The Inverse Problem Of The Strum-Liouville Operator And The AKNS Operator

Differential operator theory mainly concerns two aspects. One is the eigenvalues, eigen-functions and expanding any function in terms of series (or integration) by eigenvalues and eigenfunctions. The other is the existence, uniqueness and reconstruction of a differential op-erator. The former is called the spectra analysis of a differential operator and the latter is the inverse spectra problem. The differential operator theory has many applications in the field of mathematical physics, mechanics and so on. In this paper, we mainly consider the inverse problem for both of the Sturm-Liouville operator and the AKNS operator who have important theory and application values.As we all know, Borg has proved that a Sturm-Liouville operator or an AKNS operator (a.e. the potential and the boundary conditions) can be determined by two full spectral. Besides, there are counter-examples imply that when some eigenvalues are missed in one spectrum, the potential can not be determined uniquely. Furthermore, Hochstadt has proved that when we know one full spectrum and one partial spectrum in which finite numbers of eigenvalues are missed, the potential can be determined by the solutions of the equations.For the inverse Sturm-Liouville problem in this paper, we use the method of Hochstadt's and consider the problem of determining the potential and the boundary conditions by three spectra. That is to prove that the potential on [0,1] can be uniquely determined by a full spectrum on [0,1] and two partial spectra on [0,α] and [α,1] (0<α< 1) respectively. On both of these two subinterval, any one of the eigenvalues can be missed in each interval respectively. After these two eigenvalues being missed, the potentials can also be uniquely determined.Basing on the studying of the above problem, we consider the inverse problem of the AKNS operator defining on L2([0,1], C2). We prove that under the condition of the potential on the interval [α,1] (0<α< 1) being known, only two partial spectra can determine the potential on [0,1].