## Tunisian Journal of Mathematics

### Generic colourful tori and inverse spectral transform for Hankel operators

#### Abstract

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 $G δ$ 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.

#### Article information

Source
Tunisian J. Math., Volume 1, Number 3 (2019), 347-372.

Dates
Accepted: 11 May 2018
First available in Project Euclid: 15 December 2018

https://projecteuclid.org/euclid.tunis/1544842819

Digital Object Identifier
doi:10.2140/tunis.2019.1.347

Mathematical Reviews number (MathSciNet)
MR3907744

Zentralblatt MATH identifier
07027459

#### Citation

Gérard, Patrick; Grellier, Sandrine. Generic colourful tori and inverse spectral transform for Hankel operators. Tunisian J. Math. 1 (2019), no. 3, 347--372. doi:10.2140/tunis.2019.1.347. https://projecteuclid.org/euclid.tunis/1544842819

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