On the finiteness of the Morse Index for Schr\"odinger operators

Abstract

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.