Abstract
For the AKNS operator on L 2([0,1],C 2) it is well known that the data of two spectra uniquely determine the corresponding potential ϕ a.e. on [0,1] (Borg's type Theorem). We prove that, in the case where ϕ is a-priori known on [a,1], then only a part (depending on a) of two spectra determine ϕ 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 functions. An important ingredient in our strategy is the link between the rate of growth of an entire function and the distribution of its zeros.
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del Rio, R., Grébert, B. Inverse Spectral Results for AKNS Systems with Partial Information on the Potentials. Mathematical Physics, Analysis and Geometry 4, 229–244 (2001). https://doi.org/10.1023/A:1012981630059
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DOI: https://doi.org/10.1023/A:1012981630059