Tunisian Journal of Mathematics

Generic colourful tori and inverse spectral transform for Hankel operators

Patrick Gérard and Sandrine Grellier

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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

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

Received: 5 December 2017
Accepted: 11 May 2018
First available in Project Euclid: 15 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 37K15: Integration of completely integrable systems by inverse spectral and scattering methods 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Cubic Szegő equation action-angle variables integrable systems Hankel operators spectral analysis


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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