Momentum approach to the $1\mathrm{/}{r}^2$ potential as a toy model of the Wilsonian renormalization

The Bessel operator, that is, the Schr"odinger operator on the half-line with a potential proportional to $1\mathrm{/}{x}^2$, is analyzed in the momentum representation. Many features of this analysis are parallel to the approach `a la K. Wilson to Quantum Field Theory: one needs to impose a cutoff, add counterterms, study the renormalization group flow with its fixed points and limit cycles.