Regularized quantum motion in a bounded set: Hilbertian aspects. (English) Zbl 07915181
Summary: 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.
MSC:
| 81Sxx | General quantum mechanics and problems of quantization |
| 47Bxx | Special classes of linear operators |
| 81Rxx | Groups and algebras in quantum theory |
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