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, 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.

On the calculation of the discrete spectra of one‐dimensional Dirac operators

“…In the paper we obtained an explicit representation of monodromy matrices $$$ {L}_{j,j+1} $$$ ( $$$ j=0,\cdots, N $$$) by means of the method introduced by Gutiérrez and Torba 19 . It follows that the entries of matrices $$$ {L}_{j,j+1} $$$ were obtained in the form of power series of the spectral parameter $$$ \lambda $$$, which converge uniformly in $$$ \left[{x}_j,{x}_{j+1}\right] $$$.…”

“…The recent work of Gutiérrez and Torba 19 introduces a method for obtaining two (exact) linearly independent solutions in $$$ \left(a,b\right)\subset \mathbb{R} $$$ of the system of first order differential equations $$$$ {v}^{\prime }+{p}_1(x)u+q(x)v=\lambda \left({r}_{11}(x)u+{r}_{12}(x)v\right), $$$$ $$$$ -{u}^{\prime }+q(x)u+{p}_2(x)v=\lambda \left({r}_{21}(x)u+{r}_{22}(x)v\right), $$$$ where $$$ q,{p}_i,{r}_{ij}\in \mathcal{C}\left(\left[a,b\right]\right) $$$ ( $$$ i,j=1,2 $$$) are complex‐valued functions depending on the real variable $$$ x\in \mathbb{R} $$$, and $$$ \lambda \in \mathbb{C} $$$ is a parameter. The solutions are given as power series of the spectral parameter $$$ \lambda $$$.…”

“…The SPPS method 19 leads to the dispersion equations $$$ {\eta}_{\Omega}\left(\lambda \right)=0 $$$ in the form of a power series of the spectral parameter $$$ \lambda $$$, which makes it possible to consider their truncations and obtain approximate dispersion equations $$$ {\tilde{\eta}}_{\Omega}\left(\lambda \right)=0 $$$ that can be solved numerically. Let $$$ {\mu}_n\in \mathbb{C} $$$ be a zero of the approximate dispersion equation.…”

Relativistic collapse of Landau levels of Kane fermions in crossed electric and magnetic fields

“…( 36) cannot be represented in the form of the eigenvalue problem. A similar type of mathematical problem has recently been considered in the context of Dirac systems [40].…”

“…Thus, using of Eqs. (39)(40)(41)(42)(43), the solution of the Schrödinger equation in the boosted frame is reduced to the algebraic secular equation in Eq. (38).…”