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.

DOI: http://dx.doi.org/10.7900/jot.2022mar16.2385
Keywords: smooth fields of Hilbert spaces and operators, connections, reduction theorem, constant of motion, Weyl quantization

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