Weyl's theory applied to the Stark effect in the hydrogen atom

Abstract

A short review of Hermann Weyl's theory for singular second-order differential equations is given and its numerical aspects are discussed. It is pointed out that this method is suitable for the treatment of perturbations which make the spectrum continuous. The Stark effect on the ground state of the hydrogen atom is taken as an example. The spectral density, the imaginary part of Weyl's "$m$ function," is calculated numerically using Runge-Kutta integration and Airy integrals for the asymptotic region. Showing $\ensuremath{\delta}$-function-like behavior with poles of $m$ on the real axis for the discrete levels, the spectral density involves approximate Lorentzians for the metastable states of the continuous spectrum, corresponding to poles of $m$ in the complex plane. Trajectories of these poles for electric fields up to 0.25 a.u. are shown for the one-dimensional as well as for the full three-dimensional problem.