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A new high-accuracy method for calculating the energy values of low-lying excited states of a hydrogen atom in a strong magnetic field (0 ⩽ B ⩽ 1013G) is developed based on the Kantorovich approach to parametric eigenvalue problems and using the axial symmetry. The initial two-dimensional spectral problem for the Schrödinger equation is reduced to a spectral parametric problem for a one-dimensional equation and a finite set of ordinary second-order differential equations. The rate of convergence is examined numerically and is illustrated with a set of typical examples. The results are in good agreement with precise calculations by other authors.

The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the onedimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.

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