A Liouville type theorem for the Schrödinger operator

Abstract:

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.