV hierarchy. This Hamiltonian, as any other in the hierarchy, is of the form H = H2 + H3 + H4 + . . . , and it is in involution with the KdV Hamiltonian. Moreover, the Addendum applies, since all terms Hk are of the form (G.2). Hence, the transformation 8 of Theorem 14.2 also normalizes this Hamiltonian up to order 4. To determine this normal form write again 8 = 9 B 4 = F3∗ B F4∗ . With the usual expansions into Poisson brackets we have H B F3∗ = H2 + H3 + {H2 ,F3 } + H4 + {H3 ,F3 } +

1 2

{{H2 ,F3 },F3 } + . . .

and H B F3∗ B F4∗ = H2 + H3 + {H2 ,F3 } + H4 + {H2 ,F4 } + {H3 ,F3 } +

1 2

{{H2 ,F3 },F3 } + . . . ,

where the dots comprise all terms of order five or more. This transformed Hamiltonian is known to be in normal form up to order four. Hence, H3 + {H2 ,F3 } = 0, and thus H B 8 = H2 + H4 + {H2 ,F4 } +

1 2

{H3 ,F3 } + . . . .

The term {H2 ,F4 } does not generate any contribution to the normal form, but removes those terms not belonging to it. Thus we must have H B 8 = H2 + π N H4 + 12 π N {H3 ,F3 } + . . . , where π N denotes the projection onto the subspace of Hamiltonians in normal form.

G Two Results on Birkhoff Normal Forms

239

We now calculate the normal form for H = Hc2 . We observed in section 14 that the Hamiltonian of KdV-2 is Hc2 (v) = H 2 (v) + 10cH 1 (v) + 30c2 H 0 (v) Z
2 2 3 2 2 1 2 5 4 = 2 vx x + 5vvx + 2 v + 5cvx + 10cv + 15c v dx.

(G.3)

S1

The third order term is thus Z H3 = Using the expansion v = H3 = −5


5vvx2 + 10cv 3 dx.

S1

n6=0 γn vn e

P

X

2πinx

one finds that X

σlm γk γl3 γm3 vk vl vm + 10c

k+l+m=0

γk γl γm vk vl vm .

k+l+m=0

The coefficient in the first sum may be symmetrized with respect to k, l, m, and since (k + l + m)2 = 0 one finds σlm γk γl3 γm3 + σmk γl γm3 γk3 + σkl γm γk3 γl3 = γk γl γm (σlm γl2 γm2 + σmk γm2 γk2 + σkl γk2 γl2 ) = 4π 2 γk γl γm (lm + mk + kl) = −2π 2 γk γl γm (k 2 + l 2 + m 2 ) with σkl = sgn kl. It follows that H3 =

1 3

X

sklm γk γl γm vk vl vm

k+l+m=0

with sklm = 10π 2 (k 2 + l 2 + m 2 ) + 30c. On the other hand, in (14.4) we had F3 = −

i 3

X k+l+m=0

vk vl vm γ˜k γ˜l γ˜m

with γ˜k = σk γk . We obtain {H3 ,F3 } = i

X

σj

j6=0

=

X j6=0

=−

σj

∂ H3 ∂ F3 ∂v j ∂v− j X

k+l=− j

X k+l+m+n=0 k+l6 =0

skl j γk γl γ j vk vl

X m+n= j

1 vm vn γ˜m γ˜n γ˜− j

γk γl skl(k+l) vk vl vm vn . γ˜m γ˜n

240

IX Birkhoff Normal Forms

The normal form contribution of the last sum consists of those terms with k + m = 0 (and thus also l + n = 0) or k + n = 0 (and thus also l + m = 0). In precisely the same fashion as in section 14 one finds that the contribution of all terms with k 6= l, skl(k+l) σkl |vk |2 |vl |2 , to the normal form is zero, since the coefficients change sign with k and l. Hence, X 1 1X π N {H3 ,F3 } = − snn(n+n) |vn |4 = − (15c + 30π 2 n 2 ) |vn |4 . 2 2 n6=0

n6 =0

It remains to consider the fourth order term. Clearly, from (G.3) we get Z X 5 5 H4 = v 4 dx = γk γl γm γn vk vl vm vn . 2 S1 2 k+l+m+n=0

Counting terms correctly, π N H4 =

X 5 · 16π 2 (3 − 2δkl ) |kl| |vk |2 |vl |2 . 2 k,l≥1

Leaving to the reader the calculation of the term H2 , this proves Theorem 14.5.

H Birkhoff Normal Form of Order 6 In section 14 we calculated the Birkhoff normal form of the KdV Hamiltonian up to order four. The KdV Hamiltonian is nonresonant up to order four at the origin, so by a first transformation we could eliminate all terms of third order, and by a second transformation put the resulting fourth order terms into a unique normal form. According the Theorem G.1 this process even uniquely determines the Birkhoff normal form coefficients up to order six. For this we do not have to require additional nonresonance properties. On the other hand, by the existence of global angle-action coordinates for KdV we do have a classical Birkhoff normal form to any order for KdV. Therefore, by Theorem G.1, our construction even determines the normal form coefficients up to order six, and thus the Taylor series expansion of the KdV Hamiltonian up to order three in the actions. Formally, it is not difficult to determine the sixth order coefficients. Write the KdV Hamiltonian as H = 3 + G = H2 + H3 , dropping the subscript c from the notation, and the normalizing transformation as 8 = 9 B 4 = F3∗ B F4∗ as in appendix G. Let π N denote the projection onto normal form terms.

H Birkhoff Normal Form of Order 6

241

Lemma H.1. The sixth order terms of the Birkhoff normal form of the KdV Hamiltonian are N6 =


1 π N {{G 4 ,F3 },F3 } + 2 {G 4 ,F4 } , 4

G4 =

1 {H3 ,F3 }. 2

To shorten notation, we let G F = {G,F},

G F 2 = {{G,F},F},

and so on in the following. Proof. For H = H2 + H3 we obtain 1 1 1 H2 F32 + H2 F33 + H2 F34 2 6 24 1 1 + H3 + H3 F3 + H3 F32 + H3 F33 + . . . , 2 6

H B F3∗ = H2 + H2 F3 +

where here and below the dots stand for terms of order 7 and more. The Hamiltonian F3 was chosen so that H2 F3 + H3 = 0. Hence we have 1 1 1 H3 F3 + H3 F32 + H3 F33 + . . . 2 3 8 2 1 1 = H2 + G 4 + G 4 F3 + G 4 F32 + . . . , G 4 = H3 F3 . 3 4 2

H B F3∗ = H2 +

By solving H2 F4 + G 4 = π N G 4 in the second step we then obtain 1 2 H B F3∗ B F4∗ = H2 + G 4 + G 4 F3 + G 4 F32 3 4 1 + H2 F4 + G 4 F4 + H2 F42 + . . . 2 2 = H2 + π N G 4 + G 4 F3 3 1 1 2 + G 4 F3 + G 4 F4 + (π N G 4 − G 4 )F4 + . . . . 4 2 Since {π N G 4 ,F4 } contains no normal form terms, π N applied to the last line gives N6 =


1 π N G 4 F32 + 2G 4 F4 . 4

t u

In section 14 we calculated G4 = where B = 3 pare (14.5).

4 k≥1 |vk |

P

1 H3 F3 = −B − Q, 2

is the normal form part of −G 4 , and π N Q = 0. Com-

242

IX Birkhoff Normal Forms

Corollary H.2. The sixth order terms of the Birkhoff normal form of the KdV Hamiltonian are
1 N6 = − π N B F32 + Q F32 + 2Q F4 . 4 Before calculating the Poisson brackets appearing in N6 we collect all the ingredients. From (14.5) we have B=3

X

|vk |4 =

k≥1

3X 2 2 vk v−k 2 k6=0

and Q=

3 2

X k+l+m+n=0

3 = 2 where γk =

√

X k+l+m+n=0

γk γl vk vl vm vn γ˜m γ˜n vk vl vm vn kl, [klmn]

2π |k| and γ˜k = σk γk with σk = sgn(k) as before, and p [k] = σk |k|

for the rest of this appendix. The sum is restricted to k + l, k + m, k + n 6= 0. Symmetrizing the last series in all indices and observing that (k + l + m + n)2 = 0 gives k 2 + l 2 + m 2 + n 2 + 2(kl + km + kn + lm + ln + mn) = 0, (H.1) we have Q=−

1 8

X k+l+m+n=0 k+l,k+m,k+n6=0

vk vl vm vn 2 (k + l 2 + m 2 + n 2 ). [klmn]

Furthermore, F3 =

1 3i

X k+l+m=0

vk vl vm 1 = γ˜k γ˜l γ˜m 3i(2π)3/2

X k+l+m=0

vk vl vm [klm]

and F4 =

3i 64π 5

3i = 16π 3

X k+l+m+n=0

X k+l+m+n=0

k3

γk γl γm γn vk vl vm vn 3 3 3 mn +l +m +n

vk vl vm vn kl , 3 3 [klmn] k + l + m 3 + n 3

where the sum is restricted to the same integers as for Q. Symmetrizing in k, l, m, n

H Birkhoff Normal Form of Order 6

243

and using again (H.1), F4 =

1 8i(2π)3

X k+l+m+n=0 k+l,k+m,k+n6 =0

vk vl vm vn k 2 + l 2 + m 2 + n 2 . [klmn] k 3 + l 3 + m 3 + n 3

Finally, recall that F G = {F ,G} = i

X

σj

j6 =0

∂ F ∂G . ∂v j ∂v− j

To shorten notation, we set F˜3 = (2π)3/2 F3 and F˜4 = (2π)3 F4 . The Term B F32 One finds B F˜3 = −6

X k+l+m=0

= −2

X k+l+m=0

σm

2v vk vl vm −m [klm]

vk vl vm (σk vk v−k + σl vl v−l + σm vm v−m ) [klm]

and then B F˜32 = −6

X k+l+u+v=0 k+l6=0

vk vl vu vv 1 [kluv] k + l · (σk vk v−k + σl vl v−l − 2σk+l vk+l v−k−l ).

