Uncertainty principle, minimal escape velocities, and observability inequalities for Schrödinger Equations

We develop a new abstract derivation of the observability inequalities at two points in time for Schr\"{o}dinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\Bbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schr\"{o}dinger equation.