Tokyo Journal of Mathematics

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


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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.

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Tokyo J. Math., Volume 25, Number 2 (2002), 285-294.

First available in Project Euclid: 5 June 2009

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KONNO, Reiji. Square Integrable Solutions of $\Delta u+\lambda u=0$ on Noncompact Manifolds. Tokyo J. Math. 25 (2002), no. 2, 285--294. doi:10.3836/tjm/1244208854.

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