R. del Rio, B. Grebert
Inverse spectral results for AKNS systems with partial information on the potentials
(40K, LaTex)
ABSTRACT. For the AKNS operator on $L^2 ([0,1], \CE^2) $ it is well known that
the data of two spectra uniquely determine the corresponding potential
$\varphi$ a.e. on $ [0,1] $ (Borg's type problem).We prove that, in the
case where $\varphi$ is a priori known on $[a,1]$,then only a part
(depending on $a$) of two spectra determine $\varphi$ on $[0,1]$. Our
results include generalizations for Dirac Systems of classical results
obtained by Hochstadt and Lieberman for the Sturm Liouville case, where
they showed that half of the potential and one spectrum determine all
the potential function. An important ingredient in our strategy is the
link between the rate of growth of an entire function and the
distribution of its zeros.