Weyl Calculus and Wigner Transform on the Poincaré Disk

Abstract

The aim of this paper is to establish an analogue of Weyl calculus on the Poincaré disk. The Weyl calculus is defined by functional calculus with respect to the Schrödinger representation of the Heisenberg group, and the Wigner transform is defined to be the Fourier transform of the matrix coefficients of this representation. As in [1], [3], the Weyl calculus and the Wigner transform are closely related to each other. In this paper we will define an analogue of the Weyl calculus on the Poincaré disk by using the middle point m(z, w) on the geodesic through z and w. We will also define the Wigner transform by taking into account a formula describing a relationship between the matrix coefficients of a Weyl pseudo-differential operator and the Wigner transform on Euclidean space.