Most textbooks in quantum mechanics handle the Coulomb problem by solving the Schrödinger equation directly in the coordinate representation.

Is there any book or reference that adopts a more formal approach, not going into the coordinate representation first, but instead working in the abstract Hilbert space, treating the Coulomb potential

$$ \frac{1}{\hat{r}} := \mathrm{inverse \,\, of \,\, } \hat{r} $$

performing some formal analysis, and computing $\langle r| \frac{1}{\hat{r} } | r' \rangle$ or $\langle r| \frac{1}{ \hat{r} } | \psi \rangle$? I believe this is a straightforward concept, but I would appreciate any references that discuss this approach, especially seldom people use the notation $1/\hat{r}$