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Journal of Operator Theory

Volume 91, Issue 1, Winter 2024  pp. 203-238.

Smooth fields of operators, a smooth reduction theorem, and a derivation formula for constants of motion of the free Hamiltonian

Authors:  Fabian Belmonte (1), Harold Bustos (2), Sebastian Cuellar (3)
Author institution:(1) Departamento de Matematicas, Universidad Catolica del Norte, Antofagasta, 1240000, Chile
(2) Centro de Docencia de Ciencias Basicas para Ingenieria, Universidad Austral de Chile, Valdivia, 5090000, Chile
(3) Departamento de Matematicas, Universidad\break Catolica del Norte, Antofagasta, 1240000, Chile

Summary:  We introduce a notion of smooth fields of operators following the notion of smooth fields of Hilbert spaces defined by L. Lempert and R. Szoke. We show that, if $\nabla$ is the connection of a smooth field of Hilbert spaces, then $\widehat\nabla=[\nabla,\cdot]$ defines a connection on a suitable space of fields of operators. In order to provide examples, we prove a smooth version of the reduction theorem. We show under mild conditions that, if $h(q,p)=\|p\|^2$ and $\{u,h\}=0$ then $\mathfrak{Op}(u)$ is an operator commuting with $\mathrm e^{-\mathrm it\Delta}$ and admitting a decomposition as a smooth field of operators over the open interval $(0,\infty)$, where $\mathfrak{Op}$ denotes the canonical quantization (Weyl calculus) and $\Delta$ is the Laplace operator on $L^2(\mathbb{R}^n)$. Moreover, we prove that we can compute derivatives using the formula $\widehat\nabla_{X_0}(\mathfrak{Op}(u))=\mathfrak{Op}(\{h_0,u\})$, where $h_0(q,p)=\sum q_j p_j$ and $X_0=2\lambda\frac{\partial}{\partial \lambda}$. We also introduce a notion of smooth field of $C^*$-algebras and we give an example using Hilbert modules theory.

Keywords:  smooth fields of Hilbert spaces and operators, connections, reduction theorem, constant of motion, Weyl quantization

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