Spectral parameter power series representation for solutions of linear system of two first order differential equations

Abstract

A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter,BdYdx+P(x)Y=λR(x)Y,(*)where Y=(y1,y2)T is the unknown vector-function, λ is the spectral parameter, B=(01−10), and P is a symmetric 2 × 2 matrix, R is an arbitrary 2 × 2 matrix whose entries are integrable complex-valued functions. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a non-vanishing solution for one particular λ=λ0. The existence of such solution is shown.For a general linear system of two first order differential equationsP(x)dYdx+Q(x)Y=λR(x)Y,x∈[a,b],where P, Q, R are 2 × 2 matrices whose entries are integrable complex-valued functions, P being invertible for every x, a transformation reducing it to a system (*) is shown.The general scheme of application of the SPPS representation to the solution of initial value and spectral problems as well as numerical illustrations are provided.