Weyl pseudo-differential operator and Wigner transform on the Poincaré disk
type: Item
instance of: scholarly article
title: Weyl pseudo-differential operator and Wigner transform on the Poincaré disk
zbMATH Open document ID: 1022.58011
DOI: 10.1023/A:1016253829938
author: Tatsuya Tate
published in: Annals of Global Analysis and Geometry
publication date: 2002-08-19T00:00:00Z
review text: The author continues his study on the Weyl calculus and the Wigner transform of the Poincaré disk begun in his paper [Math. Phys. Stud. 23, 227-243 (2001)]. In the Euclidean frame, the Wigner transform is nothing but the inverse of the map which associates to a symbol on \( T ^{\ast}\mathbb{R}^n\) (with appropriate behaviour at infinity) the kernel of the associated pseudo-differential operator in the Weyl-quantization. Recall that if the symbol is \(a(x, \xi)\), this kernel is \( (2 \pi) ^{-n}\int_{\mathbb{R}^n} \exp [ i \langle x-y , \xi \rangle] a((x+y)/2, \xi) d \xi \). To define a related map on the Poincaré disk, the author replaces the Fourier-transform in \(\mathbb{R}^{n}\) by the non-Euclidean Fourier transform on the Poincaré disk and \((x+y)/2\) by the middle point \(m(x,y)\) on the geodesic in the Poincaré metric through \(x\) and \(y\). He obtains among others an asymptotic formula as \( h \rightarrow 0\) for the Wigner transform of the kernel of a given Weyl pseudo-differential operator.
reviewed by: Otto Liess
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 1783478
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Wigner transform
zbMATH Keywords: Poincaré disk
zbMATH Keywords: Weyl calculus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1783478