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

Yelda Aygar,Murat Olgun

doi:10.1186/1029-242x-2014-73

Abstract

We consider the boundary value problem (BVP) for the discrete Dirac equations where and , 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 . We also prove that the Dirac system has a finite number of real eigenvalues.

Highlights

Let us consider the BVP generated by the Sturm-Liouville equation,–y + q(x)y = λ y, x ∈ R+ = [, ∞), ( )and the boundary condition y( ) =, where q is a real valued function and λ is a spectral parameter
We find a polynomial type Jost solution of ( ) with the boundary condition y( ) =, ( )
Which is analytic in D := {z : |z| < }\{ }

Summary

Introduction

The functions e(x, λ) and e(λ) := e( , λ) are called the Jost solution and Jost function of the BVP ( ) and ( ), respectively. We find a polynomial type Jost solution of ( ) with the boundary condition y( ) = , which is analytic in D := {z : |z| < }\{ }. Theorem Under the condition ( ) for λ = –iz – (iz)– and |z| = , ( ) has the solutions fn(z) =. Due to the condition ( ), the series in the definition of Knijm and Binjm (i, j = , ) are absolutely convergent. Knijm and Binjm (i, j = , ) can uniquely be defined by pn and qn (n ∈ Z), i.e., the system ( ) for λ = –iz – (iz)– , |z| = , have the solutions fn(z) given by ( ).

Kn m

The Wronskian of the solutions