Regularized quantum motion in a bounded set: Hilbertian aspects

Abstract

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self- adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenb erg covariant integral quantization of functions or distributions