Wellposedness of the Inverse Problem for Dirac Operator

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 [1–5]. 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 [14–19]. 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 L denote a matrix operator


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][2][3][4][5].
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.
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   () (,  = 1, 2) are real functions which are defined and continuous on the interval [0, ].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  12 () =  21 () = 0,  11 () = () + ,  22 () = () − , 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].
We denote by the solution of ( 6) satisfying the initial conditions where  = | Im |.
We show the spectral characteristics of the problems ( 6)-( 8) by {  ,   } ( ∈ Z), where {  } is the spectrum (set of eigenvalues) and {  } is the norming constants of this problem.These spectral characteristics satisfy the following asymptotic expressions, respectively [9]: where ,  1 , and  1 are real numbers.Now let us consider the second eigenvalue problem with the same boundary conditions ( 7), (8), where Q() = ( p() q() q() − p() ), p() and q() ∈  1 [0, ] are real-valued functions and  is spectral parameter.We denote the spectral characteristics of the problems ( 12), (7), and (8) by {  ,   } ( ∈ Z), where {  } is the spectrum and {  } is the norming constants of this problem.These spectral characteristics satisfy the following asymptotic expressions, respectively: where   ,   1 , and   1 are real constants.

Main Results
where   > 0 and   > 0 are constants.

Conclusion
The more norming constants and spectrums which are taken as spectral characteristics of Dirac operators are close to each other, the more difference of potential functions is sufficiently small.