Long Time Estimates of Solutions to Hamiltonian Nonlinear Pdes

Long time estimates of solutions to Hamiltonian nonlinear PDEs

Patrick Gérard Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France [email protected]

UNC PDE mini-school February 2-4, 2016 Abstract

This minicourse is devoted to long time behavior of solutions to nonlinear PDEs such as nonlinear Schrödinger equation or nonlinear wave equations. More precisely, we would like to provide an introduction to the following general question, closely connected to wave turbulence: assume that such a nonlinear PDE is globally well-posed on high regularity Sobolev spaces; how big can the high Sobolev norms of generic solutions be as time goes to inﬁnity? The second part of the course will be focused on the special case of the cubic Szegő equation, which is a model of a nonlinear wave evolution and enjoys some integrable structure allowing to study its solutions in detail. Introduction

Since the eighties, the study of Hamiltonian evolution partial differential equations, such as nonlinear Schrödinger equations of nonlinear wave equations, has become a topic of growing importance in mathematical analysis. Currently, a fairly general theory of initial value problems is now available, and the main effort of the experts is now mainly directed towards the long term description of solutions, starting with the case of small initial data. Though this program is well advanced if the physical space is the whole Euclidean space, where the dispersive effects are maximal and generally lead to scattering for small data solutions, it is still widely unexplored if the physical space is a bounded domain or a closed manifold, where the dispersive effects may be much weaker. In particular, a specific important question in the latter case concerns the possibility of strong oscillations of the solution as time becomes large, or equivalently the growth of Sobolev norms of high regularity. The goal of these notes is to provide an elementary introduction to this topic, as well as a focus onto one of the few Hamiltonian evolutions for which such a description is available, thanks to a special integrability structure. The content of these notes is based on a series of three lectures given in the University of North Carolina during the Winter 2016, addressed to an audience of graduate students and postdocs in PDEs coming from various universities in the United States. I am grateful to the Mathematics Department of Chapel Hill, who made this minicourse possible, and to my student Joseph Thirouin, who wrote the first version of these notes.

Orsay, April 2016 Patrick Gérard

1 Lecture 1

Introduction to long time estimates

1.1 What is a Hamiltonian PDE?

1.1.1 Hamiltonian systems in finite dimension At the beginning of the nineteenth century, physicists were looking for an abstract setting to reformulate Newton’s laws of mechanics. The ideas of the French mathematician Joseph-Louis Lagrange (1736-1813) led to the so-called Euler-Lagrange equations and to the calculus of vari- ations, whereas the Irish scientist William Rowan Hamilton (1805-1865) introduced another formalism, which later inspired the theory of quantum mechanics, and gave rise to a whole field in the study of PDEs. Here is the finite-dimensional setting Hamilton proposed. Let n ∈ N\{0}, and consider n n the phase space R × R . An element (q, p) in this space corresponds to a set of “positions” n n q = (q1, . . . , qn) and of “momenta” p = (p1, . . . , pn). Given a smooth function H : R × R → R, the Hamilton equations associated to H are  ∂H  q˙j = (q, p),  ∂pj  ∂H  p˙j = − (q, p), ∂qj for all j ∈ {1, . . . , n}, where the dot stands for the time derivative. The function H is then called the energy of this system of ordinary differential equations, since it is conserved along the flow lines.

n An example. Let V : R → R be a smooth function (a potential), and consider the Hamilto- nian given by 1 H(q, p) := |p|2 + V (q), 2m 2 Pn 2 where |p| := j=1 pj and m > 0 is a real number. Then the Hamilton equations read mq˙j = pj, which exactly means that pj is the momentum in a physical sense, and p˙j = −∂qj V (q), i.e. mq¨j = −∂qj V (q). The Hamilton equations associated to H thus describe the motion of n bodies of same mass m according to Newton’s second law of Mechanics.

2 Complex formulation. We can reformulate the equations in an even shorter way, introducing n n n C = R ⊕ iR and a new variable z := q + ip. The evolution of z under the Hamilton equations is given by ∂H ∂ 1 ∂ ∂ iz˙j = 2 (z), where := + i . (1.1) ∂z¯j ∂z¯j 2 ∂qj ∂pj ∂ := 1 ∂ − i ∂ Similarly, we deﬁne ∂zj 2 ∂qj ∂pj . n n Note that for any smooth function F : C → R and any smooth curve γ : R → C , γ(s) = (γ1(s), . . . , γn(s)), we have

n d X dγj ∂F dγ¯j ∂F F (γ(s)) = (s) (γ(s)) + (s) (γ(s)). ds ds ∂z ds ∂z¯ j=1 j j

n ∂F ∂F Now denote by (·|·) the usual Hermitian product on C , and by ∂z (resp. ∂z¯ ) the n-vector ∂F ∂F j ∈ {1, . . . , n} F ∂F = ∂F formed by the collection of ∂zj (resp. ∂z¯j ), . Since is real-valued, ∂z¯ ∂z , so that we can write d ∂F F (γ(s)) = 2 Re γ˙ (s) (z) = Im (γ ˙ (s)|XF (γ(s))) , ds ∂z¯

∂F setting XF (z) := −2i ∂z¯ (z). The Hamilton equations (1.1) are thus equivalent to

z˙ = XH (z), (1.2) and this is how the Hamiltonian formalism can be extended to the case of inﬁnite dimensional phase spaces.

1.1.2 Extension to inﬁnite dimensional systems ∞ As a phase space, consider now E := C (M, C). In general, M could be any compact bound- aryless manifold. For instance, if dim M = 0, M is a set formed by n points, and E is then n d d isomorphic to C . In the sequel, we will restrict ourselves to M = T = (R/2πZ) , with d ≥ 1. ∞ d In that case, E is isomorphic to the set of C functions on R which are 2π-periodic with respect to each coordinate.

Deﬁnition. Let H : E → R. Assume that there exists a mapping XH : E → E such that for all smooth path γ : R → E, d H(γ(s)) = Im (γ ˙ (s)|XH (γ(s))) , (1.3) ds where Z (f|g) = f(x)g(x)dx, Td d d and dx is the Lebesgue measure renormalized by (2π) , so that T is of measure 1.

3 Then we say that XH is the Hamiltonian vector ﬁeld of H, and the corresponding equation

u˙ = XH (u) (1.4) is called the Hamiltonian system associated to H. Remark 1. A crucial remark is that, as in the ﬁnite dimensional case, H is a conservation law for the system. Indeed, by deﬁnition, d H(u(t)) = Im(u ˙(t)|X (u(t))) = Im(X (u(t))|X (u(t))) = 0. dt H H H Remark 2. Note that there is no theorem insuring that solutions of (1.4) exist, even locally in time.

1.2 Some examples

1.2.1 The nonlinear Schrödinger equation ∞ d Let λ ∈ R. For u ∈ E = C (T , C), we set Z 1 2 λ 4 H(u) = 2 |∇u(x)| + 4 |u(x)| dx. Td Let us show that the hypothesis of the above definition is satisfied. Let γ : R → E be a d smooth path (in the sequel, we write γ as a function of s ∈ R and x ∈ T ), and compute d Z H(γ(s)) = Re ∇γ˙ (s, x)∇γ(s, x) + λ|γ(s, x)|2γ˙ (s, x)γ(s, x) dx ds Td Z = Re γ˙ (s, x)[−∆γ(s, x) + λ|γ(s, x)|2γ(s, x)] dx Td Z = Im γ˙ (s, x)[i∆γ(s, x) − iλ|γ(s, x)|2γ(s, x)]dx Td 2 Thus, XH : E → E, u 7→ i(∆u − λ|u| u) is the Hamiltonian vector field of H, and (1.4) reads ( iu˙ + ∆u = λ|u|2u (1.5) u|t=0 = u0.

>From now on, we assume that λ ≥ 0 : this is called the defocusing case. The existence of solutions of (1.5) has been extensively studied, and we have the following result : ∞ d Theorem 1. Let d ∈ {1, 2, 3}. For any u0 ∈ E, there exists a unique u ∈ C (R × T , C) satisfying (1.5). The case d = 2 dates back to Brezis and Gallouët [4], and d = 3 is a consequence of the works of Bourgain [1]. In fact, more general results are available, involving less regular data.

