Schr\"odinger semigroups and the H\"ormander hypoellipticity condition

TL;DR

This paper establishes the existence, uniqueness, and dispersive properties of solutions to a class of degenerate dispersive equations with drift under H"ormander's hypoellipticity condition, extending analysis in $L^2$ and $L^p$ spaces.

Contribution

It introduces a new class of degenerate dispersive equations with drift and proves well-posedness and dispersive estimates under H"ormander's hypoellipticity condition.

Findings

Unique solvability in Schwartz class

Extension of solution operator to a strongly continuous semigroup in $L^2$

Sharp dispersive estimates and uncertainty principle for $t>0$

Abstract

We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the H\"ormander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal T(t)\}_{t\ge 0}$ in $L^2(\Rm)$. Finally, we prove that for $t>0$ the operator $\mathcal T(t)$ satisfies a sharp form of dispersive estimate in $L^p$, for any $1\le p\le 2$, and an uncertainty principle.