The hydrogen atom in strong electric fields: Summation of the weak field series expansion

The order dependent mapping method, whose convergence has recently been proven for the energy eigenvalue of the anharmonic oscillator, is applied to re-sum the standard perturbation series for the Stark effect of the hydrogen atom. We perform a numerical experiment up to the 50th order of the perturbation expansion. A simple mapping, suggested by the analytic structure and the strong field behavior, gives an excellent agreement with the exact value for an intermediate range of the electric field, 0.03 (a.u.) less than or equal to E less than or equal to 0.25 (a.u.). The imaginary part of the energy (the decay width) as well as the real part of the energy is reproduced from the standard perturbation series.