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

TL;DR

This paper establishes an elliptic gradient estimate for solutions to the Schrödinger equation on noncompact manifolds, leading to dimension-free inequalities and Liouville theorems, advancing understanding of geometric analysis in quantum mechanics contexts.

Contribution

It introduces a new elliptic type gradient estimate for Schrödinger equations on noncompact manifolds, with applications to Harnack inequalities and Liouville theorems.

Findings

Proved a dimension-free Harnack inequality for Schrödinger equations.

Established Liouville type theorems under certain conditions.

Developed an elliptic gradient estimate applicable to noncompact manifolds.

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.