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December 2002 Square Integrable Solutions of $\Delta u+\lambda u=0$ on Noncompact Manifolds
Tokyo J. Math. 25(2): 285-294 (December 2002). DOI: 10.3836/tjm/1244208854


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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Reiji KONNO. "Square Integrable Solutions of $\Delta u+\lambda u=0$ on Noncompact Manifolds." Tokyo J. Math. 25 (2) 285 - 294, December 2002.


Published: December 2002
First available in Project Euclid: 5 June 2009

zbMATH: 1028.58031
MathSciNet: MR1948665
Digital Object Identifier: 10.3836/tjm/1244208854

Rights: Copyright © 2002 Publication Committee for the Tokyo Journal of Mathematics

Vol.25 • No. 2 • December 2002
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