%0 Journal Article
%T Wellposedness of the Inverse Problem for Dirac Operator
%A Murat Sat
%A Etibar S. Panakhov
%A Kezban Tas
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/279394
%X We study the wellposedness of the inverse problem for Dirac operator. We consider two different problems (unperturbed and perturbed problem) for Dirac operator, and then we prove that if the spectral characteristics of these problems are close to each other, then the difference between their potential functions is sufficiently small. 1. Introduction Inverse problems are studied for certain special classes of ordinary differential operators. Typically, in inverse eigenvalue problems, one measures the frequencies of a vibrating system and tries to infer some physical properties of the system. An early important result in this direction, which gave vital impetus for the further development of inverse problem theory, was obtained in [1¨C5]. The Dirac equation is a modern presentation of the relativistic quantum mechanics of electrons, intended to make new mathematical results accessible to a wider audience. It treats in some depth the relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states, and the external field problem in quantum electrodynamics, without neglecting the interpretational difficulties and limitations of the theory. Inverse problems for Dirac system had been investigated by Moses [6], Prats and Toll [7], Verde [8], Gasymov and Levitan [9], and Panakhov [10, 11]. It is well known [12] that two spectra uniquely determine the matrix-valued potential function. In particular, in [13], eigenfunction expansions for one-dimensional Dirac operators describing the motion of a particle in quantum mechanics are investigated. Recently, Dirac operators have been extensively studied [14¨C19]. Mizutani showed the wellposedness problem of the Sturm-Liouville operator according to norming constants and eigenvalues [20]. The purpose of this paper is to give the wellposedness problem for Dirac operator by using Mizutani's method. Let denote a matrix operator where the are real functions which are defined and continuous on the interval . Further, let denote a two-component vector-function Then the equation where is a parameter and is equivalent to the system of two simultaneous first-order ordinary differential equations For the case in which , , , where is a potential and is the mass of a particle, the system (5) is known in relativistic quantum theory as a stationary one-dimensional Dirac system [5]. Consider the following eigenvalue problem for Dirac operator: where , , and are real-valued functions, , and is spectral parameter. We denote by the solution of (6) satisfying
%U http://www.hindawi.com/journals/cjm/2013/279394/