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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 $1/\hat{r}$

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    $\begingroup$ That simply does not exist. However, what you want exists in another form: You can use the LRLP method to algebraically extract the energy eigenvalues. There is also another method based upon ladder operators, but this is not just far more involved, but will also involve wavefunctions. $\endgroup$
    – naturallyInconsistent
    Commented May 23 at 5:31
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    $\begingroup$ @naturallyInconsistent what is LRLP? $\endgroup$
    – Tobias Fünke
    Commented May 23 at 5:52
  • $\begingroup$ In QM $1/r$ is simply the operator "multiplication by $U(r)=1/r$" defined via $(U\psi)(x) =\psi(x)/r$. $\endgroup$
    – lcv
    Commented May 23 at 21:28

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