Its contribution to the normal stems from the index matches k + u = 0, l + v = 0 or k + v = 0, l + u = 0, whereas the match k + l = 0, u + v = 0 can not arise in this sum. The two matches are different only when k 6= l. Hence we obtain π N B F˜32 = −12

X vk v−k vl v−l µkl |kl| k +l k,l

· (σk vk v−k + σl vl v−l − 2σk+l vk+l v−k−l ) where µkl = 1 for k 6= l and µkk = 12 . The Term Q F32 One finds 1 Q F˜3 = 2

X k+l+m+u+v=0

vk vl vm vu vv k 2 + l 2 + m 2 + (u + v)2 . [klmuv] u+v

244

IX Birkhoff Normal Forms

The sum is restricted to k + l, k + m, l + m, k + l + m 6= 0. After some calculations, one then obtains 3 Q F˜32 = − 2

X k+l+ p+q+u+v=0

vk vl v p vq vu vv k 2 + l 2 + ( p + q)2 + (u + v)2 [klpquv] ( p + q)(u + v)

X

+

k+l+m+ p+q+u=0

vk vl vm v p vq vu k 2 + l 2 + m 2 + (k + l + m)2 [klmpqu] (k + l + m)( p + q)

= X + Y, where the second sum is restricted to integers with k + l, k + m, l + m, k + l + m 6= 0 and p +q 6 = 0, while the first sum is restricted to integers with k +l, p +q, u +v 6= 0 and k + p + q, l + p + q 6 = 0. The two terms are now discussed separately. As to the term X , to contribute to the normal form the indices have to match in the following way. For each k and l, there are 8 ways to match with p, q and u, v, since they can not match both u and v, or both p and q at the same time. In each case there is only one way for the remaining indices to match. Due to the symmetry in p, q and u, v, each such match leads to the same coefficient. A prototypical match is k+p=0 l +u =0 q +v =0 and renaming q as m, the coefficient of −vk v−k vl v−l vm v−m /|klm| becomes −

k 2 + l 2 + (m − k)2 + (m + l)2 , (m − k)(m + l)

with the restriction that the denominator does not vanish and k + l 6= 0, k 6 = m + l. If k = l, but p 6 = q and u 6 = v, then the count of the terms has to be divided by 2. It is not possible, however, that all three indices k, l, m are equal. Hence we obtain π N X = 12

X vk v−k vl v−l vm v−m k 2 + l 2 + (k − m)2 + (l + m)2 ηklm , |klm| (k − m)(l + m)

k,l,m

where ηklm = 1, 21 , 0, depending on whether no, exactly two except k and l, or all indices are equal. The sum is restricted to integers with k + l, k − m, l + m 6 = 0 and k 6= l + m. As to the term Y , for each choice of k, l, m with k +l, k +m, l +m, k +l +m 6 = 0 we can find matching p, q, u to obtain a term in normal form. The possible matches are k+p=0 k+p=0 k+u =0 l +q =0 l +u =0 l+p=0 , or or m+u =0 m+q =0 m+q =0

H Birkhoff Normal Form of Order 6

245

and the three matches we obtain from these by interchanging p and q, which lead to the same coefficient. In the case that k, l, m are pairwise different, the coefficient of −vk v−k vl v−l vm v−m /|klm| is   k 2 + l 2 + m 2 + (k + l + m)2 1 1 1 −2 + + k +l +m k +l k+m l +m = −4

k 2 + l 2 + m 2 + kl + km + lm k 2 + m 2 + m 2 + 3(kl + km + lm) · . k +l +m (k + l)(k + m)(l + m)

If exactly two of the three indices k, l, m are equal, then the count of the terms has to be divided by two, and if all three indices k, l, m are equal, it has to be divided by 6. Hence we obtain X vk v−k vl v−l vm v−m πN Y = 4 µklm |klm| ·

k2

k,l,m + l2

+ m 2 + kl + km + lm k 2 + l 2 + m 2 + 3(kl + km + lm) · , k +l +m (k + l)(k + m)(l + m)

where µklm = 1, 21 , 16 , depending on whether no, exactly two, or all indices are equal. The Term Q F4 One finds 1 Q F˜4 = − 4 ·

k2

X

vk vl vm vu vv vw · [klmuvw]

k+l+m+u+v+w=0 + l 2 + m 2 + (k + l

+ m)2

(k + l + m)

·

u 2 + v 2 + w 2 + (u + v + w)2 , u 3 + v 3 + w 3 − (u + v + w)3

where the sum extends over integers with k + l, k + m, l + m, u + v, u + w, v + w 6= 0

and k + l + m 6 = 0.

Using u 3 + v 3 + w3 − (u + v + w)3 = −3(u + v)(u + w)(v + w) from Lemma 14.4 and expanding squares leads to 1 Q F˜4 = − 3 ·

(k 2

X

vk vl vm vu vv vw · [klmuvw]

k+l+m+u+v+w=0 + l 2 + m 2 + kl + km

+ lm)(u 2 + v 2 + w 2 + uv + uw + vw) . (u + v)(u + w)(v + w)(u + v + w)

To contribute to the normal form, each index in {k, l, m} must match with one index in {u, v, w}. Each match gives the same coefficient of −vk v−k vl v−l vm v−m /|klm|, namely (k 2 + l 2 + m 2 + kl + km + lm)2 . (k + l)(k + m)(l + m)(k + l + m)

246

IX Birkhoff Normal Forms

If all three indices are different, then there are 6 ways to match indices. If exactly two of them are equal, the number of matches is 3, and if they are all equal, there is 1 such match. Hence we obtain X vk v−k vl v−l vm v−m π N Q F˜4 = 2 µklm |klm| k,l,m

·

(k 2 + l 2 + m 2 + kl + km + lm)2 , (k + l)(k + m)(l + m)(k + l + m)

where µklm is defined as above. Before proceeding we combine the last two terms to X vk v−k vl v−l vm v−m
µklm · π N Y + 2Q F˜4 = 8 |klm| k,l,m

·

(k + l + m)(k 2 + l 2 + m 2 + kl + km + lm) . (k + l)(k + m)(l + m)

In this sum, the restriction k + l + m 6= 0 can be dropped, as the factor k + l + m appears in the numerator. These normal form expansions should now be represented in terms of the action variables I p = v p v− p for p > 0. We will not do this for all coefficients – for our purposes it suffices to determine the coefficient of I p2 Iq for p 6 = q. Let π˜ N denote the projection onto just those normal form terms. The Term π˜ N (Y + 2Q F˜4 ) To obtain the coefficient of I p2 Iq , two indices in the sum must both be equal either to p or − p, as they can not have opposite signs. The third index has to be equal to either q or −q. Adding the coefficients for k = l = p, m = q and k = l = p, m = −q we obtain (2 p + q)(3 p 2 + 2 pq + q 2 ) (2 p − q)(3 p 2 − 2 pq + q 2 ) + 2 p( p + q)2 2 p( p − q)2 3 p4 − 2 p2 q 2 + q 4 =2 · . ( p + q)2 ( p − q)2 The same contribution is obtained, if k = l = − p and m = q or m = −q. Further, for the choice of q, there are three possibilities. Thus the above coefficient has to be multiplied by 2 · 3. As µklm = 21 in our situation we thus obtain π˜ N (Y + 2Q F˜4 ) = 48

X I p2 Iq p6=q

p2 q

·

3 p4 − 2 p2 q 2 + q 4 . ( p + q)2 ( p − q)2

(H.2)

H Birkhoff Normal Form of Order 6

247

The Term π˜ N X The contributions from π N X to π˜ N X stem from the matches k = l, k = −m and l = m, as the possibilities k = −l, k = m or l = −m are excluded from the sum of π N X . Adding up the coefficients for k = l = p, m = ±q gives 2 p 2 + ( p − q)2 + ( p + q)2 2 p 2 + ( p + q)2 + ( p − q)2 2 p2 + q 2 + =4 2 . ( p − q)( p + q) ( p + q)( p − q) p − q2 Adding up the coefficients for k = p, m = − p, l = ±q gives −

5 p 2 + q 2 + ( p − q)2 5 p 2 + q 2 + ( p + q)2 p2 − = −6 2 . 2 p( p − q) 2 p( p + q) p − q2

To take account of the restriction k 6= l + m, however, we have to add the term −5δq−2 p for the case k = p, m = − p, l = q = 2 p, where δ j is one for j = 0 and zero otherwise. The same contribution arises from l = m = p, k = ±q. The count of each term has to be multiplied by 2, for we obtain the same terms by reversing the signs of all the integers k, l, m. The total coefficient is 8 As ηklm =

1 2

2 p2 + q 2 p2 − 24 − 20δq−2 p = −8 − 20δq−2 p . p2 − q 2 p2 − q 2

in the situation at hand we obtain π˜ N X = −48

X I p2 Iq p6 =q

p2 q

− 60

X I p2 I2 p p

p3

.

