Boundedness of Schr{ö}dinger operator in energy space

On a complete weighted Riemannian manifold $(M^n,g,\mu )$ satisfying the doubling condition and the Poincar{é} inequalities, we characterize the class of function $V$ such that the Schr{ö}dinger operator $\mathrm{\Delta }-V$ 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.