An Elliptic Type Gradient Estimate For the Schr\"{o}dinger Equation

Abstract

In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schr\"{o}dinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schr\"{o}dinger equation are proved.