Complex coordinates and the Stark effect

Abstract The theories of the dilatation, r → r e iθ , and translation, x → x + iq , transformations as related to the Stark problem are reviewed, and new results obtained. Results for the hydrogen atom n = 1 and n = 2 levels and the 1 P 0 , 2 s 2 p H − shape resonance in dc fields are presented, and the extension to the ac Stark effect made. Spectral estimates are made using the technique of the numerical range and via discussion of several model problems, using both coordinate rotation and coordinate translation.