4 1 d Definition. • Let f ∈ L (T ). We define the Fourier coefficients of f by Z −ik·x d fˆ(k) = f(x)e dx, ∀k ∈ Z . Td

s d 1 d • For s ≥ 1, we deﬁne the space H (T ) to be the space of all f ∈ L (T ) such that X (1 + |k|2)s|fˆ(k)|2 < +∞. k Hs is a Hilbert space, and the norm is deﬁned as follows :

1 ! 2 X 2 s ˆ 2 kfkHs := (1 + |k| ) |f(k)| . k

s α 2 d • Observe that, whenever s ∈ N, belonging to H is the same as requiring that ∂ f ∈ L (T ) for all |α| ≤ s. s d Theorem 2. The nonlinear Schrödinger equation (1.5) is globally well-posed on H (T ) for any s ≥ 1 and d ∈ {1, 2, 3}. What kind of informations do we have about the trajectories of (1.5), seen as a dynamical system ? In view of the conservation of H and of the control it gives over the H1 norm, we 1 d know that {u(t)}t∈R is bounded in H (T ). A very natural question to ask then would be the following : s Question ([3]). Let u0 ∈ E be a smooth initial data. Hence u0 ∈ H for all s, and so does the solution u(t, ·) at any time, by the previous theorem. Given some s > 1, how big can be the Hs norm ku(t, ·)kHs as t → ∞ ?

Suppose for instance that there exists s0 > 1 such that we have

lim sup ku(t, ·)kHs0 = +∞. t→∞

It means that for some sequence of times tn → ∞, we have at the same time

X 2 s0 2 (1 + |k| ) |uˆ(tn, k)| > n k∈Zd X 2 2 and (1 + |k| )|uˆ(tn, k)| ≤ C. k∈Zd 2 In other words, at t = tn, |uˆ(tn, k)| becomes big for big k’s, so that the ﬁrst series is big whereas the second one remains bounded, which reﬂects the fact that u(tn, ·) begins to oscillate. This is why such a phenomenon is called transition to high frequencies, or sometimes wave turbulence. When d = 1, we know that there is no transition to high frequencies. This is due to the integrability result of Zakharov and Shabat [17]. The equation ∂u ∂2u i + = |u|2u, x ∈ , ∂t ∂x2 T

5 admits plenty of conservation laws, which are of the form Z h (p) 2 (p−1) (p−1) i |u (x)| + Fp(u(x), u¯(x), . . . , u (x), u¯ (x)) dx, ∀p ∈ N, T where Fp is some polynomial. It is then possible to show [12] that the trajectories of smooth solutions remain bounded in every Hs, s ≥ 1. As soon as d ≥ 2, the question of wave turbulence remains widely open. The only known result is the one of Bourgain [2] and Staﬃlani [15] : for all t ∈ R, we have

(s−1)A ku(t)kHs ≤ C[u0](1 + |t|) , ∀s > 1, (1.6) where C[u0] is a constant depending only on u0, and where A can be chosen to depend only on s. To rephrase it, the growth of ku(t)kHs is at most polynomial in time. On the other hand, partial results are available, stating that, given any s > 1, there exist small data in Hs generating a solution which is big in the same space Hs at some big time, see Colliander–Keel–Staﬃlani– Takaoka–Tao [5], Guardia–Kaloshin [9], and Hani [10] for some simpliﬁed model. More recently, 2 on the unbounded manifold M = R × T , Hani–Pausader–Tzvetkov–Visciglia [11] proved the existence of unbounded trajectories in Hs — but so far these trajectories are not known to be made of smooth solutions.

1.2.2 Changing the dispersion relation In the hope of finding wave turbulence for Hamiltonian systems, a first idea is to change the dispersion relation, i.e. in our case, to replace the Laplacian by another Fourier multiplier. When M = T, we could then start from a new Hamiltonian Z 1 X α 2 1 4 H(u) = 2 |k| |uˆ(k)| + 4 |u(x)| dx, k∈Z T for u ∈ E, and some α > 0. This leads to the following PDE :  ∂u  i = |D|αu + |u|2u ∂t (1.7)  u|t=0 = u0, called the fractional Schrödinger equation. Here, the operator |D|α is defined by ! X X |D|α fˆ(k)eikx = |k|αfˆ(k)eikx. k∈Z k∈Z

s It is possible to show [16] that the fractional Schrödinger equation is globally well-posed in H (T) α 2 for s ≥ 2 , provided that 3 < α < 2. However, if in addition α 6= 1, smooth solutions satisfy polynomial estimates in the same spirit as (1.6).

6 1.2.3 The linear case The following result shows that wave turbulence only occurs in nonlinear settings. Here, d ≥ 1.

d Proposition 1.2.1. Let H : E → R such that ∀x0 ∈ T , H(u(· + x0)) = H(u). Assume that there exists XH : E → E as in (1.3), and assume XH is also linear. Then if u is a solution of d u˙ = XH (u), then |uˆ(t, k)| = |uˆ(0, k)|, for all t ∈ R and k ∈ Z . In particular, all the Hs norms are conserved along trajectories.

Proof. Indeed, because of the invariance property H(u(· + x0)) = H(u), and of the invariance by translation of the inner product, we have

XH (u(· + x0)) = XH (u)(· + x0).

d ik·x If moreover u 7→ XH (u) is linear, we infer, for every k ∈ Z , setting εk(x) := e ,

ik·x0 XH (εk)(· + x0) = e XH (εk), which implies, decomposing XH (εk) along the basis {εk}k∈Z, that

XH (εk) = ω(k)εk, for some ω(k) ∈ C. d We claim that, for every k ∈ Z , ω(k) ∈ iR. Indeed, apply the deﬁnition of XH to the path is γ(s) := e εk. We get d H(eisε ) = Im(ieisε |X (eisε )) = Im(iω(k)). ds k k H k

Integrating this identity on s ∈ [0, 2π], we conclude that 0 = 2π Im(iω(k)), which precisely means ω(k) ∈ i u = P uˆ(k)ε that R. Then, expanding k∈Zd k, we get X XH (u) = uˆ(k)ω(k)εk, k∈Zd or X\H (u)(k) =u ˆkω(k). d tω(k) This implies dt uˆ(k, t) = ω(k)ˆu(k, t), thus uˆ(k, t) = e uˆ(k, 0). Since ω(k) ∈ iR, the modulus of uˆ(k, t) is conserved, which was to prove.

Note that, in the nonlinear case, Fourier coeﬃcients may not be an appropriate representation of the solutions of the equation anymore.

7 1.3 Towards the cubic Szegő equation

∞ Consider on E = C (T, C) the following Hamiltonian : Z 1 4 H(u) := 4 |u(x)| dx . T The associated equation is iu˙ = |u|2u, and it can be solved explicitely :

2 −it|u0(x)| u(t, x) = u0(x)e .

s >From this simple formula, we infer that for s ≥ 0, the norm ku(t)kHs grows like |t| as t → ∞, 2 provided that |u0| is not constant on T. The example of the half-wave equation suggests that this transition to high frequencies could be strenghtened by breaking the symmetry and introducing a projector in the energy. So let Π be the Szegő projector onto nonnegative frequencies, such that

! +∞ X X Π fˆ(k)eikx = fˆ(k)eikx. k∈Z k=0 Now we deﬁne a new energy on E by Z 1 4 H(u) := 4 |Π(u)(x)| dx. (1.8) T This time, the associated equation reads

iu˙ = Π(|Π(u)|2Π(u)).

But observe that if u satisfies this evolution equation, and if we denote by u− := (I − Π)(u), we d − 2 have dt u = (I − Π)(iu˙) = (I − Π)(Π(|Π(u)| Π(u))) = 0, so the dynamics of the equation are in fact encoded in the “positive” part of u, i.e. u+ := Π(u). That is why it is enough to restrict ourselves to the subspace of E of functions that also belong to Ran(Π). On this space, the Hamiltonian H in (1.8) leads to the following equation : ( iu˙ = Π(|u|2u) ∞ (1.9) u|t=0 = u0 ∈ E ∩ Ran(Π) =: C+ (T), which is called the cubic Szegő equation, and was studied for the first time in [6] by Sandrine Grellier and the author. The rest of these notes will discuss the following theorem, which was proved in [6] for the first part, and in [8] for the second one :

Theorem 3 (Wave turbulence for the cubic Szegő equation). For any solution u to the Szegő 1 equation (1.9), we have, for all s > 2 ,

C0 |t| ku(t)kHs ≤ Cse s , ∀t ∈ R,

8 0 where Cs, Cs are positive constants which only depend on s and ku0kHs . ∞ Moreover, there exists a dense Gδ-subset of C+ (T) denoted by G, such that for all u solution 1 to the Szegő equation (1.9) with u0 ∈ G, for all s > 2 and all M ≥ 1, we have

ku(t)kHs lim sup M = ∞, t→∞ |t| and on the other hand lim inf ku(t)kHs ≤ ku(0)kHs . t→∞ The theorem thus gives a nearly optimal result for the Szegő equation : solutions cannot grow faster than exponentially, and most of them indeed grow faster than any polynomial in time.

