A Liouville type theorem for the Schrödinger operator

In this paper we prove that the equation $\Delta u(x)+h(x)u(x)=0$ on a complete Riemannian manifold of dimension $n$ without boundary and with nonnegative Ricci curvature admits no positive solution provided that $h$ is a $C^2$ function satisfying $\limsup _{r\rightarrow\infty}r^{-2}\inf _{x\in B_p(r)}h(x) \geq-b_na^2$ and $\Delta h(x)\geq-c_na^2$ where $0\leq a<\sup _{x\in M}h(x)$, and $b_n,c_n$ are constants depending only on the dimension, thus generalizing similar results in P. Li and S. T. Yau (Acta Math. 156 (1986), 153-201), J. Li (J. Funct. Anal. 100 (1991), 233-256) and E. R. Negrin (J. Funct. Anal. 127 (1995), 198-203) in all of which $h$ is assumed to be subharmonic. We also give a generalization in case the Ricci curvature of $M$ is not necessarily positive but its negative part has quadratic decay under the additional assumption that $h$ is unbounded from above.