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We explore the quantum Coulomb problem for two-body bound states, in $D=3$and $D=3-2\epsilon$ dimensions, in detail, and give an extensive list ofexpectation values that arise in the evaluation of QED corrections to boundstate energies. We describe the techniques used to obtain these expectationvalues and give general formulas for the evaluation of integrals involvingassociated Laguerre polynomials. In addition, we give formulas for theevaluation of integrals involving subtracted associated Laguerrepolynomials--those with low powers of the variable subtracted off--that arisewhen evaluating divergent expectation values. We present perturbative results(in the parameter $\epsilon$) that show how bound state energies and wavefunctions in $D=3-2\epsilon$ dimensions differ from their $D=3$ dimensionalcounterparts and use these formulas to find regularized expressions fordivergent expectation values such as $\big \langle \bar V^3 \big \rangle$ and$\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensionalCoulomb potential. We evaluate a number of finite $D$-dimensional expectationvalues such as $\big \langle r^{-2+4\epsilon} \partial_r^2 \big \rangle$ and$\big \langle r^{4\epsilon} p^4 \big \rangle$ that have $\epsilon \rightarrow0$ limits that differ from their three-dimensional counterparts $\big \langler^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. Weexplore the use of recursion relations, the Feynman-Hellmann theorem, andmomentum space brackets combined with $D$-dimensional Fourier transformationfor the evaluation of $D$-dimensional expectation values. The results of thispaper are useful when using dimensional regularization in the calculation ofproperties of Coulomb bound systems.

Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions