Generic non-selfadjoint Zakharov-Shabat operators

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