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

Jan Dereziński, Oskar Grocholski

Abstract

The Bessel operator, that is, the Schrödinger 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 à 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.

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

Abstract

The Bessel operator, that is, the Schrödinger operator on the half-line with a potential proportional to , is analyzed in the momentum representation. Many features of this analysis are parallel to the approach à 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.

Paper Structure

This paper contains 21 sections, 8 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Position approach
  3. Extensions of Hermitian operators
  4. Self-adjoint realizations of $L_\alpha$
  5. Point spectra
  6. Renormalization group
  7. "Wilsonian approach" to inverse square potential
  8. Momentum approach
  9. Momentum approach to Schrödinger operators on $\mathbb{R}^d$
  10. Momentum approach to Schrödinger operators on $\mathbb{R}_+$
  11. Minimal realization in the momentum approach
  12. Maximal realization in the momentum approach
  13. Self-adjoint realizations in the momentum approach I
  14. Self-adjoint realizations in the momentum approach II
  15. Conclusion
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Figures (1)

  • Figure 1: Schematic illustration of the renormalization group flow on the spaces of self-adjoint realizations of Bessel operators. It is borrowed from Derezinski3. Heavy dots represent fixed points. The letters $K$ and $F$ correspond to the Krein and Friedrichs extensions of $L_\alpha^{\min}$. Dashed lines represent realizations with a single bound state. Dotted line represents an infinite number of bound states. The arrow indicates the direction of the flow.

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem