Dispersive estimates for the Schrödinger 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 Schrödinger 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 Schrödinger operator on a space of the same dimension $k$ as the radical of the canonical skew-symmetric form, which suggests a decay rate $|t{|}^{-\left(k+p-1\right)∕2}$. 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.