THE PHASE METHOD TO SOLVE THE EIGENVALUE PROBLEM FOR THE DIRAC EQUATION
M. A. Vronskii, V. M. Povyshev, S. Yu. Polyakova, E. S. Stolmakova
VANT. Ser.: Mat. Mod. Fiz. Proc 2018. Вып.3. С. 14-31.
We propose a version of the phase method for the solution of the eigenvalue problem for the radial Dirac equation. The potential with Coulomb behavior at the origin is considered. Our method is applicable to both finite and infinite interval. The regular solution at the origin is obtained as a solution of the derived Volterra equation. We prove the comparison theorem for the solutions of the phase equation. This theorem underlies the effective algorithm to find the eigenvalues. We determine the correspondence between the quantum numbers and the right-boundary values of the phase function for the eigenstates. We highlight the eigenfunction calculation problems and describe two ways to solve them. Several test calculations are presented. Keywords: the Dirac equation, the eigenvalue problem, the Sturm-Liouville problem, the phase method, the Sturm comparison theorem, the potential with Coulomb behavior at the origin.
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