(H.3)

The Term π˜ N B F˜32 We collect the terms for k = p, l = q and k = p, l = −q. The same set of indices with all signs reversed gives the same contribution, so we obtain π N B F˜32 = −24

X I p Iq µ pq
I p + Iq − 2I p+q pq p + q p,q

X I p Iq 1
I p − Iq − 2σ p−q I| p−q| pq p − q p6 =q  X I p Iq  p I p q Iq I p+q I| p−q| = −48 + − − | p − q| pq p+q p2 − q 2 q 2 − p2 − 24

p6=q

− 12

X I p3 p

p3

+ 12

X I p2 I2 p p

p3

.

248

IX Birkhoff Normal Forms

Taking into account the special cases q = 2 p and p = 2q, this leads to π˜ N B F˜32 = −96

X I p2 I2 p p2 + 60 . p2 q p2 − q 2 p3 p

X I p2 Iq p6 =q

(H.4)

Collecting all terms from equations (H.2), (H.3) and (H.4) we arrive at X I p2 Iq
p2 q 2 π˜ N B F˜32 + Q F˜32 + 2Q F˜4 = 96 . p 2 q ( p 2 − q 2 )2 p6 =q

This has to be divided by 8π 3 in view of the definitions of F˜3 and F˜4 and by −4 in view of Corollary H.2 to give the following result. Proposition H.3. For the sixth order term of the Birkhoff normal form of the KdV Hamiltonian one obtains π˜ N N6 = −

3 X 2 q I p Iq 2 . π3 ( p − q 2 )2 p6=q

Consequently, the frequencies ω = (ωn )n≥1 of the KdV Hamiltonian satisfy ∂ 2 ωq 6 q =− 3 2 . ∂ I p2 I=0 π ( p − q 2 )2 I Kramer’s Lemma In Proposition 15.5 we verified the nondegeneracy and nonresonance conditions of the KAM Theorem 16.1 for the KdV Hamiltonian Hc by looking at the first two terms of the expansion of its frequencies ω given in (15.1), ω = λ − 6I + . . . , where λ = (λn )n≥1 with λn = 8π 3 n 3 + 12cπn depends also on c. This was not difficult except in the following case. If A ⊂ N is a given finite index set, we have to show that as a function of I A = (Ik )k∈A , we have ωn 6≡ 0, ωm + ωn 6 ≡ 0, ωm − ωn 6 ≡ 0,

m 6 = n,

for all m, n ∈ / A. As these expressions do not depend on I A up to first order, we had to resort to exclude certain values of c to establish these conditions. See Proposition 15.5 on page 131.

I Kramer’s Lemma

249

This restriction on c can be dropped by looking at the second derivative ∂ I2k ωm , which we obtained in the preceding appendix after some lengthy calculations. To verify the above three conditions, it is clearly sufficient to show that for some k ∈ A, ∂ I2k ωm 6 ≡ 0, ∂ I2k (ωm + ωn ) 6 ≡ 0, ∂ I2k (ωm − ωn ) 6 ≡ 0,

m 6 = n,

for all m, n ∈ / A. Of these the first two follow immediately from Proposition H.3, as the derivatives are all strictly negative at I = 0. It remains to verify the third condition. First we record a direct consequence of Proposition H.3. Lemma I.1. For k ∈ A and m, n ∈ / A, 6(m − n) k 4 + 2mnk 2 − (m 3 n + m 2 n 2 + mn 3 ) ∂ I2k (ωm − ωn ) I=0 = − . π3 (k 2 − m 2 )2 (k 2 − n 2 )2 Thus, to verify the third condition we have to show that the numerator above, k 4 + 2mnk 2 − (m 3 n + m 2 n 2 + mn 3 ), does not vanish for pairwise distinct, positive integers k, m, n. — We first reduce this diophantine problem to a simpler one. Lemma I.2. If the equation k 4 + 2mnk 2 − (m 3 n + m 2 n 2 + mn 3 ) = 0 has a solution in positive integers k, m, n, then up to a common factor they are of the form k = luv, m = u 4 , n = v 4 with positive integers l, u, v, such that u and v are relatively prime and l 2 = u 4 + v4 − u 2v2. Proof. We can always divide the equation by the fourth power of (k, m, n), the greatest common divisor of these three integers. Thus we can assume without loss of generality that (k, m, n) = 1. Consider then r = (m, n). If r > 1 then r - k, since (k, m, n) = 1. On the other hand, dividing the equation by r 2 we conclude that r 2 | k 4 , or r | k, a contradiction. Hence, also (m, n) = 1. Now, solving the quadratic equation for k 2 we have √ k 2 = −mn + (m + n) mn. To give an integer solution, both m and n must be square numbers. Thus, m = a 2 and n = b2 with (a, b) = 1. This then gives k 2 = ab(a 2 + b2 ) − a 2 b2 = ab(a − b)2 + a 2 b2 .

(I.1)

250

IX Birkhoff Normal Forms

This in turn implies that a and b are also square numbers, a = u2,

b = v2,

for if a or b contained a prime factor p with an odd power, then the right hand sides of (I.1) would contain p with an odd power, too, while k 2 contains p with an even power. Thus we have k 2 = u 4 v 4 + u 2 v 2 (u 2 − v 2 )2 , where u and v are relatively prime. Setting l=

k uv

⇔

k = luv,

we get l 2 = (uv)2 + (u 2 − v 2 )2 = u 4 + v 4 − u 2 v 2 .

t u

Finally we show that the diophantine equation l 2 = u 4 + v 4 − u 2 v 2 has only the obvious trivial solutions. For the following result we are indebted to J¨urg Kramer. Lemma I.3 (J. Kramer). The set of integral solutions of the equation R2 = S4 + T 4 − S2 T 2  is (±n 2 , n, 0), (±n 2 , 0, n), (±n 2 , ±n, n) : n ∈ Z .

(I.2)

Proof. Let Sa denote the affine surface defined by the equation (I.2) in affine 3-space. We will not only determine the set of integral points on Sa , but even the set of rational points Sa (Q) on Sa . First, we observe that the intersection of Sa with the plane given by T = 0 contains precisely the rational points (r, s, t) of the form (r, s, t) = (±n 2 , n, 0) with arbitrary n ∈ Q. Let now Ua denote the open subset of Sa given by the set of points having non-zero T -coordinate. Dividing (I.2) by T 4 and setting U := S/T,

V := R/T 2 ,

(I.3)

we obtain the morphism ϕ : Ua → Ca from the open subset Ua to the affine curve Ca defined by the equation V 2 = U 4 − U 2 + 1. Since we have an inclusion ϕ(Ua (Q)) ⊆ Ca (Q), hence Ua (Q) ⊆ ϕ −1 (Ca (Q)), it suffices to find the rational points on the quartic Ca . To do this, we set X := U 2 ,

Y := U V,

(I.4)

which gives rise to the morphism ψ : Ca → Ea , Ea being given as the affine cubic Y 2 = X 3 − X 2 + X.

