2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129593In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schrödinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schrödinger equation are proved.Differential GeometryAnalysis of PDEs58G11, 53C21text