The non-self-adjointness of the radial momentum operator in n dimensions

TL;DR

This paper rigorously proves that the radial momentum operator in n dimensions is not self-adjoint and cannot be extended to a self-adjoint operator, clarifying a previously noted but unproven property.

Contribution

It provides the first rigorous proof that the n-dimensional radial momentum operator is not self-adjoint and lacks self-adjoint extensions.

Findings

Radial momentum operator is not self-adjoint.

Radial momentum operator has no self-adjoint extensions.

Operator is unitarily equivalent to a non-self-adjoint momentum operator.

Abstract

The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the $n$-dimensional radial momentum operator is not self- adjoint and has no self-adjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on $L^{2}[(0,\infty),dr]$ which is not self-adjoint and has no self-adjoint extensions.