Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds

TL;DR

This paper investigates the uniqueness of solutions to the parabolic Schrödinger equation on Riemannian manifolds, demonstrating that under certain conditions on the potential, the integrability requirements for solutions can be substantially relaxed.

Contribution

It introduces a novel approach linking the decay of stationary Schrödinger solutions to relaxed uniqueness conditions, advancing the understanding of potential's influence.

Findings

Relaxed integral conditions for uniqueness under certain potentials

Decay properties of stationary solutions are key to the new conditions

Potential's behavior significantly affects solution uniqueness

Abstract

We study uniqueness for solutions to the Cauchy problem associated with the parabolic Schr\"odinger equation on complete noncompact Riemannian manifolds, under suitable integral conditions on the solution. We show that, under suitable assumptions on the potential V, the required integrability condition can be significantly relaxed compared to the case without potential. This improvement is achieved by exploiting the decay of positive solutions to the associated stationary Schrodinger equation. To the best of our knowledge, identifying how the behavior of the potential influences the uniqueness integral condition, through the decay properties of solutions to the corresponding stationary equation, constitutes a novel contribution to the theory.