I Kramer’s Lemma

251

As before, we have an inclusion ψ(Ca (Q)) ⊆ Ea (Q), that is, Ca (Q) ⊆ ψ −1 (Ea (Q)), and hence we are reduced to determine the rational points on the cubic Ea . Now, Ea is the affine part (that is, Z = 1) of the elliptic curve E defined by the equation Y 2 Z = X 3 − X 2 Z + X Z 2. This curve is well-known and listed as curve 24A in the tables given in [127, p. 83]; it is modular of conductor 24, and the rank of its Mordell-Weil group is zero. Hence the set of rational points on E consists of finitely many points, the so-called torsion points of E (Q), which correspond to the elements of Ea (Q) with the exception of the point at infinity. To determine these torsion points we use the theorem of Nagell-Lutz – see [125, p. 221]: changing the coordinates X, Y, Z to X˜ := 9X − 3,

Y˜ := 27Y,

Z˜ := Z ,

the elliptic curve E is isomorphically mapped onto the elliptic curve E˜ given by the equation Y˜ 2 Z˜ = X˜ 3 + A X˜ Z˜ 2 + B Z˜ 3 with A = 54, B = 189 and discriminant 4A3 + 27B 2 = 313 . The torsion points (x˜ : y˜ : 1) (different from the point (0 : 1 : 0) at infinity) of E˜ (Q) are then characterized by those pairs (x, ˜ y˜ ) satisfying x, ˜ y˜ ∈ Z and y˜ = 0 or y˜ 2 | 313 . A straightforward calculation shows that these conditions are only satisfied for y˜ = 0 and y˜ = ±27. This leads, up to the point at infinity, to the three torsion points (−3 : 0 : 1),

(+6 : +27 : 1),

(+6 : −27 : 1)

of E˜ (Q), and hence proves Ea (Q) = { (0, 0), (+1, +1), (+1, −1) }.

(I.5)

By means of the formulae (I.4), we then immediately find the following six rational points on the quartic Ca (observing that each one of the three rational points of Ea given above has exactly two preimages in Ca (Q) for the morphism ψ) Ca (Q) = { (0, +1), (0, −1), (+1, +1), (−1, −1), (+1, −1), (−1, +1) }. Finally, using the formulae (I.3), it is now a simple task to determine the rational points in the fibers of ϕ over the six points of Ca (Q) given in (I.5); in this way we find  Ua (Q) = (±n 2 , 0, n), (±n 2 , ±n, n) : n ∈ Q∗ , which completes the proof of the lemma.

t u

252

IX Birkhoff Normal Forms

J Nondegeneracy of the Second KdV Hamiltonian We establish the nondegeneracy properties of the second KdV Hamiltonian as stated in Lemma J.2. First we consider the Jacobian of the frequency map I A → ω2A at I A = 0. This Jacobian is represented by the matrix C A = (Ckl )k,l∈A = Q 2A I =0 , A

whose coefficients are given in Theorem 15.3. Recall that they are linear functions of the parameter c ∈ R, which we do not indicate for the sake of the simplicity of the notation. Also, we do not use bold face symbols to denote infinite dimensional vectors, since there is no danger of confusion. Lemma J.1. For every finite set A ⊂ N there exists an |A|-point set C A ⊂ R not containing 0 such that det C A = 0

⇔

c ∈ CA.

In particular, if A = { i }, then C A = { − 23 π 2 i 2 }, while for A = { i 1 < · · · < i n } one has  C A = cnA < · · · < c2A < 0 < c1A with −

14 14 2 2 2 π i ν < cνA < − π 2 i ν−1 , 3 3

2 ≤ ν ≤ n,

and c1A → ∞ as |A| → ∞. Proof. In view of Theorem 15.3 we can write the matrix under consideration as C = D − B, where D = diag(Di )i∈A and B = (Bi j )i, j∈A have coefficients Di = 280π 2 i 2 + 60c,

Bi j = 240π 2 i j.

The matrix B has rank 1, so by the multi-linearity of the determinant we have X Y det C = det D − Bii Dj. i∈A

j∈A, j6=i

If one of the D Q and we have j vanishes, say Dk = 0, then all other D j do not vanish, P det C = −Bkk j6=k D j 6 = 0. Otherwise, det C = (det D)(1 − i∈A Bii /Di ), and the determinant vanishes if and only if 1=

X Bii X 12 = , Di 14 + c f i i∈A

i∈A

fi =

3 . π2i 2

6 2 2 Each summand is a hyperbola in c. It has a single pole at c = − 14 3 π i , value 7 at c = 0, and asymptotic value 0 as c → ±∞. Also, both branches of the hyperbola

J Nondegeneracy of the Second KdV Hamiltonian

253

are monotonically decreasing. From these considerations it follows that the above equation has exactly n solutions, which for n ≥ 2 are located as cnA < cn−1 < · · · < c2A < 0 < c1A , A as described in the lemma. For n = 1 the result is also immediately read off.

t u

The lemma shows that for any given A ⊂ N the Jacobian of the frequency map I A 7 → ω2A of the second KdV Hamiltonian does become singular, at least at I A = 0 for c ∈ C A . This is in contrast to the first KdV Hamiltonian, where the Jacobian is always regular at I A = 0. Moreover, Krichever [68] and Bobenko & Kuksin [11] have shown that for c = 0 it is indeed regular everywhere. See the second reference for a complete proof. We now fix a finite set A ⊂ N and consider the frequency combinations k · ω2 as functions of I A on P` A . In view of ω2 = λ2 − C I + . . . and the symmetry of the matrix C we have k · ω2 = k · λ2 − (Ck) A I A + . . . on P` A . To prove that k · ω2 6 ≡ 0 on P` A it is thus sufficient to show that k · λ2 6 = 0

or

(Ck) A 6 = 0.

(J.1)

We first prove a general statement to this effect. Recall that C depends on the parameter c ∈ R. Lemma J.2. For each k ∈ Z∞ with 1 ≤ |k Z | ≤ 2 there exists at most one ck ∈ R such that the alternative (J.1) does not hold. This ck is a rational multiple of π 2 . Moreover, within every compact subset of R − C A there are only finitely many such ck . Proof. We have (Ck) A = C A k A + C AZ k Z , where C AZ = (Ci j )i∈A, j∈Z . The diagonal elements of C A are linear functions of c, namely 60c + 40π 2 i 2 , while all other coefficients of both matrices are integer multiples of π 2 , namely −240π 2 i j. Hence, given k the above vector (Ck) A can vanish for at most one value of c, and this value must be a rational multiple of π 2 . To prove the remaining statement suppose that (Ck) A = 0, and that c belongs to some compact subset F ⊂ R − C A . Then C A is invertible, k A = −C −1 A C AZ k Z , and we can bound C −1 A uniformly for c ∈ F. Moreover, the vector C AZ k Z has coefficients −240π 2 i p, i ∈ A, where p = k Z · λoZ ,

λoZ = ( j) j∈Z .

It follows that for any c ∈ F, o |k A | ≤ C −1 A |C AZ k Z | ≤ K k Z · λ Z ,

254

IX Birkhoff Normal Forms

where here and below, K stands for various constants bigger than 1 that depend only on A and the compact set F. Now suppose that also k · λ2 = k A · λ2A + k Z · λ2Z = 0. In view of λ2n = n˜ 5 + 10cn˜ 3 + 30c2 n˜ from Theorem 15.3 it is a routine estimate to show that for 1 ≤ |k Z | ≤ 2 one has k Z · λ2 ≥ K −1 k Z · λo 5 − K k Z · λo 3 . Z

Z

Z

Thus we obtain 5 3 K k Z · λoZ λ2A ≥ k A · λ2A = k Z · λ2Z ≥ K −1 k Z · λoZ − K k Z · λoZ . Therefore, k Z · λoZ ≤ K with a different constant K . Combining this estimate with the estimate for |k A | we find k Z · λ2 ≤ K . |k A | ≤ K , Z Thus, for c ∈ F there can be only finitely many k ∈ Z∞ with 1 ≤ |k Z | ≤ 2 for which the alternative (J.1) does not hold. Consequently, there can be at most finitely many exceptional values ck in F. This proves the last statement. u t The preceding lemma does not make explicit the value of ck . So it may happen that the mean value zero case c = 0 is excluded. We now show that this is not the case. Lemma J.3. For every finite set A ⊂ N the second KdV Hamiltonian at c = 0 satisfies (J.1) and is thus nondegenerate. Proof. As in the previous proof suppose that (Ck) A = 0. At c = 0 the coefficients of C are, up to a common multiplicative factor, (7δi j − 6)i j. Hence the coefficients of k satisfy X 7iki = 6 jk j , i ∈ A. j≥1

It follows that iki = r is independent of i for i ∈ A. Substituting these identities in the above sum we obtain X  X 7r = 6 r+ jk j = 6nr + 6 p, i∈A

where n = |A| and p =

P

j∈Z

j∈Z

jk j . Hence,

iki = r =

6p , 7 − 6n

i ∈ A.

In particular, all ki for i ∈ A are distinct and of the same sign.

