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Spectral parameter power series representation for regular solutions of the radial Dirac system

Emmanuel Roque, Sergii M. Torba

TL;DR

The paper develops a spectral parameter power series (SPPS) framework for the regular solution of the radial Dirac system with complex potentials on a finite interval, yielding a representation of the regular solution in terms of formal powers and a convergent series uv = ∑_{n=0}^{∞} (λ^n / n!) Ẋ^{(n)} Ẏ^{(n)}. It extends the method to singular and nonstandard cases, introduces a spectral shift technique to enhance numerical accuracy, and provides explicit constructive formulas for a particular solution satisfying the required asymptotics. The authors present a complete numerical implementation, demonstrate high accuracy on perturbed Bessel problems, and apply the approach to hydrogen-like atoms with finite radius, obtaining energy eigenvalues with adaptive spectral-shift strategies. Overall, the SPPS approach offers a versatile and accurate tool for solving spectral problems for radial Dirac systems and related perturbed equations with potential applications in atomic and quantum systems.

Abstract

A spectral parameter power series (SPPS) representation for the regular solution of the radial Dirac system with complex coefficients is obtained, as well as a SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. Based on the SPPS representation, a numerical method for solving spectral problems is developed. It is shown that the method is also applicable to solving spectral problems for perturbed Bessel equations. We exhibit that the proposed numerical method delivers excellent results. Additionally, an application of the method to find the energy values of hydrogen-like atoms with a finite radius is presented.

Spectral parameter power series representation for regular solutions of the radial Dirac system

TL;DR

The paper develops a spectral parameter power series (SPPS) framework for the regular solution of the radial Dirac system with complex potentials on a finite interval, yielding a representation of the regular solution in terms of formal powers and a convergent series uv = ∑_{n=0}^{∞} (λ^n / n!) Ẋ^{(n)} Ẏ^{(n)}. It extends the method to singular and nonstandard cases, introduces a spectral shift technique to enhance numerical accuracy, and provides explicit constructive formulas for a particular solution satisfying the required asymptotics. The authors present a complete numerical implementation, demonstrate high accuracy on perturbed Bessel problems, and apply the approach to hydrogen-like atoms with finite radius, obtaining energy eigenvalues with adaptive spectral-shift strategies. Overall, the SPPS approach offers a versatile and accurate tool for solving spectral problems for radial Dirac systems and related perturbed equations with potential applications in atomic and quantum systems.

Abstract

A spectral parameter power series (SPPS) representation for the regular solution of the radial Dirac system with complex coefficients is obtained, as well as a SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. Based on the SPPS representation, a numerical method for solving spectral problems is developed. It is shown that the method is also applicable to solving spectral problems for perturbed Bessel equations. We exhibit that the proposed numerical method delivers excellent results. Additionally, an application of the method to find the energy values of hydrogen-like atoms with a finite radius is presented.
Paper Structure (10 sections, 7 theorems, 122 equations, 4 figures, 2 tables)

This paper contains 10 sections, 7 theorems, 122 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The spectral parameter power series representation for the regular solution of the radial Dirac system
  3. Construction of the regular solution for the generalized radial Dirac system
  4. Spectral shift technique
  5. Construction of a particular solution
  6. The case of a singular potential
  7. Solutions with a non-vanishing first entry
  8. Numerical implementation and examples
  9. Application to spectral problems for the perturbed Bessel equation
  10. Application of the method for a hydrogen-like atom with finite radius

Key Result

Lemma 2.4

Assume that the homogeneous system eqn:dirach possesses a solution $Y_0 = (f,g)^T$ such that $f,\, g$ satisfy the asymptotic relation eqn:asymfg and $f$ does not vanish except at $x=0.$ Then, there exists a unique solution to the non-homogeneous system where $h_{1,2} \in C([0,a])$ and $h_{1,2} = O(x^{\kappa+1})$, that satisfies the condition $u(x)=O(xf(x))$ and $v(x)=O(xg(x))$ as $x\to 0$. Such s

Figures (4)

  • Figure 1: Absolute error of $\sqrt{\lambda_n}$ of Example \ref{['Ex:73']} using the spectral shift technique with a constant step size $\lambda_0^{(n)}=n \sigma+ n \tau_0 i$ for different values of $\tau_0, \, \sigma=-2$.
  • Figure 2: Absolute error of $\sqrt{\lambda_n}$ of Example \ref{['Ex:73']} using the spectral shift technique with an adaptative step size, with the seed value $\eta_0=\sigma+\tau_0 i$ for different values of $\tau_0, \, \sigma=-2$. The set of dilation constants is $A = \{ 0.9, 1, 1.1 \}$.
  • Figure 3: Absolute error of $\sqrt{\lambda_n}$ of Example \ref{['Ex:boyd']} using the spectral shift technique with an adaptive step size, with the seed value $\eta_0=\sigma+\tau_0 i$ for different values of $\tau_0, \, \sigma=-6$. The set of dilation constants is $A = \{ 0.9, 1, 1.1 \}, \, N=100, \, M=100000$.
  • Figure 4: Graph (a) and (b) corresponds to the first four large and small wave components for the hydrogen-like oxygen with finite size radius respectively.

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 11 more