Multiple singular values of Hankel operators

TL;DR

This paper develops a nonlinear Fourier transform for compact Hankel operators on the circle, enabling the solution of inverse spectral problems with arbitrary singular value multiplicities and revealing a rich foliation structure related to the cubic Szeg"o equation.

Contribution

It introduces a novel nonlinear Fourier transform for Hankel operators, solving inverse spectral problems with arbitrary multiplicities and analyzing the dynamics of the cubic Szeg"o equation.

Findings

Constructed a nonlinear Fourier transform for Hankel operators.

Solved inverse spectral problems with arbitrary singular value multiplicities.

Identified a foliation of the symbol space into tori where the Szeg"o flow acts.

Abstract

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 Szeg\"o equation acts. In particular, we infer that arbitrary solutions of the cubic Szeg\"o equation on the circle with finite momentum are almost periodic with values in H^{1/2}(S ^1).