Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Recovery of the matrix potential of the one-dimensional Dirac equation from spectral data

Emmanuel Roque, Sergii M. Torba

TL;DR

This work presents a practical inverse spectral method for reconstructing the matrix potential $Q$ in the one-dimensional Dirac equation on $[0,1]$ from spectral data. It combines the Gelfand–Levitan framework with matrix Fourier–Legendre expansions of transmutation kernels and Neumann series of Bessel functions to reduce the problem to a linear system for kernel coefficients, from which $Q$ is recovered. Two recovery pathways are described: reconstructing $Q$ from the first kernel coefficient or from associated norming-data-based relations, both supported by NSBF representations for efficient computation. Numerical experiments on smooth and non-smooth potentials illustrate high accuracy and fast performance, highlighting the method’s potential for extensions to radial Dirac problems and to broader inverse problems in Dirac-type systems.

Abstract

A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand-Levitan equation and the Fourier-Legendre series expansion of the transmutation kernel. A linear algebraic system of equations is obtained, which can be solved numerically. To the best of our knowledge, this is the first practical method for the solution of the inverse problem for the one-dimensional Dirac equation on a finite interval.

Recovery of the matrix potential of the one-dimensional Dirac equation from spectral data

TL;DR

This work presents a practical inverse spectral method for reconstructing the matrix potential in the one-dimensional Dirac equation on from spectral data. It combines the Gelfand–Levitan framework with matrix Fourier–Legendre expansions of transmutation kernels and Neumann series of Bessel functions to reduce the problem to a linear system for kernel coefficients, from which is recovered. Two recovery pathways are described: reconstructing from the first kernel coefficient or from associated norming-data-based relations, both supported by NSBF representations for efficient computation. Numerical experiments on smooth and non-smooth potentials illustrate high accuracy and fast performance, highlighting the method’s potential for extensions to radial Dirac problems and to broader inverse problems in Dirac-type systems.

Abstract

A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand-Levitan equation and the Fourier-Legendre series expansion of the transmutation kernel. A linear algebraic system of equations is obtained, which can be solved numerically. To the best of our knowledge, this is the first practical method for the solution of the inverse problem for the one-dimensional Dirac equation on a finite interval.
Paper Structure (10 sections, 19 theorems, 104 equations, 6 figures)

This paper contains 10 sections, 19 theorems, 104 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier-Legendre series for the integral kernels
  4. Neumann series of Bessel functions representation for solutions of the Dirac equation
  5. The Gelfand-Levitan equation
  6. A method of solution of the Gelfand-Levitan equation
  7. Reduction of the Gelfand-Levitan equation to a linear system of algebraic equations
  8. Two possibilities for the recovery of the potential
  9. Numerical implementation
  10. The Bubnov-Galerkin process for stationary problems

Key Result

Theorem 2.2

Let $Q\in L^2([0,1], \mathcal{M}_2)$ and let $I$ denote the $2\times 2$ identity matrix. Then there exists a matrix-valued function $K(x,t)$ such that the matrix solution $U(\lambda,x)$ of the Cauchy problem can be represented in the form and the relations hold for almost all $x \in [0,1].$ Moreover, $K(x,\cdot) \in L^2([-x,x],\mathcal{M}_2)$.

Figures (6)

  • Figure 1: Absolute error of the method based on spline differentiation (left) vs the method based on the Goursat characteristic equation (right) for example \ref{['ex:example1']}.
  • Figure 2: Absolute error of the recovery of the potential \ref{['eqn:example1']} based on spline differentiation using 2001 exact eigenvalues with no asymptotic eigenvalues (AEVs). This is compared to the absolute error of the recovery using 51 and 11 exact eigenvalues and generating enough AEVs to have 10,001 spectral data in total.
  • Figure 3: Left: Absolute error of the recovery of the potential \ref{['eqn:example2']} based on spline differentiation using 21 exact eigenvalues and 4980 AEVs to have 5,001 spectral data in total. Right: Exact and recovered potential.
  • Figure 4: Absolute error of the recovery of the potential \ref{['eqn:example3']} based on spline differentiation using 41 exact eigenvalues and complemented with 9960 AEVs to have 10,001 spectral data in total. Right: Exact and recovered potential.
  • Figure 5: Left: Absolute error of the recovery of the potential \ref{['eqn:example3']} based on spline differentiation using 41 exact eigenvalues and 9960 AEVs to have 10,001 spectral data in total. The parameter $\beta=\pi/4$ is assumed to be unknown, $\vert \beta - \beta_{rec} \vert = 9.1\cdot 10^{-4}.$ Right: Exact and recovered potential.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Remark 2.1
  • Theorem 2.2: diracTOPdiracbook
  • Remark 2.3
  • Theorem 2.4: diracbook
  • Remark 2.5
  • Corollary 2.6
  • Theorem 2.7: diracbooklevitanHryniv
  • Theorem 2.8: diracbookHryniv
  • Remark 2.9
  • Proposition 3.1
  • ...and 32 more