On the finiteness of the Morse Index for Schrödinger operators

2010/11/15 by Baptiste Devyver, Devyver, Baptiste · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1011.3390

openalex publication_date 2010/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H=Δ+V be a Schrödinger on a complete non-compact manifold. It is known since the work of Fischer-Colbrie and Schoen that the finiteness of the negative spectrum of H implies the existence of a function ϕ solution of Hϕ=0 outside a compact set. This has consequences for minimal surfaces and for the finiteness of spaces of harmonic sections in the Bochner method. Here we show that the converse statement also holds: if there exists ϕ solution of Hϕ=0 outside a compact set, then H has a finite number of negative eigenvalues.

Cited by

• Caccioppoli's inequalities on constant mean curvature hypersurfaces in\n Riemannian manifolds
• Maximizing higher eigenvalues in dimensions three and above

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