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

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 \ T(t)≥ 0 in L²(). Finally, we prove that for $t>0$ the operator T(t) satisfies a sharp form of dispersive estimate in Lᵖ, for any 1≤ p≤ 2, and an uncertainty principle.