Homogeneous Rank One Perturbations and Inverse Square Potentials

Abstract

Following [2,4,6], I describe several exactly solvable families of closed operators on L 2 [0, ∞[. Some of these families are defined by the theory of singular rank one perturbations. The remaining families are Schrödinger operators with inverse square potentials and various boundary conditions. I describe a close relationship between these families. In all of them one can observe interesting "renormalization group flows" (action of the group of dilations).Mathematics Subject Classification (2010). 34L40, 33C10.

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

“…Here is an illustration of these flows borrowed from [3], hopefully self-explanatory. The names of the various phases can be treated as jokes.…”