The hydrogen atom in $D=3-2\epsilon$ dimensions

Abstract

The nonrelativistic hydrogen atom in $D=3-2\epsilon$ dimensions is the reference system for perturbative schemes used in dimensionally regularized nonrelativistic effective field theories to describe hydrogen-like atoms. Solutions to the $D$-dimensional Schr\"odinger-Coulomb equation are given in the form of a double power series. Energies and normalization integrals are obtained numerically and also perturbatively in terms of $\epsilon$. The utility of the series expansion is demonstrated by the calculation of the divergent expectation value $\langle (V')^2 \rangle$.