On the finiteness of the Morse Index for Schrödinger operators

Abstract

Let H=$\Delta +V$ 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 $\varphi$ solution of $H\varphi =0$ 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 $\varphi$ solution of $H\varphi =0$ outside a compact set, then $H$ has a finite number of negative eigenvalues.

17 pages