9 Lecture 2

The cubic Szegő equation on the circle and its special structure

2.1 The setting

We ﬁrst introduce the Hardy space of holomorphic functions on the open unit disc of C, whose 2 1 1 ix trace on the circle is in L (S ). Note that, because of the mapping T = R/2πZ → S , x 7→ e , 2 2 1 we identify L (T) and L (S ) from now on. 2 In the theory of Hardy spaces, much developped by F. & M. Riesz, that space is called H (D), but to avoid any confusion with Sobolev spaces, we name it

2 2 L+ := u ∈ L (T) | ∀k < 0, uˆ(k) = 0 , or alternatively : ( ) ∞ ∞ 2 ix X ikx X 2 L+ = u(e ) = uˆ(k)e |uˆ(k)| < ∞ , k=0 k=0 ( ∞ Z 2π ) 2 X k ix 2 L+ = u(z) = uˆ(k)z , |z| < 1 sup |u(re )| dx < ∞ . r<1 k=0 0 s s 2 s For s ≥ 0, we also deﬁne H+ := H (T) ∩ L+, so that for u ∈ H+,

∞ 2 X 2 s 2 kukHs = (1 + k ) |uˆ(k)| < ∞. k=0

2 1 2 Finally, we recall the deﬁnition of the Szegő projector Π mapping L (S ) onto L+, by cutting Π P c eikx = P c eikx negative frequencies : k∈Z k k≥0 k . The cubic Szegő is the following PDE :  ∂u  i = Π(|u|2u), ∂t (2.1) s  u(0) = u0 ∈ H+.

10 From the Hamiltonian structure given in lecture1, we know that Z 1 4 H(u) = 4 |u| T 2 P 2 is a conservation law for (2.1), as well as kukL2 and M(u) := (−i∂xu|u)L2 = k≥0 k|uˆ(k)| . The existence of the last two laws is a consequence of the invariance of H under two groups of 2 iθ isometries of L+ given by θ 7→ e u and x 7→ u(· + x) respectively. kuk2 + M(u) = P (1 + k)|uˆ(k)|2 ' kuk2 H1/2 Observe that L2 k≥0 H1/2 . This implies that the norm of (sufficiently smooth) solutions of the Szegő equation remains uniformly bounded in time. This is the cornerstone of the proof of s 1 ∞ s Proposition 2.1.1. Let u0 ∈ H+ with s > 2 . There exists a unique u ∈ C (R,H+) such that (2.1) is satisfied. Proof. The local existence is simple consequence of the Cauchy-Lipschitz theory for ODEs, since s 1 H is an algebra when s > 2 . Moreover, the time of existence of this solution only depends on the norm of the data in Hs. As a consequence, to show that local solutions extend globally in time, it suffices to show that the Hs norm does not blow up in finite time. So let us compute d kDsu(t)k2 = 2 Re(Dsu˙|Dsu) = 2 Im(Ds(|u|2u)|Dsu). dt L2 2 2 Due to tame estimates, the right hand side can be bounded by kukHs kukL∞ . Now let us recall a celebrated lemma : 1 Lemma 2.1.2 (Brezis-Gallouët, see [6, Appendix 2]). Let s > 2 . There exists a constant Cs > 0 s such that for all v ∈ H (T), s kvkHs kvkL∞ ≤ CskvkH1/2 log 1 + kvkH1/2

Using this inequality, as well as the control on the H1/2 norm, and applying a Gronwall-type argument then allows us to bound

0 eC t ku(t)kHs ≤ Ce , which proves that the Hs norm of solutions remain ﬁnite in ﬁnite time. Hence local solutions are in fact global. Uniqueness follows from a standard connexity argument.

Remark 3. The above proposition can in fact be extended, even if we will not use it : given 1/2 1/2 u0 ∈ H+ , there exists a unique u ∈ C(R,H+ ) solution to (2.1). The existence is obtained by standard compactness methods. As for the uniqueness, the proof is based on an inequality of Trudinger : there exists C > 0 such that for any v ∈ H1/2 and 1 ≤ p < +∞, √ kvkLp ≤ C pkvkH1/2 .

Again, see [6] for more details.

11 2.2 Hankel operators and the Lax pair structure

The principle of Lax pairs, introduced by Peter Lax in a seminal paper on the KdV equation [13], is to translate a PDE in terms of the evolution of a family of operators. That is why, before stating the Lax pair theorem for the cubic Szegő equation (2.1), we must deﬁne a class of operators, called the Hankel operators.

2.2.1 Hankel operators on sequences 2 We denote by ` (N) the set of square-summable sequences of complex numbers. On this space, there is a scalar product given by ∞ X (x|y) = xnyn. n=0 2 Deﬁnition. Let c = {cn} ∈ ` (N). The Hankel operator of symbol c is 2 2 Γc : ` (N) −→ ` (N), x = {xn} 7−→ y = {yn}, where {yn} is deﬁned by an “anti-convolution” product : ∞ X yn = cn+pxp, ∀n ∈ N. p=0

As Γc is an operator given by a kernel k(n, p) := cn+p, it is easy to compute its Hilbert- Schmidt norm : it is simply the L2 norm of the kernel in the product space. More precisely, ∞ X X kΓ k2 = kk(n, p)k2 = |c |2 = (1 + l)|c |2. c HS `2(N×N) n+p l n,p≥0 l=0

In the sequel, we make the assumption that kΓckHS < ∞. 2 Deﬁnition. The shift operator on ` (N) is the following (not onto) isometry

S :(x0, x1, x2,... ) 7−→ (0, x0, x1, x2,... ). The anti-shift is its adjoint with respect to the scalar product (·|·) : ∗ S :(x0, x1, x2,... ) 7−→ (x1, x2, x3,... ). The following identities are direct consequences of the deﬁnitions : S∗S = I, ∗ SS = I − (·|e0)e0, where e0 = (1, 0, 0,... ), ∗ S Γc = ΓcS = ΓS∗c. 2 The last identity even characterizes the Hankel operators among continuous operators on ` (N). ∗ We call S Γc the shifted Hankel operator associated to Γc. We immediately infer a link between a Hankel operator and its associated shifted operator. ∗ ∗ Lemma 2.2.1. ΓS∗cΓS∗c = ΓcΓc − (·|c)c. ∗ ∗ ∗ Proof. The left hand side equals ΓcSS Γc , and we apply the above formula on SS .

12 2.2.2 Back to the Szegő equation 1/2 Theorem 4. If u is a solution to (2.1), at least in H+ , then the eigenvalues of

∗ ∗ ΓuˆΓuˆ and ΓS∗uˆΓS∗uˆ are conserved quantities.

∗ ∗ Deﬁnition. The square root of the eigenvalues of ΓuˆΓuˆ (resp. ΓS∗uˆΓS∗uˆ) are called the singular values of Γuˆ (resp. ΓS∗uˆ).

Proof of theorem4. The proof divides into several steps.

Step 1. We reformulate the problem, using the isometric isomorphism

2 ∼ 2 L+ −→ ` (N) u 7−→ {uˆ(n)}n≥0.

1/2 2 2 ¯ For u ∈ H+ , deﬁne Hu : L+ → L+, h 7→ Π(uh). Observe that

ˆ H\u(h) = Γuˆ(h),

∗ and since Γc = Γc¯, 2 ∗ ˆ H\u(h) = ΓuˆΓuˆ(h). 2 Notice that Hu is C-antilinear, whereas Hu is C-linear and self-adjoint positive. We similarly identify the conjugate of S, u 7→ eixu, and the conjugate of its adjoint S∗, u 7→ Π(e−ixu) (also denoted by S and S∗ respectively). In this framework, the shifted Hankel operator is

∗ Ku := S Hu = HuS = HS∗u.

Lemma 2.2.1 can be restated as 2 2 Ku = Hu − (·|u)u. (2.2)

Step 2. We also need another class of operators, called Toeplitz operators. ∞ 1 2 Deﬁnition. Let b ∈ L (S ). For h ∈ L+, we deﬁne Tb(h) := Π(bh). 2 2 The C-linear operator Tb : L+ → L+ is called the Toeplitz operator of symbol b. It is bounded, ∗ and Tb = T¯b.

