Jost Solution and the Spectrum of the Discrete Dirac Systems

Elgiz Bairamov,Yelda Aygar,Murat Olgun

doi:10.1155/2010/306571

Abstract

We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.

Highlights

Let us consider the boundary value problem BVP generated by the Sturm-Liouville equation−y q x y λ2y, 0 ≤ x < ∞1.1 and the boundary condition y 0 0, 1.2 where q is a real-valued function and λ ∈ is a spectral parameter
Boundary Value Problems will be denoted by e x, λ
The collection of quantities {s λ, λ ∈ Ê; λk, mk, k 1, 2, . . . , n} that specify to as the behaviour of the radial wave functions E x, λ and E x, iλk at infinity is called the scattering of the BVP 1.1 and 1.2

Summary

Introduction

Let us consider the boundary value problem BVP generated by the Sturm-Liouville equation. 1.1 and the boundary condition y 0 0, 1.2 where q is a real-valued function and λ ∈ is a spectral parameter. The solution e x, λ satisfies the integral equation e x, λ eiλx. The functions e x, λ and e λ : e 0, λ are called Jost solution and Jost function of the BVP 1.1 and 1.2 , respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory 1–4. The scattering date of the BVP 1.1 and 1.2 is defined in terms of Jost solution and Jost function. Let iλk, k 1, 2, 3, . . . , n, be the zeros of the Jost function, numbered in the order of increase of their moduli 0 < λ1 < λ2 < · · · < λn and m−k1 : e2 x, iλk dx

The functions

Now let us consider the discrete Dirac system

It is clear that