(J.2)

J Nondegeneracy of the Second KdV Hamiltonian

255

Now consider k · λ2 = (2π)5

X

X

i 5 ki +

i∈A

 j 5k j .

j∈Z

Solving equation (J.2) for i and for ki we have −

X i∈A

6p X 4 i ki = i = 6n − 7 5



i∈A

6p 6n − 7

5 X i∈A

1 . ki4

To show that k · λ2 6 = 0 it thus suffices to show that the two terms I=

X 1 k4 i∈A i

and  II =

6n − 7 6p

5 X

j 5k j ,

p=

j∈Z

X

jk j > 0

j∈Z

are not equal. In most cases, this can be done by straightforward estimates. For n = 1, the terms I and II have opposite sign. For n ≥ 3, we note that with 1 ≤ |k Z | ≤ 2, X j∈Z

j 5k j ≥

1 24

X

5 jk j

j∈Z

=

1 5 p . 24

It thus suffices to show that   X 1 6n − 7 5 1 < . 6 24 k4 i∈A i This is obviously true for n ≥ 4, and for n = 3 it follows from X 1 1 1 115 1 + < · . ≤ 1 + 24 34 65 24 k4 i∈A i For n = 2 the above estimates still suffice in the case where k Z has only one nonzero component, so that k Z = le j with 1 ≤ l ≤ 2. Then X

j 5k j =

j∈Z

p5 , l4

and so  5 5 1 II = IIl = , 6 l4

l = 1, 2.

256

IX Birkhoff Normal Forms

On the other hand, depending on whether mini∈A (−ki ) is 1, 2, or ≥ 3, let I = Im =

X 1 , k4 i∈A i

m = 1, 2, 3,

respectively. One then checks that I3 < II2 < I2 < II1 < I1 . It remains to discuss the case n = 2 and k Z = e j1 ±e j2 with j1 > j2 . By equation (J.2) we have 6p iki = − , i ∈ A. 5 In particular, 5 | p. Assuming to the contrary that k · λ2 = 0, we have I = II, which is equivalent to X X 6p i4 = 5 j 5 k j = 5( j15 ± j25 ). i∈A

j∈Z

We now show that this equation has no integer solutions. We have p = j1 ± j2 . Let q = j1 ∓ j2 > 0. Solving for j1 and j2 , taking powers and adding up one obtains 24 ( j15 ± j25 ) = p( p 4 + 2 · 5 p 2 q 2 + 5q 4 ). So it suffices to show that 25 · 3 · (i 14 + i 24 ) = 5( p 4 + 2 · 5 p 2 q 2 + 5q 4 ) can not have an integer solution, if p is divisible by 5. For every integer n, n 4 is congruent to 0 or 1 modulo 5. Therefore, 5 | i 14 + i 24

⇔

5 | i1

and

5 | i2 .

If the above equation holds, then 5 | i 14 + i 24 , hence 5 | i 1 and 5 | i 2 , and so the left hand side is divisible by 54 . On the right hand side the first two terms are divisible by 54 , since 5 | p. Hence also 54 | 52 q 4 , and thus 5 | q, too. Thus, every integer in the equation is divisible by 5, the whole equation is divisible by 54 , and so we get 25 · 3 · (˜ı 14 + ı˜24 ) = 5( p˜ 4 + 2 · 5 · p˜ 2 q˜ 2 + 5q˜ 4 ), where p˜ = p/5 and so on. The preceding argument can now be repeated ad infinitum. But this is absurd. So also in the remaining cases we have k · λ2 6 = 0, and the lemma is proven. u t

X Some Technicalities

K Symplectic Formalism We describe some basic notions of symplectic geometry in infinite dimensions to the extent as they are needed here. We follow Kuksin’s exposition in [72], with a few simplifications and modifications. A scale of Hilbert spaces, or a Hilbert scale, is a sequence E = (E p ) p=0,1,... of Hilbert spaces E p with inner products h · , · i p and norms k · k p such that E0 ⊃ E1 ⊃ E2 ⊃ . . . ,

k · k0 ≤ k · k1 ≤ . . . ,

and such that all E p are dense in E 0 . A linear isomorphism of order d between two Hilbert scales E and F, where d is an integer, is a linear isomorphism L : E q → Fq−d ,

q = max(0, d),

whose restrictions L|E p give rise to linear isomorphisms E p → F p−d for each p ≥ q. Recall that a constant symplectic form on a finite dimensional Hilbert space E with inner product h · , · i is a bilinear, anti-symmetric, nondegenerate form α on E. It can be represented as α(ξ, η) = hξ ,K ηi,

ξ, η ∈ E,

with an anti-symmetric isomorphism K of E. In view of our applications to the KdV equations we extend this notion to infinite dimensional spaces as follows. A constant symplectic form α on a Hilbert scale E is given by a linear anti-symmetric isomorphism K of the scale E of some order −dα ≤ 0 such that α(ξ, η) = hξ ,K ηi0 ,

ξ, η ∈ E 0 .

The pair (E, α) is called a symplectic Hilbert scale. Note that α is anti-symmetric, nondegenerate and closed, since it is constant.

258

X Some Technicalities

Consider an open domain U p ⊂ E p and a function H ∈ C 1 (U p ; C). A continuous map X H : U p → E0 is called the Hamiltonian vector field of the Hamiltonian H on U p , if α(X H , · ) = d H at every point of U p . That is, α(X H (x), ξ ) = dx H (ξ ) for all x ∈ U p and ξ ∈ E p . The vector field is said to be of order d, where d ≤ p is an integer, if in fact X H maps into E p−d ⊂ E 0 : X H : U p → E p−d . This definition requires some explanations. First, X H is not a vector field in the strict sense, since it is not required to be a section of the tangent bundle over U p . This allows us to deal with unbounded operators. On the other hand, not every differentiable Hamiltonian H admits a Hamiltonian vector field X H in this sense, since in general dx H is an element of the dual space E ∗p only and thus lacks the required regularity. But if it exists, then it is unique. To be more precise, given H ∈ C 1 (U p ; C) we always have a representation d H (ξ ) = h∇ H ,ξ i0 ,

ξ ∈ E p,

with a continuous map ∇ H : U p → E ∗p , if we identify the dual space E ∗p with the completion of the space E 0 with respect to the norm k · k− p = sup |h · ,ξ i p |. kξ k p ≤1

Then the equation α(X H , · ) = d H is equivalent to − hK X H , · i0 = h∇ H , · i0 . This equation can be solved for X H in E 0 , if ∇ H is in the domain of K −1 , that is, if ∇ H : U p → E dα ⊂ E 0 , since K is an anti-symmetric isomorphism of the scale E of order −dα ≤ 0. In this case we have X H = J ∇ H, J = −K −1 , where J is an anti-symmetric isomorphism of the Hilbert scale E of order dα ≥ 0. This is the situation we encounter in our applications. For the sake of completeness we mention the notion of a strong solution. Let I be an interval. A differentiable map u : I → U p is called a strong solution of the Hamiltonian vector field X H on U p , or more precisely of the evolution equation u˙ = X H (u), if u˙ = du/dt satisfies this equation in E 0 for every t ∈ I .

K Symplectic Formalism

259

We are interested in constructing strong solutions of perturbed KdV equations. However, we will not do this directly, nor will we construct local flows or semi-flows. Rather, we transform the vector fields into a form such that the existence of certain strong solutions is evident. We consider the relevant transformation rule. Let (E, α E ) and (F, α F ) be two symplectic Hilbert scales. Fix p. A differentiable map 8 : Fp ⊃ V p → E p is symplectic, if 8∗ α E = α F on V p . That is, in every point of V p one has α E (8∗ ξ, 8∗ η) = α F (ξ, η),

ξ, η ∈ F p ,

where 8∗ denotes the push forward map between tangent spaces induced by 8. Using representations of α E , α F by isomorphisms K E , K F , respectively, this is equivalent to h8∗ ξ ,K E 8∗ ηi E 0 = hξ ,K F ηi F0 , or 8∗ K E 8∗ = K F , where 8∗ is the adjoint of 8∗ . In a finite dimensional setting a symplectic transformation takes Hamiltonian vector fields into Hamiltonian vector fields – see section 2. Our definition of Hamiltonian vector fields guarantees that this is also true in infinite dimensions. Transformation Rule. Suppose H admits a Hamiltonian vector field X H on a domain U p ⊂ E p of order d ≤ p, and 8 : Fp ⊃ V p → U p ⊂ E p is a symplectic transformation, such that at every point in V p , D8 : F p → E p extends to a linear isomorphism F p−d → E p−d . Then the transformed vector field 8∗ X H = D8−1 X H B 8 is a Hamiltonian vector field on V p of order d with Hamiltonian H B 8. That is, 8∗ X H = X H B8 : V p → F p−d ⊂ F0 . Proof. Clearly, 8∗ X H = D8−1 X H B 8 : V p → F p−d ⊂ F0 , since by assumption, 8

XH

D8−1

V p −→ U p −−→ E p−d −−−−→ F p−d .

