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Representation of solutions of the one-dimensional Dirac equation in terms of Neumann series of Bessel functions

Emmanuel Roque, Sergii M. Torba

TL;DR

This work develops a rigorous NSBF representation for the matrix solution of the one-dimensional Dirac equation, achieved through a transmutation-operator framework and a Fourier-Legendre expansion of the transmutation kernel. The coefficients $K_n(x)$ are computed from a recursive differential system via $\theta_n(x)=x^nK_n(x)$, with proven uniform convergence in the spectral parameter $\lambda$ and explicit approximation bounds under Sobolev regularity of the potential $Q$. The NSBF approach is extended to the ZS-AKNS system, enabling complex-valued potentials and half-line inverse scattering, and a practical numerical scheme is provided for initial-value and spectral problems that reliably yields large eigendata with stable accuracy. Overall, the method offers an exact, computationally efficient framework for Dirac-type spectral problems and related Zakharov-Shabat systems, with strong convergence guarantees and a clear path to inverse problems.

Abstract

A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of Bessel functions. The representations are shown to be uniformly convergent with respect to the spectral parameter. Explicit formulas for the coefficients are obtained via a system of recursive integrals. The result is based on the Fourier-Legendre series expansion of the transmutation kernel. An efficient numerical method for solving initial-value and spectral problems based on this approach is presented with a numerical example. The method can compute large sets of eigendata with non-deteriorating accuracy.

Representation of solutions of the one-dimensional Dirac equation in terms of Neumann series of Bessel functions

TL;DR

This work develops a rigorous NSBF representation for the matrix solution of the one-dimensional Dirac equation, achieved through a transmutation-operator framework and a Fourier-Legendre expansion of the transmutation kernel. The coefficients are computed from a recursive differential system via , with proven uniform convergence in the spectral parameter and explicit approximation bounds under Sobolev regularity of the potential . The NSBF approach is extended to the ZS-AKNS system, enabling complex-valued potentials and half-line inverse scattering, and a practical numerical scheme is provided for initial-value and spectral problems that reliably yields large eigendata with stable accuracy. Overall, the method offers an exact, computationally efficient framework for Dirac-type spectral problems and related Zakharov-Shabat systems, with strong convergence guarantees and a clear path to inverse problems.

Abstract

A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of Bessel functions. The representations are shown to be uniformly convergent with respect to the spectral parameter. Explicit formulas for the coefficients are obtained via a system of recursive integrals. The result is based on the Fourier-Legendre series expansion of the transmutation kernel. An efficient numerical method for solving initial-value and spectral problems based on this approach is presented with a numerical example. The method can compute large sets of eigendata with non-deteriorating accuracy.

Paper Structure

This paper contains 9 sections, 16 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transmutation operators
  3. The NSBF representation
  4. A recursive system of equations
  5. Solving the system of recursive equations
  6. The NSBF representation of the solution of the ZS-AKNS system
  7. Approximation results
  8. Numerical implementation and examples
  9. The mapping property and its connection to the NSBF representation

Key Result

Theorem 2.2

Nelson-analytic Suppose that $Q$ is a continuously differentiable matrix-valued function on $[-b,b]$. Then a transmutation operator $T: E \to E, \, E:=C([-b,b], \mathbb{C}^2)$, relating the operators $A_0$ and $A_Q$ for all $Y \in E_1:=C^1([-b,b],\mathbb{C}^2 )$, can be realized in the form of a Vol where $K(x,t)$ is a $2\times 2$ matrix-valued function satisfying the partial differential equation

Figures (1)

  • Figure 1: Absolute error of the eigenvalues $\lambda_n$ of example \ref{['ex:AT']}, $N=16$.

Theorems & Definitions (38)

  • Definition 2.1: levitan
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Lemma 3.6
  • ...and 28 more