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.

No abstract

No abstract

A revised version of the concavity method of Levine, based on a new ordinary differential inequality, is proposed. Necessary and sufficient condition for nonexistence of global solutions of the inequality is proved. As an application, finite time blow up of the solution to Klein-Gordon equation with arbitrary positive initial energy is obtained under very general structural conditions.