Step 3. We can now prove a crucial algebraic lemma : s 1 Lemma 2.2.2. Let a, b, c ∈ H+, s > 2 . We have

HΠ(a¯bc) = Ta¯bHc + HaTbc¯ − HaHbHc. (2.3)

13 2 Indeed, for h ∈ L+, ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ HΠ(a¯bc)(h) = Π(Π(abc)h) = Π(abch − (I − Π)(abc)h) = Π(abch), because (I − Π)(a¯bc)h¯ has only negative frequencies. Hence ¯ ¯ ¯ ¯ ¯ ¯ HΠ(a¯bc)(h) = Π(abch) = Π(abΠ(ch)) + Π(ab(I − Π)(ch)) = Ta¯bHc(h) + Ha(f), where f := b(I − Π)(ch¯). Since (I −Π)(ch¯) has only negative frequencies, f has only nonnegative frequencies, hence ¯ f = Π(f) = Π(bch¯ ) − Π(bΠ(ch)) = Tbc¯(h) − HbHc(h), and the proof of the lemma is complete.

Step 4. Now we establish an equation for Hu when u is a solution of the Szegő equation (2.1) s 2 in H+. Assuming iu˙ = Π(|u| u), and applying lemma 2.3 with a = b = c = u, we get d H = H = −iH 2 dt u u˙ Π(|u| u) 3 = −i(T|u|2 Hu + HuT|u|2 − Hu) i 2 i 2 = −iT|u|2 Hu − Hu(−iT|u|2 ) + 2 HuHu − Hu( 2 Hu) i 2 = [−iT|u|2 ,Hu] + [ 2 Hu,Hu], where [A, B] stands for the commutator AB − BA. Notice that we used the antilinearity of the i 2 operator Hu. Finally, setting Bu := −iT|u|2 + 2 Hu, we end up with the following identity

d Hu = [Bu,Hu] (2.4) dt

∗ which is called a Lax pair identity. In addition here, Bu is anti-self-adjoint (i.e. Bu = −Bu). Similarly, we can prove that d Ku = [Cu,Ku], (2.5) dt i 2 where Cu := −iT|u|2 + 2 Ku is also anti-self-adjoint.

2 2 Step 5. It remains to infer from (2.4) and (2.5) informations on the spectrum of Hu and Ku. Notice that the following argument remains valid as soon as we have a Lax pair (H,B) with anti-self-adjoint operators B. 2 We deﬁne U(t) ∈ L(L+) to be the global solution of the following linear ODE :   U(0) = I, dU(t)  = B U(t), dt u(t)

14 whose existence (and uniqueness) is guaranteed by the linear Cauchy theory. Let us ﬁrst show that U(t) is unitary for all t ∈ R. On the one hand, d U(t)∗U(t) = U(t)∗B∗U(t) + U(t)∗B U(t) = 0 dt u u which means that for all times, U(t)∗U(t) = U(0)∗U(0) = I. Secondly, d U(t)U(t)∗ = B U(t)U(t)∗ + U(t)U(t)∗B∗ = [B ,U(t)U(t)∗]. dt u u u In other words, t 7→ U(t)U(t)∗ is a solution to the following Cauchy problem : d F = [B ,F ],F (0) = I, dt u but by the uniqueness of the solution of such a linear problem, we must have UU ∗ ≡ I. Hence 2 U is unitary on L+. Now we compute d U(t)∗H U(t) = −U(t)∗B H U(t) + U(t)∗[B ,H ]U(t) + U(t)∗H B U(t) = 0. dt u u u u u u u ∗ Then for all t ∈ R, we have Hu(t) = U(t)Hu0 U(t) , and squaring this identity, we also get

2 2 ∗ Hu(t) = U(t)Hu0 U(t) ,

H2 H2 H which implies that u(t) and u0 have the same spectrum, or in other words, that u(t) and

Hu0 have the same singular values.

Similarly, Ku(t) and Ku0 have the same singular values.

2.3 The role of singular values

2.3.1 The interlacement property 2 2 We begin with establishing a simple lemma about the eigenvalues of Hu and Ku.

1/2 2 02 Lemma 2.3.1. For u ∈ H+ , let (sj )j≥1 and (sk )k≥1 be the decreasing sequence formed by the 2 2 eigenvalues of Hu and Ku respectively, written with multiplicities. Then we have

0 0 s1 ≥ s1 ≥ s2 ≥ s2 ≥ · · · ≥ 0.

Proof. The proof relies on the formula (2.2.1) and on a use of the min-max formula for compact 2 self-adjoint operators : if A is a compact positive self-adjoint operator on L+, and if (ρj)j∈N∗ denotes the decreasing sequence of its eigenvalues, then

ρj = min max kA(h)k. 2 ⊥ F ⊂L+ h∈F dim F ≤j−1 khk=1

15 2 2 We are going to apply this for Hu and Ku. First of all, notice that 2 2 max kKu(h)k = max (Ku(h)|h). h∈F ⊥ h∈F ⊥ khk=1 khk=1

2 2 2 2 We compute (Ku(h)|h) = (Hu(h)|h) − |(u|h)| ≤ (Hu(h)|h). Taking the maximum and then the 02 2 minimum over subspaces F on both sides ﬁnally yields sj ≤ sj . 2 Next, if F is a subspace of L+ of dimension at most j − 1, we have max kK2(h)k ≥ max kK2(h)k = max kH2(h)k, ⊥ u ⊥ u ⊥ u h∈F h∈(F +Cu) h∈(F +Cu) khk=1 khk=1 khk=1 because of formula (2.2.1). Now, F + Cu is of dimension at most j. Hence, taking the minimum over F on both sides yields 02 2 2 2 sk ≥ min max kHu(h)k ≥ min max kHu(h)k ≥ sj+1. 2 ⊥ ˜ 2 ˜⊥ F ⊂L+ h∈(F +Cu) F ⊂L+ h∈F khk=1 dim F ≤j−1 khk=1 dim F˜≤j

2.3.2 The inverse spectral formula 2 2 Now that we know, because of (2.4) and (2.5), that eigenvalues of Hu and Ku are conserved, there is a natural question to ask : given a set of positive interlaced real numbers, is it possible to 1/2 2 2 find u ∈ H+ such that the corresponding eigenvalues of Hu and Ku are precisely these numbers? The following theorem provides a positive answer in the finite rank case, when eigenvalues are distinct from one another. 1/2 Theorem 5 (the finite rank case, no multiplicity). Let u ∈ H+ . Assume that ∃N ∈ N such that rk Hu = N and rk Ku = N. We write s := s1 > ··· > sN > 0 the singular values of Γuˆ, and 0 0 0 0 0 s := s1 > ··· > sN > 0 the ones of ΓS∗uˆ. Suppose that we have s1 > s1 > s2 > s2 > ··· > sN > 0 sN > 0, according to the previous lemma. 0 0 0 0 2N Then there exists a set of angles (ψ, ψ ) := (ψ1, ψ2, . . . , ψN , ψ1, ψ2, . . . , ψN ) ∈ T such that the N × N matrix C (z) given by

0 iψj 0 iψ sje − ske k z C (z)j,k = 2 02 sj − sk for 1 ≤ j, k ≤ N is invertible for any z in D (the closed unit disc of C), and such that −1 u(z) = hC (z) (1N )|1N iCN , ∀|z| ≤ 1, N where 1N is the vector of C each component of which is 1, and where h·, ·i denotes the standard N scalar product on C . 0 0 1/2 Conversely, given (s, s , ψ, ψ ), there exists a unique u ∈ H+ , deﬁned through the above 0 1 formula, such that s (resp. s ) is the set of singular values of Γuˆ (resp. ΓS∗uˆ) . 1The mapping between u and the corresponding (s, s0, ψ, ψ0) turns out to be even a diﬀeomorphism.

16 In other words, theorem5 gives an explicit set of coordinates for the 2N-torus of functions 1/2 in H+ whose associated Hankel and shifted Hankel operators have prescribed singular values, with an additional non-degeneracy condition contained in the strict inequalities.

