Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups

In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesies project to circles on the contact plane at the identity.