Dunkl-Schrödinger semigroups and applications

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.

Keywords:

• Dunkl–Schrödinger semigroups
• Strichartz estimates
• Nonlinear Dunkl–Schrödinger equation
• local and global well-posedness

AMS Subject Classifications:

• Primary: 35L05
• Secondary: 22E30

Acknowledgements

The author gratefully acknowledges the Deanship of Scientific Research at the University of King Faisal on material and moral support in the financing of this research project No. 130172. The author is deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article.