Abstract: Uniform resolvent estimates play a fundamental role in the study of spectral and scattering theory for Schrödinger equations. In particular, they are closely connected to global-in-time contrast with the Euclidean setting, a peculiar fact of the Schrödinger evolution equation associated with the sublaplacian on the Heisenberg group is that it fails to be dispersive, as shown by Bahouri, Gérard, and Xu. In fact, Strichartz or $L^p-L^q$ estimates cannot hold in general. In this talk, we will discuss uniform resolvent estimates on the Heisenberg group and their application to obtain certain smoothing effects for Schrödinger equations.

Joint work with Luca Fanelli, Haruya Mizutani, and Nico Michele Schiavone.