On Hansen's version of spectral theory and the Moyal product
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 presented technique extends the work of Antonets and must be made compatible with that of Daubechies. The authors study how the space \(D'_b\) can be brought into the formalism of this extended product. Finally, the authors succeed to combine the two approaches into a single new one. The authors conclude ``there is, however, the possibility that classes of more appropriate distributions do exist and could be discovered''.
scientific article; zbMATH DE number 5673115
scientific article