On Hansen's version of spectral theory and the Moyal product
type: Item
instance of: scholarly article
title: On Hansen's version of spectral theory and the Moyal product
zbMATH Open document ID: 1204.81102
author: M. A. Hennings
author: Daniel A. Dubin
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2010-02-19T00:00:00Z
review text: The paper under review deals with the problem of a rigourous treatment of the Weyl quantization and the Moyal product. The main difficulty of extending the definition of the Moyal product from functions on \(L^2(E)\), \(E=\mathbb{R}^d\), to tempered distributions on \({\mathcal S}(\mathbb R^{2d})\), where in order to simplify the notation, \(d=1\), comes from the fact this cannot be done for all distributions. Previously, \textit{M. A. Antonets} [``Classical limit of the Weyl quantization'', Theoret. Math. Fiz. 38, No. 3, 331--344 (1979)] considered the \(^*\)-algebra \({\mathcal W}_{\hbar}\), here denoted \(N\cap\bar{N}\), of the \(^*\)-multipliers of test functions, while \textit{I. Daubechies} [Commun. Math. Phys. 75, 229--238 (1980; Zbl 0451.47059)] considered the class of \(D'_b\) distributions whose Weyl quantizations are bounded, which cannot handle the basic position and momentum distributions. In the paper under review, both classes \(N\cap\overline{N}\) and \(D'_b\) are extended to a larger class \(Q\cap\overline{Q}\) on which it is possible to define a Moyal product, the price being that with respect to this Moyal product, \(Q\cap\overline{Q}\) is only a partial *-algebra [\textit{J.-P. Antoine, A. Inoue} and \textit{C. Trapani}, Rev. Math. Phys. 8, No. 1, 1--42 (1996; Zbl 0857.47029)]. Several Moyal-type products are introduced, including different left and right products thereby denying associativity, several resolvent candidates, none of them analytic in the resolvent. In [Publ. Res. Inst. Math. Sci. 26, No. 6, 885--933 (1990; Zbl 0748.46043)], \textit{F. Hansen} proposed a method for transferring spectral theory form operators on a Hilbert space to distributions on the phase space themselves. Hansen suggested that it is possible to extend the Moyal product sufficiently so that one could effectively determine a spectral representation for a symmetric, but not necessarily self-adjoint operator. Hansen's program is pushed on without associativity. The pr…
reviewed by: Stefan Berceanu
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 81S30
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5673115
zbMATH Keywords: Weyl quantization
zbMATH Keywords: Moyal product
zbMATH Keywords: Wigner transform
zbMATH Keywords: tempered distributions
zbMATH Keywords: phase space
zbMATH Keywords: spectral theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1260476653
OpenAlex ID: W2035109146
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DOI: 10.2977/PRIMS/1260476653
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description: scientific article; zbMATH DE number 5673115