On the finiteness of the Morse Index for Schrödinger operators

Details On the finiteness of the Morse Index for Schrödinger operators On the finiteness of the Morse Index for Schrödinger operators

2010-11-15

arxiv.org/abs/1011.3390v2

Mathematics

Differential Geometry

17 pages

Scientific paper

Let H=$\Delta +V$ be a Schr\"odinger 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 $\phi$ solution of $H\phi=0$ 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 $\phi$ solution of $H\phi=0$ outside a compact set, then $H$ has a finite number of negative eigenvalues.

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