Generic non-selfadjoint Zakharov-Shabat operators

TL;DR

This paper develops analytical tools for non-selfadjoint operators, specifically Zakharov-Shabat operators in the focusing NLS, showing that potentials with simple small eigenvalues form a dense, path-connected set.

Contribution

It introduces new methods to analyze non-selfadjoint operators and proves the density and path-connectedness of potentials with simple small eigenvalues in Sobolev spaces.

Findings

Potentials with simple small eigenvalues are dense in Sobolev spaces.

The set of such potentials is path connected.

Tools are developed for studying spectra of non-selfadjoint operators.

Abstract

In this paper we develop tools to study families of non-selfadjoint operators $L(\varphi), \varphi \in P$, characterized by the property that the spectrum of $L(\varphi)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(\varphi)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $\varphi $ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(\varphi)$ are simple, is path connected and dense.