Proof of the ﬁrst part of theorem5. Since Ku = HuS, we have Ran Ku ⊆ Ran Hu, and in fact, both spaces coincide, for their dimension is N. We call this space W . Setting

2 2 Eu(sj) := ker(Hu − sj I), 0 2 02 Fu(sk) := ker(Ku − sk I),

0 we infer from the hypothesis that for all 1 ≤ j, k ≤ N, dim Eu(sj) = dim Fu(sk) = 1, and besides, N N M M 0 W = Eu(sj) = Fu(sk), (2.6) j=1 k=1 where the direct sums are orthonormal sums. 0 0 Let also uj (resp. uk) be the orthogonal projection of u on Eu(sj) (resp. Fu(sk)). Observe that we must have uj 6= 0 for all j ∈ {1,...,N}. Indeed, in view of (2.2.1), if u was orthogonal to some Eu(sj), we would have

2 2 2 2 Ku(h) = Hu(h) − (h|u)u = Hu(h) = sj h,

2 2 for all h ∈ Eu(sj). But this cannot happen, since sj is not an eigenvalue of Ku. For the same 0 reason, uk 6= 0. Now compute, for 1 ≤ j, k ≤ N :

Consequently, 0 2 2 0 kukk kujk (uj|uk) = 2 02 . sj − sk Finally using (2.6), we get  N 0  2 X uk  uj = kujk ,  s2 − s02  k=1 j k N  X uj  u0 = ku0 k2 .  k k s2 − s02  j=1 j k

Consider now the action of Hu on the subspaces Eu(sj), and note that Hu(uj) ∈ Eu(sj). Since it is a one-dimensional space, we know that there exists λj ∈ C such that Hu(uj) = λjuj. 2 2 2 iψj Applying Hu to the equality, we get Hu(uj) = |λj| uj = sj uj, so that we can write λj = sje

17 0 for some ψj ∈ T. The same holds for Ku(uk), hence for all 1 ≤ j, k ≤ N, we have found angles 0 ψj, ψk ∈ T, such that ( iψj Hu(uj) = sje uj, 0 0 iψ0 0 Ku(uk) = ske k uk. ∗ Fix j ∈ {1,...,N}. Noting that SKu = SS Hu = Hu − (u|·)1, we write, using complex notation :

iψj 2 sje uj(z) = Hu(uj)(z) = SKu(uj)(z) + (u|uj) = zKu(uj)(z) + kujk N 0 N 0 iψ0 0 X Ku(u ) X s e k u = zku k2 k + ku k2 = zku k2 k k + ku k2, j s2 − s02 j j s2 − s02 j k=1 j k k=1 j k

On the other hand, we know that

N X u0 s eiψj u = ku k2 k , j j j s2 − s02 k=1 j k

2 so we simplify by kujk and eventually,

N iψ 0 iψ0 X sje j − s e k z k u0 (z) = 1, s2 − s02 k k=1 j k

2 which holds in fact for any j ∈ {1,...,N}. This implies that ∀z ∈ C such that |z| ≤ 1,

 0    u1(z) 1  .  −1  .  −1  .  = C (z)  .  = C (z) (1N ). 0 uN (z) 1

PN 0 To conclude, it remains to observe that u = Hu(1), so u ∈ W . Thus u(z) = k=1 uk(z), and the formula −1 u(z) = hC (z) (1N )|1N i is proved.

2.3.3 The evolution in new coordinates 1/2 On this peculiar subset of H+ , and in these new coordinates, it turns out that solving the Szegő equation becomes trivial. In the literature, those (s, s0, ψ, ψ0)-coordinates are called action-angle variables : the actions s, s0 are constant in time, and the angles ψ, ψ0 evolve linearly.

2For the sake of brevity, we assume here without further explanation that C (z) is invertible. See section 2.3.4 for a proof of this non-trivial fact.

18 1/2 Proposition 2.3.2. Suppose u0 ∈ H+ satisﬁes the hypothesis of theorem5 and corresponds 0 0 2N 2N to a set (s0, s0, ψ0, ψ0) ∈ R × T . If u0 is assumed to be the initial data for the evolution 0 0 problem (2.1), then for all t ∈ R, u = u(t) corresponds to a set (s(t), s (t), ψ(t), ψ (t)) satisfying ds ds0 j = 0, k = 0, dt dt dψ dψ0 j = s2, k = s02. dt j dt k 0 Proof. We already know that the sj’s and sk’s are conserved. Let us turn to ψj, for some j ∈ {1,...,N}. Using the notations of the previous proof, we have for all t ∈ R

iψj (t) Hu(t)(uj(t)) = sje uj(t). (2.7)

But recall the Lax pair (2.4) : it even implies that for any Borel function f : R+ → R,

d 2 2 2 f(H ) = [B , f(H )] = [−iT 2 , f(H )], dt u u u |u| u

i 2 2 2 because Bu = −iT|u|2 + 2 Hu, and Hu commutes to f(Hu). We are going to apply this identity 2 to u with f = 1{sj }. This will give us the evolution of uj, since uj = 1{sj }(Hu)u. We thus have

d d 2 2 2 2 u = f(H )u = [−iT 2 , f(H )]u + f(H )(−iΠ(|u| u)) = −iT 2 u . dt j dt u |u| u u |u| j We are ready to diﬀerentiate (2.7). The left hand side gives

d dt l.h.s. = [Bu,Hu](uj) + Hu(−iT|u|2 uj) i 3 i 2 = −iT|u|2 Hu(uj) + 2 Hu(uj) − Hu( 2 Hu(uj)) 2 = −iT|u|2 Hu(uj) + isj Hu(uj), whereas for the right hand side :

d ˙ iψj iψj ˙ dt r.h.s. = iψjsje uj + sje (−iT|u|2 uj) = iψjHu(uj) − iT|u|2 Hu(uj).

˙ 2 Since Hu(uj) 6= 0, we have ψj = sj . 0 Starting from (2.5) similarly leads to the law of the evolution of the ψk’s. 2 2 Remark 4. Since Hu − Ku is a rank-one operator, and because the inclusion Ran Ku ⊆ Ran Hu always holds true, we see that rk Ku ∈ {rk Hu, rk Hu − 1} whenever Hu has ﬁnite rank. The above results only deal with the case of equality, i.e. Ran Ku = Ran Hu, but in fact, when 0 Ran Ku = Ran Hu − 1, the inverse spectral formula remains valid, simply setting sN = 0 in the matrix C (z).

19 2.3.4 Complement 1 : C (z) is invertible We prove the following proposition, which was included in the statement of theorem5:

Proposition 2.3.3. Under the hypothesis of theorem5, the matrix C (z) is invertible for any z ∈ D. Proof. Set Q(z) := det C (z). The polynomial Q is of degree N exactly. Indeed, its dominant coeﬃcient is ! 0 0 1 N 0 0 i(ψ1+···+ψN ) (−1) s1 . . . sN e det 2 0 2 6= 0, sj − (sk) because of the formula for the Cauchy determinant : Q Q 1 i

Therefore Q has N zeroes in C, counted with multiplicities. Assume one of these zeroes, say z0, −1 belongs to D. Since, by the Cramer formulae for C (z) whenever Q(z) 6= 0,

−1 P (z) u(z) = hC (z) (1N )|1N i N = , C Q(z)

2 where P is a polynomial of degree at most N −1, and since u ∈ L+, it is necessary that P (z0) = 0. Therefore, after a ﬁnite number simpliﬁcations,

P˜(z) u(z) = , Q˜(z) where Q˜ is a polynomial of degree at most N − 1 with no zeroes in the closed unit disc, and P˜ is a polynomial of degree at most N − 2. We can therefore decompose

X αj u(z) = , (1 − p z)mj j j P with 0 < |pj| < 1 and j mj ≤ N − 1. Now we use the following elementary lemma. Lemma 2.3.4. If 1 v(z) = (1 − pz)m with m ≥ 1 and 0 < |p| < 1, then

1 Ran(H ) = span , k = 1, . . . , m . u (1 − pz)k

20 2 Let us prove the lemma. Given h ∈ L+, we can expand

m−1 m−1 X k m z h(z) = ck(z − p) + (z − p) g(z), k=0

2 2 1 with g ∈ L+ and |z| < 1. Taking the L trace on S , conjugating and multiplying by v, we infer

m−1 h (eix) X ei(m−1−k)x = c + e−ixg (eix) . (1 − peix)m k (1 − peix)m−k k=0

Consequently, m−1 X zm−k−1 Π(vh) = c , k (1 − pz)m−k k=0 and it is clear that one can ﬁt the value of each coeﬃcient ck by an appropriate choice of h.

Using the lemma, we conclude that X dim RanHu = mj ≤ N − 1, j which contradicts the assumption that Hu has rank N.

