Dirac System with Discontinuities at Two Points

Fatma Hıra,Nihat Altınışık

doi:10.1155/2014/927693

Abstract

We deal with a regular Dirac system which has discontinuities at two points and contains eigenparameter in a boundary condition and two transmission conditions. We investigate asymptotic behaviour of eigenvalues and corresponding eigenfunctions of this Dirac system and construct Green’s function.

Highlights

We consider the Dirac system u2󸀠 (x) − p1 (x) u1 (x) = λu1 (x), (1)u1󸀠 (x) + p2 (x) u2 (x) = −λu2 (x), x ∈ I, B1 (u) := β1u1 (a) + β2u2 (a) = 0, (2)B2 (u) := u1 (b) − u2 (b) = 0, (3)T1 (u) := u1 (c1−) − δu1 (c1+) = 0, (4)T2 (u) := u2 (c1−) − δu2 (c1+) + λu1 (c1−) = 0, (5)T3 (u) := δu1 (c2−) − γu1 (c2+) = 0, (6)T4 (u) := δu2 (c2−) − γu2 (c2+) + λu1 (c2−) = 0
Sampling theories associated with discontinuous Dirac systems were investigated in [11, 12]
We examine Dirac system which has two points of discontinuity and contains at the same time an eigenparameter in a boundary condition and two transmission conditions in this paper

Summary

Introduction

B; λ ∈ C; the real valued functions p1(⋅) and p2(⋅) are cpo1n(ct1±in)u:o=ulsiminx[→a,c1c±1p)1,. References [3,4,5] are examples of works with boundary conditions depending linearly on an eigenparameter and transmission conditions at the point of discontinuity for Sturm Liouville problem. Sampling theories associated with discontinuous Dirac systems were investigated in [11, 12] ( the problem in [12] contains eigenparameter in a boundary condition). Dirac operators with eigenparameter dependent on both boundary conditions and one of the discontinuity conditions were investigated in [10]. In these discontinuous works, Dirac systems had only one point of discontinuity. Integrating the above equation through [a, c1], [c1, c2], and we construct Green’s function of the problem (1)–(7)

Spectral Properties

Asymptotic Approximate Formulas

Green’s Function