Weyl pseudo-differential operator and Wigner transform on the Poincaré disk
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.
scientific article; zbMATH DE number 1783478