2.3.5 Complement 2 : Surjectivity of the spectral transform We conclude this section by giving a proof of the fact that the mapping u 7→ (s, s0, ψ, ψ0) is onto (under the hypothesis of theorem5). The proof of the surjectivity of the mapping

u 7→ (s, s0, ψ, ψ0) in the most general case is given in [7], [8], by showing that this map is open and closed for appropriate topologies. Here we give a diﬀerent, purely algebraic proof, based on a recent work in collaboration with A. Pushnitski.

First of all, the above calculations imply that, if u exists, its orthogonal projections uj and 0 2 2 uk onto the eigenspaces of Hu and Ku satisfy

N 2 0 2 X kujk ku k k = δ . (s2 − (s0 )2)(s2 − (s0 )2) k` j=1 j k j `

2 0 2 In view of the above formula (2.8) for the Cauchy determinants, this imply that kujk and kukk 0 can be expressed in terms of the numbers si, s` as

2 0 2 2 0 2 Y sj − (si) Y s − (s ) ku k2 = (s2 − (s0 )2) , ku0 k2 = (s2 − (s0 )2) ` k . j j j s2 − s2 k k k (s0 )2 − (s0 )2 i6=j j i `6=k ` k

21 0 0 0 Now, we ﬁx once and for all s1 > s1 > s2 > s2 > ··· > sN > sN > 0, and we set similarly

2 0 2 2 0 2 Y sj − (si) Y s − (s ) τ 2 := (s2 − (s0 )2) , κ2 := (s2 − (s0 )2) ` k , j j j s2 − s2 k k k (s0 )2 − (s0 )2 i6=j j i `6=k ` k so that we also have, for 1 ≤ k, l ≤ N,

N 2 2 X τj κk = δk`. (2.9) (s2 − (s0 )2)(s2 − (s0 )2) j=1 j k j `

Then we consider the following two N–dimensional Hermitian spaces. We denote by E and by 0 N E the space C equipped respectively with the inner products

N N X 2 X 2 (˜z|z)E := τj z˜jzj, (˜z|z)E0 := κkz˜kzk. j=1 k=1

Consider the linear operator

N 2 X zkκ Ω: E0 → E, (Ωz) := k . j s2 − (s0 )2 k=1 j k

We claim that Ω is a unitary operator. Indeed, for w ∈ E, z ∈ E0,

N 2 2 X zkwjτj κk ∗ (Ωz|w) = = (z|Ω w) 0 E s2 − (s0 )2 E j,k=1 j k with N 2 X wjτj (Ω∗w) := , k s2 − (s0 )2 j=1 j k and (2.9) precisely means that Ω∗Ω = I. Also notice that

Ω(1N ) = 1N . (2.10)

Indeed, this is equivalent to

N X κ2 k = 1, j = 1,...,N, (2.11) s2 − (s0 )2 k=1 j k a property which can be established as follows : consider the rational function

N 0 2 Y x − (sj) R(x) = . x − s2 j=1 j

22 Since R(x) tends to 1 at inﬁnity, we can expand

N X cj R(x) = 1 + , x − s2 j=1 j

2 and the computation of the residue cj yields cj = τj . Similarly,

N N 1 Y x − s2 X κ2 = k = 1 − k . R(x) x − (s0 )2 x − (s0 )2 k=1 k k=1 k

From this we infer that

N ! N ! X τ 2 X κ2 1 + i 1 − k = 1. x − s2 x − (s0 )2 i=1 i k=1 k

2 2 Multiplying both sides of the above identity by x − sj and letting x tend to sj , we obtain (2.11). 0 0 0 2N Then, given (ψ1, ψ1, ψ2, ψ2, . . . , ψN , ψN ) ∈ T , we consider the antilinear operators

H : E → E,K0 : E0 → E0, deﬁned by 0 iψj 0 0 iψ (Hz)j = sje zj, (K z)k = ske k zk. 0 0 0 Notice that H, K satisfy (Hz˜|z)E = (Hz|z˜)E, and (K z˜|z)E0 = (K z|z˜)E0 . In particular they are R–linear operators which are symmetric with respect to the real scalar products deﬁned by

(˜z, z)E = Re(˜z|z)E, (˜z, z)E0 = Re(˜z|z)E0 .

2 0 2 0 Moreover, H , (K ) are positive selfadjoint C–linear operators on E,E respectively. We set

K := ΩK0Ω∗ : E → E.

The following lemma establishes a crucial identity between the operators K2 and H2 on E.

2 2 Lemma 2.3.5. K = H − ( · |1N )E1N .

Proof. We compute

N 2 0 2 X κ (s ) zk (Ω(K0)2z) = k k j s2 − (s0 )2 k=1 j k N 2 2 N X κksj zk X = − κ2z s2 − (s0 )2 k k k=1 j k k=1 2 = (H Ωz)j − (z|1N )E0 1N .

23 Because of formula (2.10), this can be written as 0 2 2 Ω(K ) = (H − ( . |1N )E1N )Ω, or 2 2 K = H − ( . |1N )E1N , by setting K := ΩK0Ω∗ : E → E as above. We now set Σ := KH−1 : E → E. Since 2 2 2 2 2 2 kKzkE = (K z|z)E = (H z|z)E − |(z|1N )E| ≤ (H z|z)E = kHzkE, we have ∀z ∈ E, kΣzkE ≤ kzkE. In fact, this contraction map Σ enjoys the following asymptotic stability property. Lemma 2.3.6. n ∀z ∈ E, kΣ zkE −→ 0. n→+∞ Proof. Since E is ﬁnite dimensional and since Σ is a contraction, the theory of the Jordan decomposition of matrices shows that it is enough to prove that Σ has no eigenvalue on the unit 1 circle. Let ω ∈ S , and consider F := ker(Σ − ωI). We claim that F = ker(Σ∗ − ωI). Indeed, since Σ is a contraction, the Hermitian form B(z, z˜) := ((I − Σ∗Σ)z|z˜) is non negative, hence ∗ kΣzkE = kzkE ⇐⇒ Σ Σz = z, as a consequence of the Cauchy–Schwarz inequality for B. This implies F ⊂ ker(Σ∗ − ωI). The reverse inclusion follows from a similar argument applied to the contraction Σ∗. In order to study the space F , we recall that by deﬁnition K = ΣH. Hence, by symmetry of H and K for the real scalar product on E, K = HΣ∗. Therefore Lemma 2.3.5 can be reformulated as 2 ∗ 2 ΣH Σ = H − ( . |1)E1. (2.12) Given z ∈ F , we infer 2 2 2 kHzkE = kHzkE − |(z|1)E| , hence F is orthogonal to 1N . Coming back to (2.12), we conclude that, if z ∈ F , ωΣH2z = H2z, 2 2 2 hence H z ∈ F . Therefore F is a stable subspace for H , which is the diagonal matrice of sj , j = 1,...,N. Hence F has to be the direct sum of one dimensional eigenspaces of H2. On the other hand, none of these lines is orthogonal to 1N . The only possibility is therefore F = {0}.

24 1 2 2 At this stage we are in position to construct u ∈ H+ and a linear isometry U : E → L+ such that ∗ 0 ∗ ∗ UHU = Hu,UΩK Ω U = Ku. (2.13) Notice that property (2.13) implies that (s, s0, ψ, ψ0) corresponds to u via Theorem5. First we give another reformulation of Lemma 2.3.5. Deﬁne q ∈ E by

eiψj qj = , j = 1,...,N, sj

∗ 2 2 so that Hq = 1N . Plugging K = HΣ = ΣH in K = H − ( · |1N )E1N , we obtain HΣ∗ΣH = H2 − ( · |Hq)Hq, hence, recalling that H is invertible,

Σ∗Σ = I − ( · |q)q . (2.14)

Identity (2.14) combined with Lemma 2.3.6 has an important consequence. Indeed, iterating (2.14) yields, for every z ∈ E,

n−1 X (Σ∗)nΣnz = z − (z|(Σ∗)kq)(Σ∗)kq. k=0 Taking the scalar product with z in E, we obtain

n−1 n 2 2 X ∗ k 2 kΣ zkE = kzkE − |(z|(Σ ) q)E| . k=0 Passing to the limit as n → ∞, and using Lemma 2.3.6, we ﬁnally conclude ∞ 2 X ∗ k 2 ∀z ∈ E, kzkE = |(z|(Σ ) q)E| . (2.15) k=0

2 It allows us to deﬁne U : E → L+ by ∞ X ∗ k ikx Uz = (z|(Σ ) q)E e . k=0

2 In view of (2.15), the operator U is an isometry, and, for every h ∈ L+, ∞ X U ∗h = hˆ(k)(Σ∗)kq . k=0

Consider ∞ X ∗ k ikx u := U(1N ) = (1N |(Σ ) q)E e . k=0

25 Since Σ∗ has eigenvalues only in the open unit disc — see the proof of Lemma 2.3.5—, the Jordan decomposition implies that ∗ k k |(z|(Σ ) q)E| ≤ Cβ

1/2 3 ∗ for some β < 1. Hence u ∈ H+ , so we may study Hu and Ku = S Hu = HuS. Given z ∈ E, consider

2 Consequently, for every h ∈ L+, ! ∞ ∞ ∗ X ˆ ∗ k+` X ˆ UHU\h(k) = 1N h(`)(Σ ) q = uˆ(k + `)h(`) . `=0 E `=0 We conclude ∗ UHU = Hu . Furthermore, since

∗ k ∗ k+1 ∗ U[Σz(k) = Σz|(Σ ) q = (z|(Σ ) q)E = S\Uz(k), E we also have ∗ ∗ ∗ ∗ ∗ UKU = UΣHU = S UHU = S Hu = Ku, or equivalently 0 ∗ ∗ UΩK Ω U = Ku.

