On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus

TL;DR

This paper analyzes the spectral properties of the Lax operator associated with the Benjamin-Ono equation on the torus, focusing on complex potentials with small imaginary parts and their impact on the spectral data and moment map.

Contribution

It provides new insights into the spectral analysis of the Lax operator for complex potentials in low regularity Sobolev spaces and establishes analytic properties of the moment map.

Findings

Spectral analysis of Lax operator for complex potentials in H^{-s}

Analytic properties of the moment map in spectral data

Extension of spectral theory to low regularity potentials

Abstract

We investigate the spectrum of the Lax operator $L_u$ of the Benjamin-Ono equation on the torus for complex valued potentials $u$ in the Sobolev space $H^{-s}(\mathbb{T},\mathbb{C})$, $0 \le s < 1/2$, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of $L_u$.