260

X Some Technicalities

This vector field is Hamiltonian, since d(H B 8)(ξ ) = (d H B 8)(D8 · ξ ) = α E (X H B 8, D8 · ξ ) = α E (D8 · 8∗ X H , D8 · ξ ) = 8∗ α E (8∗ X H , ξ ) = α F (8∗ X H , ξ ) for all ξ ∈ F p . Hence, 8∗ X H is the Hamiltonian vector field for H B 8 as defined above. u t

L Infinite Products We collect some estimates concerning infinite products of complex numbers that are quite elementary but used repeatedly in the construction of the Birkhoff coordinates. Lemma L.1. Suppose amn , m, n ≥ 1, are complex numbers satisfying   1 , |amn | = O 2 m 6 = n. m − n 2 Then Y

(1 + amn ) = 1 + O

m≥1 m6=n



 log n , n

n ≥ 1,

where the implicit constant in the conclusion depends only on the implicit constant in the assumption. Proof. By assumption, X

|amn | ≤ C

m≥1 m6=n

X m≥1 m6 =n

1 m 2 − n 2

with some positive constant C. The sum on the right can be estimated by X m≥1 m6=n

1 = m 2 − n 2 ≤

X 1≤m≤2n m6 =n

X 1 1 1 + 2 |m − n| m + n m − n2 m>2n

2 X 1 X 1 + n k k2 1≤k≤n

k>n

2 1 ≤ (1 + log n) + . n n

L Infinite Products

261

Hence we obtain Y Y (1 + amn ) − 1 ≤ (1 + |amn |) − 1 m≥1 m6=n

m≥1 m6 =n

≤ exp

X 

≤ exp

 |amn | − 1

C 0 log n n

with a different constant C 0 depending only on C.



 −1= O

log n n



t u

For the sake of reference we separately state an important special case of the preceding lemma. Lemma L.2. Suppose ξm and ζm are complex numbers satisfying ξm , ζm = m 2 π 2 + O(1) for m ≥ 1. Then   Y ζm − λ log n =1+O ξm − λ n

m6 =n

uniformly for (n − 12 )2 π 2 ≤ |λ| ≤ (n + 12 )2 π 2 with n sufficiently large, where the implicit constant in the conclusion depends only on the implicit constant in the assumption. Proof. For λ = n 2 π 2 + O(n) and m 6= n,   ζm − λ ζm − ξm = =O 1 , − 1 ξ − λ ξ −λ m 2 − n 2 m m and the preceding lemma applies.

t u

Lemma L.3. Suppose ξm are complex numbers with ξm = m 2 π 2 + O(1). Then the infinite product Y ξm − λ m 2 π2 m≥1

defines an entire function of λ, whose roots are precisely ξm , m ≥ 1, and which satisfies √    Y ξm − λ sin λ log n = 1 + O √ n m 2 π2 λ m≥1

uniformly on the circles |λ| = (n + 12 )2 π 2 .

262

X Some Technicalities

Proof. We just prove the last statement. To this end recall that √ Y m 2 π2 − λ sin λ = . √ m 2 π2 λ m≥1

√ √ Hence the quotient of the given product and sin λ/ λ is Y m≥1

ξn − λ Y ξm − λ ξm − λ = 2 2 . 2 2 m π −λ n π −λ m 2 π2 − λ m6=n

By the preceding lemma, the infinite product on the right is 1+ O(log n/n) uniformly on the circle |λ| = (n + 21 )2 π 2 , while on the same circle,   ξn − λ 1 =1+O . n n2 π2 − λ From this the result follows.

t u

M Auxiliary Results Here we collect a few frequently used estimates for analytic maps and a result about extensions of Lipschitz functions. Lemma M.1. Let 0 be a circuit around the interval [a, b] within some complex neighbourhood U of it. If f is analytic on U , then Z 1 f (λ) ≤ max | f (t)|. dλ √ 2π a≤t≤b 0 (b − λ)(λ − a) Proof. If a = b, the given integral turns into a Cauchy integral, and the result follows immediately. If a < b, then we may shrink the contour of integration to the interval [a, b] to obtain Z b Z 1 | f (λ)| f (λ) ≤ 1 dλ dλ √ √ 2π + π a (b − λ)(λ − a) 0 (b − λ)(λ − a) Z 1 b dλ ≤ max | f (t)|. √ π a + (b − λ)(λ − a) a≤t≤b The last integral is 1, giving the result.

t u

Lemma M.2. Let u j , j ≥ 1, be complex functions on Tn that are real analytic on D(s) = { |Im x| < s }. Then X  X  1/2 2 1/2 4n u j (x) 2 sup u j (x) ≤ n sup σ x∈D(s) x∈D(s−σ ) j≥1

for 0 < σ < s ≤ 1.

j≥1

M Auxiliary Results

263

Proof. First consider the case n = 1. For each j ≥ 1 there exists a point x j in the rectangle Q = { x : |Re x| ≤ π, |Im x| ≤ s − σ } such that sup u j (x) ≤ u j (x j ) . x∈D(s−σ )

By the Cauchy integral formula, 1 u j (x j ) = 2πi

Z 0

u j (ζ ) dζ, ζ − xj

where 0 describes a rectangle with distance 0 < ρ < σ around Q, independent of j. By the triangle inequality we then get X

sup

j≥1 x∈D(s−σ )

u j (x) 2

1/2

2 1/2 X Z 1 u j (ζ ) ≤ dζ 2πi 0 ζ − xj j≥1

1 ≤ 2π ≤

Z X u j (ζ ) ζ − x 0

j

j≥1

2 1/2 |dζ |

X  1/2 4 u j (x) 2 sup . ρ x∈D(s) j≥1

Letting ρ → σ the estimate follows for n = 1. For n > 1 the single Cauchy integral is replaced by an n-fold Cauchy integral. u t We also need a version of this lemma for bounded operators on `2 . Lemma M.3. Let A = (Ai j )i, j≥1 be a bounded operator on `2 which depends on x ∈ Tn such that all coefficients are analytic on D(s) = { |Im x| < s }. Suppose B = (Bi j )i, j≥1 is another operator on `2 depending on x whose coefficients satisfy sup Bi j (x) ≤ x∈D(s)

1 sup Ai j (x) , |i − j| x∈D(s)

i 6 = j,

and B j j = 0 for j ≥ 1. Then B is a bounded operator on `2 for every x ∈ D(s), and 4n+1 sup kB(x)k ≤ n sup kA(x)k σ x∈D(s) x∈D(s−σ ) for 0 < σ ≤ s ≤ 1. Proof. For x ∈ D(s − σ ) we have by the Schwarz inequality and the preceding lemma X X Bi j (x) ≤ sup Bi j (x) j≥1 x∈D(s−σ )

j≥1

≤

X

2 sup Ai j (x)

j≥1 x∈D(s)

1/2 X j6=i

1 |i − j|2

1/2

264

X Some Technicalities

X  2 1/2 4n+1 ≤ n sup Ai j (x) σ x∈D(s) j≥1

≤

4n+1 σn

The same estimate applies to kB(x)vk2 ≤

sup kA(x)k. x∈D(s)

P

i≥1

Bi j (x) . Hence, for x ∈ D(s − σ ),

 X X 2 Bi j (x) v j i≥1

j≥1

  X X X Bi j (x) Bi j (x) v j 2 ≤ i≥1

j≥1

j≥1

  X   X X 2 vj ≤ sup Bi j (x) sup Bi j (x) i

 ≤

4n+1 σn

j

j≥1

2 sup kA(x)k

i≥1

j≥1

kvk2 .

x∈D(s)

From this the final estimate follows.

t u

Let V be an open domain in a complex Banach space E with norm k · k, 5 an arbitrary subset of parameters in Rn , and X: V ×5 → E a parameter dependent vector field on V , which is analytic on some neighbourhood W ⊃ V and Lipschitz on 5. Let 8t denote the flow of X , and assume that there is a subdomain U ⊂ V such that 8t : U × 5 → V,

−1 ≤ t ≤ 1. sup

Lemma M.4. If, with the preceding assumptions, kD X kV ≤ 1, then

t

8 − id sup ≤ kX ksup , V ×5 U ×5

t

lip lip

8 − id ≤ 3 kX kV ×5 , U ×5 sup

for −1 ≤ t ≤ 1. Moreover, if | · | is any operator norm on E with |D X |V ×5 ≤ 1, then D8t − I sup ≤ 3 |I | |D X |sup , V ×5 U ×5 D8t − I lip ≤ 9 |I | |D X |lip + 27K |I | |D X |sup , V ×5 W ×5 U ×5 lip

where K = kX kV ×5 / dist(V, ∂ W ), and I is the identity operator.

