Spectral radius, index estimates for Schrödinger operators and geometric applications

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• 作者：Bruno Bianchini ; Luciano Mari ; Marco Rigoli
• 关键词：Spectral radius ; Index estimates ; Minimal surfaces ; Positioning of zeroes
• 刊名：Journal of Functional Analysis
• 出版年：2009
• 出版时间：15 March 2009
• 年：2009
• 卷：256
• 期：6
• 页码：1769-1820
• 全文大小：401 K

文摘

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation (vz^′)^′+Avz=0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.