On the finiteness of the Morse Index for Schrödinger operators

Baptiste Devyver (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: Bochner; harmonic functions; parabolicity; Schrödinger operator; Morse Index; negative eigenvalues

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Baptiste Devyver. On the finiteness of the Morse Index for Schrödinger operators. 2010. ⟨hal-00535982v1⟩