Square Integrable Solutions of $\Delta u+\lambda u=0$ on Noncompact Manifolds

Abstract

The Schr\"odinger-type equation $-\Delta u+Vu=\lambda u$ on a noncompact Riemannian manifold $\mathcal{M}$ has no nontrivial square integrable solution $u$ for any positive constant $\lambda$, if the metric and the function $V$ satisfy certain conditions near the infinity. A set of conditions of that kind was given by the author in the case that the metric is rotationally symmetric. It contained a condition which required smallness of the curvatures of $\mathcal{M}$ in the distance. But we have had a question whether the set could remain sufficient even if we remove that condition. The present paper answers it negatively by constructing a square integrable solution for a metric which satisfies all the conditions except the one in question.