Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials

Abstract

We consider the AKNS (Ablowitz-Kaup-Newell-Segur) operator on the unitinterval with potentials belonging to Sobolev spaces in the framework ofinverse spectral theory. Precise sets of eigenvalues are given in orderthat they, together with the knowledge of the potentials onthe side $(a,1)$ and partial informations on the potential on $(a-\varepsilon,a)$ for some arbitrary small$\varepsilon>0$,determine thepotentials entirely on $(0,1)$. Naturally, the smaller is $a$ andthe more partial informations are known, the less is the number of the neededeigenvalues.