Observability of a 1D Schr\"odinger equation with time-varying boundaries

We discuss the observability of a one-dimensional Schr"odinger equation on certain time dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we provide exact boundary observability when the curve satisfies some certain conditions . By duality theory, we establish the controllability of adjoint system.