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
Bagarello F., Gazeau J.P., Trapani C. (2024). Regularized quantum motion in a bounded set: Hilbertian aspects. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 540(1) [10.1016/j.jmaa.2024.128631].
Regularized quantum motion in a bounded set: Hilbertian aspects
Bagarello F.
;Trapani C.
2024-01-01
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 distributionsFile | Dimensione | Formato | |
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