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

TL;DR

This paper analyzes the $1/x^2$ potential using a momentum-space approach, drawing parallels to Wilsonian renormalization in quantum field theory, including cutoff implementation, counterterms, and renormalization group flow.

Contribution

It introduces a momentum-space analysis of the $1/x^2$ potential as a toy model for Wilsonian renormalization, highlighting key features like fixed points and limit cycles.

Findings

Identifies renormalization group flow features in the $1/x^2$ potential.

Establishes parallels between quantum mechanics and quantum field theory renormalization.

Demonstrates the necessity of counterterms in the momentum representation.

Abstract

The Bessel operator, that is, the Schr\"odinger operator on the half-line with a potential proportional to $1/x^2$, 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.