The non-self-adjointness of the radial momentum operator inndimensions

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, ), dr which is not self-adjoint and has no self-adjoint extensions.