3 In fact u is even an analytic function on T

26 Lecture 3

Long time transition to high frequencies

In our search of wave turbulence for the Szegő equation, the formula given by theorem5 does not help directly. On the contrary, it shows, together with proposition 2.3.2, that initial data belonging to the ﬁnite-dimensional manifold {rk Hu = N} ∩ {rk Ku = N} (or {rk Hu = N} ∩ {rk Ku = N − 1}) give rise to a motion which is quasi-periodic in time : there exists D ≥ 1, a D ∞ 1 D function F : T → C+ (T) , and a ω = (ω1, . . . , ωD) ∈ R such that

u(t) = F (ω1t, ω2t, . . . , ωDt), ∀t ∈ R.

Such an orbit t 7→ u(t) remains bounded in every Hs. However, this formula will allow us to infer the existence of transition to high frequencies for most solutions of the cubic Szegő equation, as we will now explain.

3.1 A crucial example

Fix a small ε > 0, and consider the following set of parameters :

0 0 s1 = 1 + ε, s1 = 1, s2 = 1 − ε, s2 = 0, 0 ψ1 = 0, ψ1 = 0, ψ2 = π.

Through the formula of lecture2, this corresponds to a matrix  1 + ε − z 1  2  (1 + ε) − 1 1 + ε  Cε(z) = −(1 − ε) − z 1  , − (1 − ε)2 − 1 1 − ε and by theorem5, the function

2 −1 2z(1 − ε ) + 3ε uε(z) := hCε(z) (1)|1i 2 = C 2 − εz

1 ∞ ∞ 2 where C+ (T) := C (T) ∩ L+.

27 2 2 2 2 is such that rk Huε = 2, rk Kuε = 1. Besides, Huε has s1, s2 as positive eigenvalues, and Kuε has 02 2 1 s1 only. We notice that the only pole of uε(z) is z = ε , far away from the unit circle S . Now modify the parameters in the following way :

0 0 s1 = 1 + ε, s1 = 1, s2 = 1 − ε, s2 = 0, 0 ψ1 = 0, ψ1 = 0, ψ2 = 0.

Only ψ2 has changed, but now the matrix  1 + ε − z 1  2 ˜ (1 + ε) − 1 1 + ε Cε(z) =  (1 − ε) − z 1  , (1 − ε)2 − 1 1 − ε and the corresponding function is

2 2 ˜ −1 2 + ε − 2z(1 − ε ) vε(z) := hCε(z) (1)|1i 2 = , C 2 − (2 − ε2)z

2 1 with a pole for z = 2−ε2 , this time dramatically close to S . A simple computation shows that because of this pole, 12s−1 kvεkHs ' 1. + ε

So if one considers the solution t 7→ u(t) of the cubic Szegő equation (2.1) with uε as an 2 2 initial data, then by proposition 2.3.2, we have ψ1(t) = t(1+ε) and ψ2(t) = π +t(1−ε) . There is a time tε such that ψ1(tε) = ψ2(tε) : π t = . ε 4ε 1 2s−1 2s−1 1 ku(t )k s ' ( ) ' (t ) s > Then the previous computation shows that ε H+ ε ε whenever 2 . Thus we found a solution with growing Sobolev norms, but the growth is polynomial in time, and the trajectories are bounded (even by great bounds) in any Sobolev space.

An improvement. First of all, it is possible to extend the previous to any power of tε. Indeed, for N ∈ N, pick real numbers ξ1 > η1 > ξ2 > η2 > ··· > ξN−1 > ηN−1 > ξN > 0. If we take as actions 1 + εξ1 > 1 + εη1 > 1 + εξ2 > 1 + εη2 > . . . , and 0 as angles, we ﬁnd a matrix ˜ (1 + εξj) − (1 + εηk)z Cε(z) = 2 2 . (1 + εξj) − (1 + εηk) 1≤j,k≤N

The corresponding u given by theorem5 then satisﬁes, for a generic choice of (ξ, η),

1 (N−1)(2s−1) kuk s ' ' (t ) . H ε(N−1)(2s−1) ε

28 As before, it is also possible to ﬁnd, changing the angles along the Szegő trajectory, a matrix Cε(z) such that the associated function is a rational fraction with no pole close to the unit disc. In the next section, we brieﬂy explain how we can deduce for those growing but bounded solutions a weak turbulence theorem. In other words, we explain how to pass from families of solutions to the behaviour of one precise solution.

3.2 Passing from families of solutions to turbulent solutions

The main result of [8] is the following :

Theorem 6 (Wave turbulence for the cubic Szegő equation). There exists a dense Gδ-subset of ∞ C+ (T) denoted by G, such that for all u solution to the Szegő equation (2.1) with u(0) ∈ G, for 1 all s > 2 and all M ≥ 1, we have

ku(t)kHs lim sup M = ∞, t→∞ |t| and on the other hand lim inf ku(t)kHs ≤ ku(0)kHs . t→∞ Remark 5. From the Lax pair identity (2.4), it is possible to show that any solution of (2.1) ∞ ∞ starting from an initial data u0 ∈ C+ remains uniformly bounded in L , and it implies that the s Cs|t| 1 H -norm of a solution grows at most like e for s > 2 and a constant Cs > 0 — see appendix A. Hence the theorem above proves that this exponential estimate is almost optimal. Using a strategy of Hani in [10], theorem6 is a consequence of the following proposition : ∞ 1 n Proposition 3.2.1. For any u0 ∈ C+ (S ), and M ≥ 1, there exists a sequence {u0 } converging ∞ n to u0 in C+ , and sequences of times tn and tn → ∞ such that u , the solution to the cubic Szegő n n equation with u (0) = u0 , satisﬁes n ku (tn)kHs M −→ +∞, |tn| and n ∞ 1 u (tn) −→ u0 in C+ (S ), as n → +∞.

Let us ﬁrst assume that proposition 3.2.1 is proved, and we show how it implies the turbulence theorem.

∞ Proof of theorem6. Let p ≥ 1 be an integer. We deﬁne Op to be the set of u0 ∈ C+ such that there exists t, t > p such that the solution u of the Szegő equation with u(0) = u0 satisﬁes

p ku(t)k 1 1 > p(t) , H 2 + p 1 ku(t) − u k p < . 0 H p

29 ∞ 1 It is clear that Op is an open set, and by the proposition 3.2.1, Op is dense in C+ (S ). By the Baire category theorem, ∞ \ G := Op p=1 is a dense Gδ set, and elements of G satisfy the conclusions of theorem6.

It remains to prove proposition 3.2.1. The complete proof is given in [8] and is quite technical. Hereafter, we only give a quick outline of it. First of all, notice that it is enough to approximate ∞ 1 initial data u0 belonging to a dense subset of C+ (S ), so we only consider u0 such that for some

N ∈ N, rk Hu0 = rk Ku0 = N, with simple singular values which we denote by

0 0 0 s1 > s1 > s2 > s2 > ··· > sN > sN .

The idea is to approximate u0 by adding new singular values, in the spirit of the examples of the previous section. We ﬁx an integer L ≥ 1, we pick 2L − 1 real numbers ξ1 > η1 > ··· > ξL−1 > ε,δ ηL−1 > ξL > 0, and for ε, δ > 0, we set u0 , with singular values given by

0 0 0 s1 > s1 > s2 > s2 > ··· > sN > sN > δ(1 + εξ1) > δ(1 + εη1) > ··· > δ(1 + εξL).

