Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds

We study uniqueness for solutions to the Cauchy problem associated with the parabolic Schrödinger 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.