M Auxiliary Results

265

Proof. In the following we suppress 5R from the notation. Fix 0 ≤ t ≤ 1. The t first estimate follows from 8t − id = 0 X B 8s ds. To prove the second one, let t t t t 18 = 8ξ − 8ζ for ξ, ζ ∈ 5, where 8ξ = 8t ( · , ξ ). Then 18 = t

t

Z

1(X B 8 ) ds = s

Z

0

t

1X

0

B 8sξ

Z

t

ds + 0

(X B 8sξ − X B 8sζ ) ds,

sup

hence, as kD X kV ≤ 1 by assumption, t

Z



18t sup ≤ U

0

sup

k1X kV ds +

Z 0

t



18s sup ds. U

sup sup With Gronwall’s inequality we get 18t U ≤ 3 k1X kV . Dividing by |ξ − ζ | and taking the supremum over 5 the Lipschitz estimate follows. sup Now let | · | be any operator norm on E, and assume that |D X |V ≤ 1. We have Z t t D8 = I + D X B 8s · D8s ds, 0

sup from which D8t U ≤ 3 |I | and the third estimate of the lemma follow with Gronwall’s inequality. For the Lipschitz estimate we write Z t Z t 1D8t = 1(D X B 8s ) · D8sξ ds + D X B 8sξ · 1D8s ds. 0

0

sup

sup

With |D8s |U ≤ 3 |I | from the preceding estimate and |D X |V ≤ 1 this gives 1D8t sup ≤ 3 |I | U

Z 0

t

1(D X B 8s ) sup ds + U

Z

t

1D8s sup ds.

0

U

Hence, by Gronwall, 1D8t sup ≤ 9 |I | U

Z 0

t

1(D X B 8s ) sup ds. U

Again, we write 1(D X B 8s ) = (1D X ) B 8sξ + D X B 8sξ − D X B 8sζ . To estimate the second term, we use the Cauchy inequality for the operator norm of D(D X ) with the norms k · k and | · | in the source and target space, respectively. With ρ = dist(V, ∂ W ) we get

D X B 8s − D X B 8s sup ≤ |D(D X )|sup 18s sup ξ ζ U V U

lip |ξ − ζ | sup |D X |W 8s − id U . ≤ ρ

266

X Some Technicalities lip

lip

It follows with k8s − idkU ≤ 3 kX kV that 1D8t sup ≤ 9 |I | |1D X |sup + 27 |I | |ξ − ζ | |D X |sup kX klip . V W V U ρ lip From this the estimate for D8t − I follows as stated. u t Lemma M.5 (Lipschitz Extension). Let F ⊂ Rn be a closed set and u : F → R a bounded Lipschitz continuous function. Then there exists an extension U : Rn → R of u, which preserves minimum, maximum and Lipschitz semi-norm of u. lip

Proof. Let λ = |u| F , and define u(x) ˜ = sup (u(ξ ) − λ |x − ξ |) ξ ∈F

for x ∈ Rn . This is an extension of u to all of Rn . By the triangle inequality, u(x) ˜ ≥ u(ξ ) − λ x 0 − ξ − λ x − x 0 for all ξ ∈ F and hence u(x) ˜ ≥ u(x ˜ 0 ) − λ x − x 0 . Interchanging x and x 0 , we get u(x) ˜ − u(x ˜ 0 ) ≤ λ. |x − x 0 | lip

lip

It follows that |u| ˜ Rn = |u| F . Setting U = (u˜ ∧ max u) ∨ min u F

lip

lip

F

we note that |U |Rn = |u| ˜ Rn . Thus the function U has all the required properties.

t u

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Index

A-gap potential, 6, 113 Abel map, 59 actions, 5, 54, 56, 64, 113 admissible path, 70 algebra Banach, 121 KdV, 101, 207 Poisson, 6, 29 almost-periodic, 6 motion, 133 solution, 6, 45, 46, 48 analytic map, 187 real, 193 subvariety, 193 weakly, 187 angle-action coordinates, 4, 26, 30, 31, 43 angles, 7, 59, 69, 73 approximation function, 46 Arnold, 8, 30, 41 asymptotics, frequency, 135, 146 average, 3 Banach algebra, 121 bands, spectral, 53 basis, 91 canonical, 223 Bellissard, 46 Bernstein, 42 Birkhoff coordinates, 37, 54 integrable, 37 invariants, 14 normal form, 36, 233

resonant, 234 up to order m, 36 Bobenko, 13, 253 Bourgain, 17, 48 Boussinesq, 1 bracket, 85 Lie, 23 Poisson, 2, 20, 21 canonical basis, 223 coordinates, 5, 56 diffeomorphism, 24 Cantor set, 8, 41 Casimir function, 23, 52, 112 Cauchy’s estimate, 188 formula, 188 Cheng, 42 Chierchia, 46, 48 coarse structure, 138, 172 collapsed gap, 2, 53, 113 compact function, 66, 195 complete set of independent integrals, 114 completely integrable, 114 complexification, 119 condition diophantine, 40, 177 Kolmogorov, 13, 40, 47, 117 Lyapunov, 43, 47 Melnikov, 13, 44, 47, 117 nondegeneracy, 13, 117, 136, 147 nonresonance, 13, 117

276

Index

regularity, 136, 147 small divisor, 40 contraction, 20 coordinates angle-action, 4, 26, 30, 31, 43 Birkhoff, 37, 54 canonical, 5, 56 Darboux, 26 Craig, 17, 48 Darboux coordinates, 26 de Vries, 1 density of finite gap potentials, 54, 113 diffeomorphism canonical, 24 symplectic, 24 diophantine condition, 40, 177 frequencies, 40 Dirichlet divisor, 56 eigenfunction, 196 eigenvalues, 55, 196 spectrum, 196 theorem, 46 discriminant, 55, 194 divisor, Dirichlet, 56 Dubrovin, 7 eigenfunction Dirichlet, 196 periodic, 198 eigenvalues Dirichlet, 55, 196 periodic, 2, 53, 112, 198 Eliasson, 8, 43 elliptic equilibrium, 13, 35, 45, 118 elliptic invariant tori lower dimensional, 39 equilibrium elliptic, 13, 35, 45, 118 solution, 15 exponents, Lyapunov, 10, 114 external frequencies, 39, 117, 146 Faddeev, 1 Fermi, 40 fine structure, 138, 172 finite gap

potential, 54, 113, 204 solution, 8, 133 Flaschka, 57, 63, 64 Floquet matrix, 194 form, symplectic, 19, 257 Fr¨ohlich, 46 frequencies, 31, 129 diophantine, 40 external, 39, 146 internal, 39, 117, 146 nonresonant, 34, 233 rationally dependent, 34 rationally independent, 34 resonant, 34 strongly nonresonant, 40 frequency asymptotics, 135, 146 map, 34 nondegenerate, 34 module, 32 spectrum, 33 function almost-periodic, 45 Casimir, 23, 52, 112 compact, 66, 195 independent -s, 27 quasi-periodic, 33 functionally independent, 37 fundamental solution, 55, 194 gap, 2, 112 collapsed, 2, 53, 113 length, 2, 53, 112 squared, 3, 53 open, 2 spectral, 53 Gardner, 1–3, 111 Garnett, 54 generalized normal form, 152 gradient, 9, 62 Greene, 3 Hamiltonian, 20, 21 KdV, 2, 52, 119, 129, 229 nondegenerate, 44 second KdV, 10, 115, 126, 229 vector field, 20, 258 zero-th KdV, 5, 223, 227, 238 hierarchy, KdV, 60, 101, 109, 207

Index Hilbert scale, 257 symplectic, 257 hull, 33 independent functionally, 37 functions, 27 linearly, 91 independent integrals complete set of, 114 inner product, 20 integrable Birkhoff, 37 in the sense of Liouville, 27 system, 31 integrable system nondegenerate, 8 integrals, 52, 113 internal frequencies, 39, 117, 146 invariants, Birkhoff, 14 involution, 23 isolating neighbourhood, 64 isospectral deformation, 3, 53 set, 3, 54, 55, 113 Ito, 37, 235 Its, 6, 7 Jacobi identity, 21 Jost, 30 KAM theorem, 41, 133, 148 theory, 8, 39 Katok, 42 KdV algebra, 101, 207 equation, 1, 51, 111 second, 10, 12, 115 Hamiltonian, 2, 52, 119, 129, 229 second, 10, 115, 126, 229 zero-th, 5, 223, 227, 238 hierarchy, 60, 101, 109, 207 Kolmogorov, 8, 40, 41 condition, 13, 40, 47, 117 Korteweg, 1 Kramer, 132, 250 Krichever, 7, 13, 131, 253 Kronecker torus, 32

277

Kruskal, 1, 3 Kuksin, 8, 13, 16, 47, 114, 116, 137, 151, 178, 253 lemma, 177 L 2 -gradient, 62 Lagrangian submanifold, 28 Lax, 2, 3, 52 pair formalism, 52, 113 Leibniz rule, 21 lexicographic ordering, 62 Lie bracket, 23 Lindstedt series, 40 linearization, 45 linearized equation, 152 linearly independent, 91 stable, 9, 10, 114, 134, 146 Liouville’s Theorem, 30 Liouville-Arnold-Jost theorem, 4, 30, 54 Lipschitz extension, 266 lower dimensional elliptic invariant tori, 39 Lyapunov, 43 condition, 43, 47 exponents, 10, 114 manifold, Poisson, 21 map analytic, 187 nondegenerate, 131 proper, 96 symplectic, 259 Matveev, 6, 7 McKean, 3, 7, 16, 54, 61, 102 McLaughlin, 57, 63, 64 mean value, 52, 152 Melnikov, 8, 43 condition, 13, 44, 47, 117 Miura, 3 moments, 25 Moser, 8, 41, 43 motion almost-periodic, 133 quasi-periodic, 35, 38, 114 neighbourhood, isolating, 64 Nikolenko, 45 nondegeneracy condition, 13, 117, 136, 147 nondegenerate, 39

