Harnack Estimates For The Schr(?)dinger Equation On Riemannian Manifolds

In this paper, On one hand, We establish the Gradient estimate about the positive solution of the Schrodinger equation when the metric is fixed on Rimannian manifold. The method is based on Bakry-Qian that solved the heat equation and extend the conclusion of Bakry-Qian. The Gradient estimate is different from the local estimate of Li-Yau, Because it is the global Gradient estimate. As applications, We conculd the fundmental solution's estimate of the Schrodinger equation and Liouville theorem about the Schrodinger operator by using the Harnack inequalities. On the other hand, We consider the Gradient estimate about the positive solution of the Schrodinger equation when the metric is evolved by Ricci flow. In the same way, We derived the Harnack inequality about the positive solution of the Schrodinger equation under Ricci flow.