Abstract: This paper studies the well-posedness and regularity of Schr?dinger equation with variable coefficients.Under Dirichlet boundary control and collocated observation, it is proved that, with the help of the multiplier method on Riemannian manifolds, the system is well-posedness in the sense of D.Salamon and regular in the sense of G.Weiss.The feedthrough operator is found to be zero.