Simplicity and Finiteness of Discrete Spectrum of the Benjamin-Ono Scattering Operator

TL;DR

This paper analyzes the spectral properties of the Lax pair operator for the Benjamin-Ono equation, establishing the simplicity and finiteness of its discrete spectrum, which are essential for inverse scattering methods.

Contribution

It introduces a new identity linking eigenvector norms to scattering potentials and extends Birman-Schwinger bounds to prove spectrum finiteness.

Findings

Discrete spectrum is finite and simple.

New identity connects eigenvector norms to scattering potential.

Finiteness proof extends Birman-Schwinger bound techniques.

Abstract

A spectral analysis is done on the $L$ operator of the Lax pair for the Benjamin-Ono equation. Simplicity and finiteness of the discrete spectrum are established as are needed for the Fokas and Ablowitz inverse scattering transform scheme. A crucial step in the simplicity proof is the discovery of a new identity connecting the $L^2$ norm of the eigenvector to its inner product with the scattering potential. The proof for finiteness is an extension of the ideas involved in the Birman-Schwinger bound for Schr\"odinger operators.