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Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation

P. J. Rijken, Th. A. Rijken

Abstract

Eigenvalues of the Breit equation, in which only the static Coulomb potential is considered, have been found. Over the past decades several authors have analyzed the Breit equation to obtain numerically or by approximation an estimation of the energy levels. Various approaches have been used and no determination of the energy levels currently exists that is directly based on the second order Heun differential equation derived. The aim of this work is to provide a method of calculation that can be used to numerically calculate the energy levels for various spin states to high accuracy. From the Breit equation, we derive the corresponding second-order Heun differential equation and continued fraction from which the eigenvalues can be determined very accurately. Next, we present a novel method based on the Green function method, which leads to a semi-infinite determinant from which we are able to obtain the numerical values of the eigenvalues by direct calculation. Using suitable numerical methods for the direct calculation of the continued fraction and the semi-infinite determinant, we show that both methods are consistent within 25 digits of accuracy. We show that the correct energy levels for the Dirac equation follow from our results by a suitable mapping of the variables. The results are in total agreement with earlier calculations found in the literature and extend this by several digits of additional accuracy. The condition on the determinant giving the energy levels provides a rich structure that is promising in extending the results of this work.

Novel method for evaluating the eigenvalues of the Heun differential equation with an application to the Breit equation

Abstract

Eigenvalues of the Breit equation, in which only the static Coulomb potential is considered, have been found. Over the past decades several authors have analyzed the Breit equation to obtain numerically or by approximation an estimation of the energy levels. Various approaches have been used and no determination of the energy levels currently exists that is directly based on the second order Heun differential equation derived. The aim of this work is to provide a method of calculation that can be used to numerically calculate the energy levels for various spin states to high accuracy. From the Breit equation, we derive the corresponding second-order Heun differential equation and continued fraction from which the eigenvalues can be determined very accurately. Next, we present a novel method based on the Green function method, which leads to a semi-infinite determinant from which we are able to obtain the numerical values of the eigenvalues by direct calculation. Using suitable numerical methods for the direct calculation of the continued fraction and the semi-infinite determinant, we show that both methods are consistent within 25 digits of accuracy. We show that the correct energy levels for the Dirac equation follow from our results by a suitable mapping of the variables. The results are in total agreement with earlier calculations found in the literature and extend this by several digits of additional accuracy. The condition on the determinant giving the energy levels provides a rich structure that is promising in extending the results of this work.
Paper Structure (45 sections, 31 theorems, 163 equations, 6 figures, 9 tables)

This paper contains 45 sections, 31 theorems, 163 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction and the Breit equation
  2. The spin-singlet state: Breit radial differential equation
  3. Basic equation
  4. The spin-singlet case
  5. The case of equal masses $\bm{ m=M}$
  6. Expansion in Associated Laguerre Polynomials
  7. Series expansion of the eigenfunctions
  8. Convergence of the series expansion
  9. Numerical determination of the eigenvalues
  10. Normalization and calculation of the eigenfunctions
  11. The spin-triplet states
  12. The spin-triplet $J=0$ case (${}^3P_0$)
  13. The spin-triplet $J=L$ case (${}^3J_J$)
  14. The spin-triplet $J=L\pm1$ case (${}^3(L\pm 1)_{L}$)
  15. Modeling of the parameters of the differential equation for $k(\rho)$
  16. ...and 30 more sections

Key Result

Theorem 1

The recurrence relation for $g_n$ in equation eq:heunttrr has two linearly independent solutions $g_n^{(1)}$ and $g_n^{(2)}$ with the properties:

Figures (6)

  • Figure 1: Continued fraction for the range $z=0...6$. The dots mark the location of the eigenvalues.
  • Figure 2: Detailed graphs of Figure \ref{['fig:ctdfraction']} around the zeros and asymptotic behaviour of \ref{['eq:eigenvalues']}. The dots mark the location of the eigenvalues.
  • Figure 3: The scaled radial components $F(r)$, $K(r)$, and $G(r)$ for the spin singlet case $N=1$. The position $r$ is expressed in units $2a_0$.
  • Figure 4: The scaled radial components $F(r)$, $K(r)$, and $G(r)$ for the spin triplet case $N=2$. The position $r$ is expressed in units $2a_0$.
  • Figure 5: The scaled radial components $F(r)$, $K(r)$, $G(r)$, and $\tilde{G}$ for the spin triplet case $N=3, L=2, S=1, J=2$. The position $r$ is expressed in units $2a_0$.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Theorem 2: Pincherle Pin94
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Definition 1
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • ...and 39 more