Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis

Abstract

We consider the boundary value problem (BVP) for the discrete Dirac equations {y(2)n+1 - y(2)n+ p ny(1)n = λy(1)ny(1)n-1 - yn + qny (2)n = λy(2)n, n ε Z = {0,±1,±2, . . .}; y(1)0= 0, where (p n) and (qn), n ε Z are real sequences, and λ is an eigenparameter. We find a polynomial type Jost solution of this BVP. Then we investigate the analytical properties and asymptotic behavior of the Jost solution. Using the Weyl compact perturbation theorem, we prove that a self-adjoint discrete Dirac system has a continuous spectrum filling the segment [-2, 2]. We also prove that the Dirac system has a finite number of real eigenvalues. © 2014 Aygar and Olgun; licensee Springer.