Regularized quantum motion in a bounded set: Hilbertian aspects

TL;DR

This paper demonstrates that by applying a symmetric weighting function to the momentum operator, essential self-adjointness can be restored for a quantum particle in a bounded interval, using a specific quantization method.

Contribution

It introduces a novel approach of weighting the momentum operator to recover self-adjointness in bounded domains via Weyl-Heisenberg covariant quantization.

Findings

Weighted momentum operator is essentially self-adjoint with bounded weights.

The approach connects classical weighted momentum to quantum operators.

Method applies to bounded quantum systems with boundary conditions.

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-Heisenberg covariant integral quantization of functions or distributions.