Linear response of the hydrogen atom in Stark states to a harmonic uniform electric field

The influence of a weak harmonic uniform electric field, switched on adiabatically, on a nonrelativistic hydrogenlike atom is examined. Each of the φ- and A-gauge first-order corrections to the wave function of a stationary state ‖N〉 is determined by a vector function that we denote ${\mathrm{v}}_N$ and ${\mathrm{w}}_N$, respectively. The absolute starting point of our calculations is Schwinger’s formula for the Coulomb Green’s function in momentum space. In the case of a bound state with definite angular momentum, we report a compact integral representation and also an explicit expression of the φ-gauge vector ${\mathrm{v}}_{\mathrm{nlm}}$, which are analogous to those of the corresponding A-gauge vector ${\mathrm{w}}_{\mathrm{nlm}}$ studied previously. We have derived compact analytic expressions of the linear-response vectors ${\mathrm{v}}_{n_{\xi }{n}_{\eta }m}$ and ${\mathrm{w}}_{n_{\xi }{n}_{\eta }m}$ associated to an arbitrary Stark state. These are written first as contour integrals, and then explicitly in terms of a new generalized hypergeometric function with five variables, ${}_2{}^{}{}_{}{}^{}\mathrm{\phi }_{\mathrm{H}}^{}{}_{}{}^{}$, which is a finite sum of Humbert functions ${\phi }_1$. We have calculated the static limit of the regular part of the vector ${\mathrm{v}}_{n_{\xi }{n}_{\eta }m}$. Also discussed are the Sturmian-function expansions of the linear-response vectors for angular momentum states.