Boundedness of Schr{\"o}dinger operator in energy space

TL;DR

This paper characterizes the conditions on potential functions V that ensure the Schrödinger operator Δ−V is bounded between specific Sobolev spaces on weighted Riemannian manifolds satisfying certain geometric conditions.

Contribution

It extends known Euclidean results to weighted Riemannian manifolds, providing a characterization of potentials V for Schrödinger operators in this setting.

Findings

Characterization of potential functions V for bounded Schrödinger operators on manifolds.

Analysis of weighted L^2-boundedness of the Hodge projector.

Extension of Euclidean results to geometric settings.

Abstract

On a complete weighted Riemannian manifold $(M^n,g,\mu)$ satisfying the doubling condition and the Poincar{\'e} inequalities, we characterize the class of function $V$ such that the Schr{\"o}dinger operator $\Delta-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.