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Dispersive estimates for the Schrödinger operator on step-2 stratified Lie groups

2016; Mathematical Sciences Publishers; Volume: 9; Issue: 3 Linguagem: Inglês

10.2140/apde.2016.9.545

2157-5045

Hajer Bahouri, Clotilde Fermanian Kammerer, Isabelle Gallagher,

Mathematical Analysis and Transform Methods

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| -(k+ p-1)/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.