Generic non-selfadjoint Zakharov-Shabat operators

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.