Observability of the Schrödinger equation with subquadratic confining potential in the Euclidean space

We consider the Schrödinger equation in ‫ޒ‬ d , d ≥ 1, with a confining potential growing at most quadratically.Our main theorem characterizes open sets from which observability holds, provided they are sufficiently regular in a certain sense.The observability condition involves the Hamiltonian flow associated with the Schrödinger operator under consideration.It is obtained using semiclassical analysis techniques.It allows us to provide an accurate estimation of the optimal observation time.We illustrate this result with several examples.In the case of two-dimensional harmonic potentials, focusing on conical or rotationinvariant observation sets, we express our observability condition in terms of arithmetical properties of the characteristic frequencies of the oscillator.1. Introduction and main results 1147 2. Study of the classical dynamics 1171 3. Proof of the main theorem 1182 4. Proofs of observability results from conical sets 1187 5. Proofs of observability results from spherical sets 1201 Appendix A. Reduction to a weaker observability inequality 1219 Appendix B. Pseudodifferential operators 1222