Generic colourful tori and inverse spectral transform for Hankel operators

TL;DR

This paper studies the regularity of an inverse spectral transform for Hankel operators, revealing how smooth and irregular functions coexist on tori and establishing conditions for uniform regularity related to wave turbulence phenomena.

Contribution

It introduces a detailed analysis of the regularity properties of spectral tori for Hankel operators, connecting them to wave turbulence and small gap phenomena.

Findings

Generic smooth functions coexist with irregular functions on the same torus.

Rapidly decreasing actions lead to uniform analytic regularity.

Small gaps between actions are linked to wave turbulence phenomena.

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-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szeg\"o 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 $\delta$ 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.