Equivalence of renormalization with self-adjoint extension in Green's function formalism

TL;DR

This paper demonstrates that renormalization and self-adjoint extension methods produce equivalent Green's functions for certain quantum systems when a specific relation between parameters is satisfied, unifying two approaches in quantum mechanics.

Contribution

It establishes the equivalence of renormalization and self-adjoint extension in Green's function formalism for delta-function plus harmonic oscillator potentials.

Findings

Both methods yield identical Green's functions under a specific parameter relation.

The work unifies renormalization and self-adjoint extension approaches.

Provides explicit derivation for 2D and 3D systems.

Abstract

Energy-dependent Green's functions for the two and three dimensional $\delta$-function plus harmonic oscillator potential systems are derived by incorporating the renormalization and the self-adjoint extension into the Green's function formalism, respectively. It is shown that both methods yield an identical Green's function if a certain relation between the self-adjoint extension parameter and the renormalized coupling constant is imposed.