arXiv subjects
Publications and source records attributed to Patrick Gerard.
We give a short proof of the main result of our previous paper [2]: every Schmidt subspace of a Hankel operator is the image of a model space by an isometric multiplier. This class of subspaces is closely related to nearly $S^*$-invariant subspaces, and our proof uses Hitt's theorem on the structure of such subspaces. We also give a formula for the action of a Hankel operator on its Schmidt subspace.
For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weighted model spaces.
We characterize all bounded Hankel operators $Γ$ such that $Γ^*Γ$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.
This paper explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szegö equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a G $δ$ dense set of irregular functions do coexist on the same torus. On the other hand, we establish some uniform analytic regularity for tori corresponding to rapidly decreasing actions which satisfy some specific property ruling out the phenomenon of small gaps.
This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic Szeg{ö} equation on the circle ${\mathbb S} ^1$,$$ i\partial \_t u=Π(\vert u\vert ^2u)\ .$$Here $Π$ denotes the orthogonal projector from $L^2({\mathbb S} ^1)$ onto the subspace $L^2\_+({\mathbb S} ^1)$ of functions with nonnegative Fourier modes.We construct a nonlinear Fourier transformation on $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$ allowing to describe explicitly the solutions of this equationwith data in $H^{1/2}({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$. This explicit description implies almost-periodicity of every solution in $H^{\frac 12}\_+$. Furthermore, it allows to display the following turbulence phenomenon. For a dense $G\_δ$ subset of initial data in $C^\infty ({\mathbb S} ^1)\cap L^2\_+({\mathbb S} ^1)$, the solutions tend to infinity in $H^s$ for every $s\textgreater{}\frac 12$ with super--polynomial growth on some sequence of times, while they go back to their initial data on another sequence of times tending to infinity. This transformation is defined by solving a general inverse spectral problem involving singular values of a Hilbert--Schmidt Hankel operator and of its shifted Hankel operator.
For a sequence $\{α_n\}_{n=0}^\infty$, we consider the Hankel operator $Γ_α$, realised as the infinite matrix in $\ell^2$ with the entries $α_{n+m}$. We consider the subclass of such Hankel operators defined by the "double positivity" condition $Γ_α\geq0$, $Γ_{S^*α}\geq0$; here $S^*α$ is the shifted sequence $\{α_{n+1}\}_{n=0}^\infty$. We prove that in this class, the sequence $α$ is uniquely determined by the spectral shift function $ξ_α$ for the pair $Γ_α^2$, $Γ_{S^*α}^2$. We also describe the class of all functions $ξ_α$ arising in this way and prove that the map $α\mapstoξ_α$ is a homeomorphism in appropriate topologies.
The goal of this paper is to construct a nonlinear Fourier transformation on the space of symbols of compact Hankel operators on the circle. This transformation allows to solve a general inverse spectral problem involving singular values of a compact Hankel operator, with arbitrary multiplicities. The formulation of this result requires the introduction of the pair made with a Hankel operator and its shifted Hankel operator. As an application, we prove that the space of symbols of compact Hankel operators on the circle admits a singular foliation made of tori of finite or infinite dimensions, on which the flow of the cubic Szegö equation acts. In particular, we infer that arbitrary solutions of the cubic Szegö equation on the circle with finite momentum are almost periodic with values in H^{1/2}(S ^1).
This paper is concerned with the cubic Szegő equation $$ i\partial_t u=Π(|u|^2 u), $$ defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb T$, where $Π: L^2(\mathbb T)\rightarrow L^2_+(\mathbb T)$ is the Szegő projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time $t\in (-\infty,\infty)$. In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the $\ell^1$ norm of Fourier transforms (the Wiener algebra).
Given two arbitrary sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ of real numbers satisfying $$|λ_1|>|μ_1|>|λ_2|>|μ_2|>...>| λ_j| >| μ_j| \to 0\ ,$$ we prove that there exists a unique sequence $c=(c_n)_{n\in\Z_+}$, real valued, such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ of symbols $c=(c_{n})_{n\ge 0}$ and $\tilde c=(c_{n+1})_{n\ge 0}$ respectively, are selfadjoint compact operators on $\ell^2(\Z_+)$ and have the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$ respectively as non zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of $Γ_c$ and of $Γ_{\tilde c}$ in terms of the sequences $(λ_j)_{j\ge 1}$ and $(μ_j)_{j\ge 1}$. More generally, given two arbitrary sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ of positive numbers satisfying $$ρ_1>σ_1>ρ_2>σ_2>...> ρ_j> σ_j \to 0\ ,$$ we describe the set of sequences $c=(c_n)_{n\in\Z_+}$ of complex numbers such that the Hankel operators $Γ_c$ and $Γ_{\tilde c}$ are compact on $\ell ^2(\Z_+)$ and have sequences $(ρ_j)_{j\ge 1}$ and $(σ_j)_{j\ge 1}$ respectively as non zero singular values.
We consider the following degenerate half wave equation on the one dimensional torus $$\quad i\partial_t u-|D|u=|u|^2u, \; u(0,\cdot)=u_0. $$ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system-- the cubic Szegö equation. As a consequence, we prove an instability result for large $H^s$ norms of solutions of this wave equation.
In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation by using Zakharov-Shabat's inverse scattering method.