Large-order behavior of the perturbation energies for the hydrogen atom in magnetic field

Abstract

Large-order behavior for the perturbation energies of the hydrogen atom in magnetic field is derived. By means of the dispersion relations, the large-order behavior of the series is determined by calculating the lifetime of the quasistationary states in an imaginary magnetic field. This problem is treated by means of the modified multidimensional WKB method. The asymptotic formula for the perturbation energies derived by Avron is generalized to the states with an arbitrary degeneracy. The first order correction to the resulting formula is also found. Thus, the multidimensional WKB method is for the first time explicitly carried out beyond the leading approximation. The analytical results are verified numerically and an excellent agreement between the two is found. The connection between our and conventional semiclassical approximation is also briefly discussed.