Abstract
We obtain dispersive estimates for the linear Dunkl-Schrödinger equations with and without quadratic potential. As a consequence, we prove the local well-posedness for semilinear Dunkl-Schrödinger equations with polynomial nonlinearity in certain magnetic field. Furthermore, we study many applications: as the uncertainty principles for the Dunkl transform via the Dunkl-Schrödinger semigroups, the embedding theorems for the Sobolev spaces associated with the generalized Hermite semigroup. Finally, almost every where convergence of the solutions of the Dunkl-Schrödinger equation is also considered.