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

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 $L_u$.

arXiv commentIn this revised version, we have corrected various misprints, updated and added some references, slightly simplified the exposition in Section 6, and made minor corrections to the estimate (50), the proof of Proposition 6, and the statement and proof of Lemma 18