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Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity and satsifies some regularity assumptions. For example under such rather general assumptions existence of $S$-matrix is proved in Ch. XIV of Hormander's "The Analysis of Linear Partial Differential Operators II".

However the Coulomb potential does not satisfy these assumptions of fast decay at infinity. Moreover my impression is that this is not just a technical issue, but the whole theory should look different for the Coulomb potential.

I am wondering if there is any notion of $S$-matrix for the Coulomb potential and how it looks like.

A reference would be helpful. Apologies if the question is not advanced enough- I am not an expert.

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    $\begingroup$ One way to define wave operators in the long range case is to modify the free motion appropriate, see e.g. Lars Hörmander, "The Analysis of Linear Partial Differential Operators IV pp 276-331" , link.springer.com/chapter/10.1007/978-3-642-00136-9_7 and J. Dollard, Quantum-mechanical scattering theory for short-range and Coulomb interactions, Rocky Mountain Journal of Mathematics 1, 5-88 (1971) . $\endgroup$
    – jjcale
    Commented Feb 2, 2020 at 15:44
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    $\begingroup$ Another standard reference that discusses scattering theory of long range potentials at length is the third volume of Reed-Simon. $\endgroup$
    – Christian Remling
    Commented Feb 2, 2020 at 19:10

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The $1/r$ Coulomb potential needs to be regularized, typically this is done by studying the Yukawa potential $e^{-\alpha r}/r$ and taking the limit $\alpha\rightarrow 0$ at the end. A recent critical examination of this procedure can be found in Regularization of the Coulomb scattering problem (2004).

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