ε,δ Then it can be shown that there is time tε,δ such that u the solution of the Szegő equation ε,δ starting from u0 satisﬁes ε,δ δ ku (t )k s ' , ε,δ H ε(L−1)(2s−1) −M which is bigger than ε for some good choice of L. Finding appropriate δn and n converging n εn,δn to 0 ﬁnally yields a sequence of initial data u0 := u0 as in the statement of proposition 3.2.1.

30 Appendix A

The L∞ estimate and its consequences

In this section, we provide a complete proof of the ﬁrst part of theorem3, and of the following result : s Theorem 7 ([6]). Assume u0 ∈ H+ for some s > 1. Then the corresponding solution u of the cubic Szegő equation satisﬁes sup ku(t)kL∞ < +∞ . t∈R The proof of this theorem relies on the following proposition. 1 2 Proposition A.0.2. Given u ∈ H+, we denote by {sj(u)}j≥1 the sequence of singular values of Hu, repeated according to multiplicity. The following double inequality holds :

1 ∞ ∞ ∞ ∞ ! 1 X X X X 2 |uˆ(n)| ≤ s (u) ≤ |uˆ(n + `)|2 2 j n=0 j=1 n=0 `=0 Furthermore, the right hand side is controlled by

CskukHs for every s > 1. Remark 6. This proposition can be interpreted as a double inequality for the trace norm of the Hankel operator Hu. A more complete characterization of functions u such that the trace norm of Hu is ﬁnite is given in Peller [14]. Here we provide an elementary proof. Assuming the proposition, let us show the theorem. From the second equality in the proposition, we have ∞ X sj(u0) < +∞. j=1

Since each sj(u) is a conservation law, we have, for every t ∈ R, ∞ ∞ X X sj(u(t)) = sj(u0). j=1 j=1

31 Finally, by the ﬁrst inequality in the proposition,

∞ ∞ X X X sup |uˆ(t, n)| ≤ 2 sup sj(u(t)) = 2 sj(u0). t∈R n≥0 t∈R j=1 j=1

The proof is completed by the elementary observation that X kukL∞ ≤ |uˆ(n)|. n≥0

We now pass to the proof of the proposition.

2 Proof of proposition A.0.2. We denote by {ej}j≥1 an orthonormal basis of Ran(Hu) = Ran(Hu) 2 2 2 such that Huej = sj u. Such a basis exists because Hu is a compact selfadjoint operator. Notice that 2 2 2 kHu(ej)k = (Hu(ej)|ej) = sj . inx We set, for every n ≥ 0, εn(x) = e . Let us prove the ﬁrst inequality. Observe that

uˆ(2n) = (Hu(εn)|εn), uˆ(2n + 1) = Ku(εn|εn).

∞ X X |uˆ(2n)| ≤ |(Hu(ej)|εn)(ej|εn)| n=0 n,j 1 1 ∞ ! 2 ∞ ! 2 X X 2 X 2 ≤ |(Hu(ej)|εn)| |(ej|εn)| j n=0 n=0 X X = kHu(ej)kkejk = sj. j j

Arguing similarly with Ku, we obtain

∞ X X 0 |uˆ(2n + 1)| ≤ sk. n=0 k

Summing up, we have proved

∞ X X X 0 X |uˆ(n)| ≤ sj + sk ≤ 2 sj. n=0 j k j

32 We now pass to the second inequality. Notice that 2 2 (Hu(ej),Hu(ej0 )) = (Hu(ej0 ), ej) = sj δjj0 .

In other words, the sequence {Hu(ej)/sj} is orthonormal. We then deﬁne the following antilinear 2 operator on L+, X Hu(ej) Ω (h) = (e , h) . u j s j j

Notice that, due to the orthonormality of both systems {ej} and {Hu(ej)/sj}, kΩu(h)k ≤ khk. Similarly, we deﬁne X (Hu(ej), h) tΩ (h) = e , u s j j j so that 0 2 0 t 0 ∀h, h ∈ L+, (Ωu(h)|h ) = ( Ωu(h )|h). We next observe that ∞ X sj = (Hu(ej)|Ωu(ej)) = (Hu(ej)|εn)(εn|Ωu(ej)) n=0 ∞ ∞ X X = uˆ(n + `)(ε`|ej)(εn|Ωu(ej)) n=0 `=0

≤ CskukHs ,

33 and proposition A.0.2 is proved.

s Corollary A.0.3. For s > 1 and u0 ∈ H+, the corresponding solution t 7→ u(t) of the Szegő equation satisﬁes C0 |t| ku(t)kHs ≤ Cse s , ∀t ∈ R, 0 where Cs, Cs are positive constants which only depend on s and ku0kHs . d s 2 Proof. Proceeding as in the proof of proposition 2.1.1, we compute dt kD u(t)kL2 , but this time, the boundedness of the L∞ norm allows to write

d s 2 2 kD u(t)k 2 ≤ Ckuk s , dt L H

s for an appropriate constant C depending on the norm of u0 in H+. A Gronwall inequality then completes the proof of the corollary.

34 Contents

1 Introduction to long time estimates2 1.1 What is a Hamiltonian PDE?...... 2 1.1.1 Hamiltonian systems in ﬁnite dimension...... 2 1.1.2 Extension to inﬁnite dimensional systems...... 3 1.2 Some examples...... 4 1.2.1 The nonlinear Schrödinger equation...... 4 1.2.2 Changing the dispersion relation...... 6 1.2.3 The linear case...... 7 1.3 Towards the cubic Szegő equation...... 8

2 The cubic Szegő equation on the circle and its special structure 10 2.1 The setting...... 10 2.2 Hankel operators and the Lax pair structure...... 12 2.2.1 Hankel operators on sequences...... 12 2.2.2 Back to the Szegő equation...... 13 2.3 The role of singular values...... 15 2.3.1 The interlacement property...... 15 2.3.2 The inverse spectral formula...... 16 2.3.3 The evolution in new coordinates...... 18 2.3.4 Complement 1 : C (z) is invertible...... 20 2.3.5 Complement 2 : Surjectivity of the spectral transform...... 21

3 Long time transition to high frequencies 27 3.1 A crucial example...... 27 3.2 Passing from families of solutions to turbulent solutions...... 29

A The L∞ estimate and its consequences 31

35 Bibliography

[1] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geometric and Functional Analysis, 3(3):209–262, 1993.

[2] J. Bourgain. On the growth in time of higher Sobolev norms of smooth solutions of Hamil- tonian PDE. Internat. Math. Res. Notices, (6):277–304, 1996.

[3] J. Bourgain. Problems in Hamiltonian PDE’s. Geom. Funct. Anal., (Special Volume, Part I):32–56, 2000. GAFA 2000 (Tel Aviv, 1999).

[4] H. Brezis and T. Gallouët. Nonlinear Schrödinger evolution equations. Nonlinear Anal., 4(4):677–681, 1980.

[5] J. Colliander, M. Keel, G. Staﬃlani, H. Takaoka, and T. Tao. Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math., 181(1):39– 113, 2010.

[6] P. Gérard and S. Grellier. The cubic Szegő equation. Ann. Sci. Éc. Norm. Supér. (4), 43(5):761–810, 2010.

[7] P. Gérard and S. Grellier. Invariant tori for the cubic Szegö equation. Inventiones mathe- maticae, 187(3):707–754, 2012.

[8] P. Gérard and S. Grellier. The cubic Szegő equation and Hankel operators. arXiv preprint arXiv:1508.06814. 2015.

[9] M. Guardia and V. Kaloshin. Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. Journal of the European Mathematical Society, 17(1):71–149, 2015.

[10] Z. Hani. Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations. Arch. Ration. Mech. Anal., 211(3):929–964, 2014.

[11] Z. Hani, B. Pausader, N. Tzvetkov, and N. Visciglia. Almost sure global well-posedness for fractional cubic Schrödinger equation on torus. Forum Math. Pi3, 2015.

[12] T. Kappeler, B. Schaad, and P. Topalov. Scattering-like phenomena of the periodic defocusing NLS equation. arXiv preprint arXiv:1505.07394, 2015.

36 [13] P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.

[14] V. Peller. Hankel operators and their applications. Springer Science & Business Media, 2012.

[15] G. Staﬃlani. On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations. Duke Math. J., 86(1):109–142, 1997.

[16] J. Thirouin. On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle. to appear in Annales de l’IHP, 2016.

[17] V. E. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one- dimensional self-modulation of waves in nonlinear media. Ž. Èksper. Teoret. Fiz., 61(1):118– 134, 1971.