Pseudodifferential Phase-Space Dynamics for SU(1,1) Systems and Numerical Evaluation Using Oscillatory Integrals

We study the phase-space dynamics of quantum systems with SU ( 1 , 1 ) group symmetry using coherent-state representations on the Poincaré disk. The resulting evolution equation combines transport terms with nonlocal contributions generated with the spectral functions of the Casimir operator, which admit a natural interpretation as pseudodifferential operators associated with the hyperbolic Laplace–Beltrami operator. Using this pseudodifferential structure, we classify the phase-space generators according to the type of the underlying PDE: compact quadratic dynamics ( H ^ ∝ K ^ 0 2 ) yield a degenerate hyperbolic operator of the transport type, and noncompact dynamics ( H ^ ∝ K ^ 2 2 ) give rise to a mixed-order differential–pseudodifferential operator. For numerical evaluation, we reformulate the propagator as an oscillatory integral and develop two complementary strategies: a Fourier-series reduction exploiting the periodicity of compact orbits and a Levin-type spectral collocation method for the noncompact case. Both approaches are stable, accurate, and free of the stiffness issues that afflict direct PDE evolution on the Poincaré disk.