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

TL;DR

This paper analyzes the hydrogen atom in fractional dimensions near three, providing solutions to the Schrödinger-Coulomb equation as power series, with energies and integrals computed numerically and perturbatively, aiding effective field theory calculations.

Contribution

It offers a detailed solution to the D-dimensional hydrogen atom problem in fractional dimensions, including series expansions and perturbative methods for energies and expectation values.

Findings

Series expansion of solutions in fractional dimensions

Numerical and perturbative energy calculations

Evaluation of divergent expectation values

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 $\langle (V')^2 \rangle$.