The Physics of Self-Adjoint Extensions: One-Dimensional Scattering Problem for the Coulomb Potential

We consider a one-dimensional single-center scattering problem on the entire axis with the original potential $\alpha|x|^{-1}$. This problem reduces to seeking admissible self-adjoint extensions. Using conservation laws at the singularity point as necessary conditions and taking the analytic structure of fundamental solutions into account allows obtaining exact expressions for the wave functions (i.eḟor the boundary conditions), scattering coefficients, singular corrections to the potential, and also the corresponding spectrum of bound states. It then turns out that pointlike $\delta$-corrections to the potential must necessarily be involved for any choice of the admissible self-adjoint extension. The form of these corrections corresponds to the form of the renormalization terms obtained in quantum electrodynamics. The proposed method therefore indicates a 1:1 relation between boundary conditions, scattering coefficients, and $\delta$-like additions to the potential and demonstrates the general possibilities arising in the analysis of self-adjoint extensions of the corresponding Hamilton operator. In the part pertaining to the renormalization theory, it can be considered a generalization of the renormalization method of Bogoliubov, Parasyuk, and Hepp.