Vol. 1, No. 3, 2019

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Generic colourful tori and inverse spectral transform for Hankel operators

Patrick Gérard and Sandrine Grellier

Vol. 1 (2019), No. 3, 347–372

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-angle variables for an integrable infinite dimensional Hamiltonian system: the cubic Szegő 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 Gδ 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.

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Cubic Szegő equation, action-angle variables, integrable systems, Hankel operators, spectral analysis
Mathematical Subject Classification 2010
Primary: 35B65
Secondary: 37K15, 47B35
Received: 5 December 2017
Accepted: 11 May 2018
Published: 8 August 2018
Patrick Gérard
Laboratoire de Mathematiques D’Orsay
Université Paris Sud XI
Sandrine Grellier
Universite d’Orleans
Institut Denis Poisson
Bâtiment de Mathematiques