Numerical algorithms without saturation for the Schrödinger equation of hydrogen atom

Abstract

Mathematically, the problem under consideration is reduced to the eigenvalue problem for the Laplace operator in the entire space with the Coulomb potential. The new mathematical apparatus developed by the author is applied to the numerical solution of the reduced problem. This problem is reduced to the eigenvalue problem in the unit ball punctured at the center after inversion with respect to the unit sphere. The null boundary condition at infinity is transformed to the condition at the center of the unit sphere. In the sphere it is possible to split off the periodic variable φ and to construct the discretization inheriting the property of the separation of variables of the differential operator (the h-matrix). Eleven points is chosen based on the values of φ. The blocks Λ0, Λ1, Λ2, Λ3, Λ4, and Λ5 of the h-matrix correspond to the Lyman, Balmer, Paschen, Brackett, Pfund, and Humphreys lines. From the obtained numerical results, it follows that the Lyman-alpha line is determined with the accuracy equal to 5.43%. Thus, the coincidence of the numerical results with the theoretical values is satisfactory.