High-order perturbation theory of the imaginary part of the resonance eigenvalues of the Stark effect in hydrogen and of the anharmonic oscillator with negative anharmonicity

The "perturbation theory" for the imaginary part of the resonance energies of the hydrogen atom in the Stark effect and of the two-dimensional anharmonic oscillator with negative anharmonicity, which is a separation constant in the Stark problem, is solved to high order. The solution is based on the Langer-Cherry generalization of the JWKB method, which can be carried out in closed form, order by order. The numerical results should be useful both in interpreting experimental measurements of excited-state lifetimes and in understanding the analytic properties of the Stark and anharmonic-oscillator resonances.