278

Index

frequency map, 34 Hamiltonian, 44 integrable system, 8 map, 131 Poisson structure, 22 nonresonance condition, 13, 117 nonresonant frequencies, 34, 233 Poisson algebra, 37 up to order m, 35 normal form, 233 Birkhoff, 36, 233 generalized, 152 regular, 152 Novikov, 7 observables, 21 open gap, 2 order, 136, 258 oscillatory integral, 179, 181 parallel torus, 32 path, admissible, 70 perfect set, 41 Perfetti, 46 periodic eigenfunction, 198 eigenvalues, 2, 53, 112, 198 solution, 43, 179 spectrum, 53, 197 phase space, 20 Poincar´e, 40 Poisson algebra, 6, 29 nonresonant, 37 bracket, 2, 20, 21 manifold, 21 structure, 22, 111 nondegenerate, 22 system, 21 positive cone, 128 potential, 2, 112 A-gap, 6, 113 M-gap, 97 finite gap, 54, 113, 204 projection, 9 proper map, 96 Qiu, 42

quasi-periodic function, 33 motion, 35, 38, 114 solution, 138 R¨ussmann, 37, 42, 45, 46, 154 rationally dependent frequencies, 34 independent frequencies, 34 Rayleigh, 1 real analytic, 119, 193 solution, 114 regular normal form, 152 regularity condition, 136, 147 resonance set, 136, 147 resonant Birkhoff normal form, 234 frequencies, 34 Riemann bilinear relations, 223 rotational torus, 32, 146 Russell, 1 sampling formula, 222 Saut, 2 scale of Hilbert spaces, 257 second KdV Hamiltonian, 10, 115, 126, 229 semi-linear, 136 set isospectral, 3, 54, 55, 113 perfect, 41 resonance, 136, 147 Sevryuk, 42 Siegel, 45 small divisor condition, 40 solitary wave solution, 1 solution almost-periodic, 6, 45, 46, 48 equilibrium, 15 finite gap, 8, 133 fundamental, 55, 194 periodic, 43, 179 quasi-periodic, 138 real analytic, 114 solitary wave, 1 spatially periodic, 2 strong, 258 spatially periodic solution, 2 spectral bands, 53

Index gap, 53 spectrum, 112 Dirichlet, 196 frequency, 33 periodic, 53, 197 Spencer, 46 squared gap lengths, 3, 53 stable, linearly, 9, 10, 114, 134, 146 strong solution, 258 strongly nonresonant frequencies, 40 structure Poisson, 22, 111 symplectic, 20 Sun, 42 symplectic diffeomorphism, 24 form, 19, 257 Hilbert scale, 257 map, 259 structure, 20 symplectomorphism, 24 system, Poisson, 21

Kronecker, 32 parallel, 32 rotational, 32, 146 with linear flow, 32 trace formula, 223 transformation rule, 259 Trubowitz, 3, 7, 54

Taylor series, 189 Temam, 2 theorem KAM, 41, 133, 148 Liouville-Arnold-Jost, 4, 30, 54 torus

You, 42, 45, 48

279

unimodular matrix, 32 van Moerbeke, 7 Vaninsky, 7, 16, 61, 102 vector field, Hamiltonian, 20, 258 Vey, 37 Vittot, 46 Ware, 45 Wayne, 17, 46–48 weakly analytic, 187 Wronskian, 90 identity, 95, 194 Xiu, 42

Zabusky, 1 Zakharov, 1 Zehnder, 45 zero-th KdV Hamiltonian, 5, 223, 227, 238

Notations Spaces, Norms and Meanvalue

Spectral Theory

L 2 = L 2 (S 1 , R)  L 20 = q ∈ L 2 : [q] = 0  H N = q ∈ L 2 : kqk N < ∞  H0N = q ∈ L 20 : kqk N < ∞  `r2 = x ∈ `2 (N, R) : kxkr < ∞ hr = `r2 × `r2

y1 , y2

fundamental solution

m i = yi (1) 1 = m 1 + m 02 µn

n-th Dirichlet eigenvalue

λn

n-th periodic eigenvalue

τn = (λ2n + λ2n−1 )/2

hr• → p. 119 2 kqk2N = |q(0)| ˆ +

L = −d 2 /dx 2 + q

2N k∈Z |k|

P

2 |q(k)| ˆ

kqk = kqk0 P 2 2r kxkr2 = n≥1 n |x n | P |k| = n |kn | for integer vectors |x| = supn |xn | for non-integer vectors R [q] = S 1 u(x) dx Sets

γn = λ2n − λ2n−1 gn

n-th normalized Dirichlet

fn λ˙ n

and periodic eigenfunction ˙ n-th root of 1   p µ∗n = µn , ∗ 12 (µn ) − 4

Birkhoff Coordinates 1

S = R/Z Tn = Rn /2πZn TI = Tn × { I } T0 = Tn × {0} × {0} × {0}  T I = (q, p) : qn2 + pn2 = 2In , n ≥ 1 A(q) = { n ∈ N : γn (q) > 0 } spec(q) Iso(q) = 6(q) = Dn = GA = hA =

0n → p. 64 R ˙ In = π1 0n c√λ1(λ) dλ 12 (λ)−4 p ξn = + 8In /γn2 σmn −λ 2 Q ψn = πn m6 =n m 2 π 2 R µ∗k √ ψn (λ) dλ βkn = λ2k−1 2

periodic spectrum of q  ηn p ∈ L 20 : spec( p) = spec(q)  (λ, z) : z 2 = 12 (λ, q) − 4 ⊂ C2 θn  2 q ∈ L 0 : γn (q) = 0 1λ  2 [8,9] q ∈ L 0 : γn (q) > 0 iff n ∈ A  2 2 (x, y) : xn + yn > 0 iff n ∈ A

1 (λ)−4

R µ∗n

√ ψn (λ) dλ mod 2π 12 (λ)−4 P = ηn + k6=n βkn =

λ2n−1

= 1(λ, · ) = h∂8,∂x ∂9i

Perturbed KdV Equations

KAM Theorem and Proof

I A = (In )n∈A  ` A = w ∈ `2 : wNXA = 0 ' R A

2 `p,a =

r2 } ×

× { kuk p + kvk p < r } → p. 137 Rkl sup kFk N ;U kW kr,q

= { ξ : hk,ω(ξ )i + hl,(ξ )i = 0 }

kWu kq +

1 r

kwk p = kwk p,a kX krλ = kX kr

sup

sup

lip

sup k1ξ ζ W kr,q;U

|ξ −ζ |

ξ 6=ζ

lip

˜ || −δ;5 = supξ,ζ ∈5

Roots p √ s + 1 − λ2 = iλ 1 − λ−2

|1ξ ζ ˜ |−δ

for |λ| > 1 p c i 12 (λ) − 4 > 0 for λ ∈ (λ0 , λ1 ) q ∗ 12 (µn ) − 4 = y1 (1, µn ) − y20 (1, µn )

|ξ −ζ |

ξ 6=ζ

1ξ ζ W = i ξ W − i ζ W |λ|β = supn≥1 n β |λn | KdV Hamiltonians R H 0 = S 1 12 u 2 dx R
H 1 = S 1 21 u 2x + u 3 dx R
H 2 = S 1 12 u 2x x + 5uu 2x + 52 u 4 dx

Miscellaneous 0

‘−1’

∂ =

∂ ∂x ∂ ∂λ ∂ ∂q

a l b ⇔ a ≤ cb with c ≥ 1

‘−1’

‘−i’ λ0

=

˙ =

Hc = H 1 + 6cH 0

‘+1’

lip

+ λ kX kr

hki = max(1, |k|) P P
[l]δ = max 1, j jl j · j j δ l j
1/β
Bσ = σab exp σc

kWv kq

kW kr,q;U ×5 = sup(w,ξ )∈U ×5 kW (w, ξ )kr,q kW kr,q;U ×5 = supξ,ζ ∈5

λ1



n S np = Sp,a P |k|s kuks = k |uˆ k |e P τ |k|s kuks,τ = k |uˆ k | |k| e P 2 p 2an |w |2 kwk2p,a = n nn e

= supu∈U kF(u)k N = |Wx | + r12 W y + 1 r

w ∈ `2 : kwk p,a < ∞

n 2 × `2 Sp,a = Tn × Rn × `p,a p,a

D(s) = { |Im x| < s } D(s, r ) = { |Im x| < s } × { |y| <



‘+1’

Figure 10 Signs of

‘+1’

‘+i’ λ2 p c

λ3

‘−1’

12 (λ) − 4 for real q

λ4