Dispersive estimates for the Schr\"odinger operator on step 2 stratified Lie groups

Abstract

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schr\"odinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schr\"odinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay with exponant -(k+p-1)/2. In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples.