Observability of Schrödinger equations in Euclidean space

In this paper we introduce a new dynamical condition, the comb geometric control condition, which is sufficient for observability of the Schrödinger equation in Euclidean space. We provide examples which show this condition is strictly weaker than the observation set being open and periodic. We also prove for the fractional Schrödinger equation that for observation functions which are uniformly continuous, the geometric control condition is equivalent to observability and implies arbitrary time observability. The proofs rely on uncertainty principles for frequency localized functions which are proved using a semiclassical propagation of singularities approach.