On the finiteness of the Morse index for Schrödinger operators (Q451991)

Roughly speaking, this paper is devoted to the proof that on a complete Riemannian manifold \(M\), if \(H= -\Delta+ V\) is a Schrödinger operator, the existence of a positive solution to the problem \(Hu=0\) outside a compact subset is equivalent to the finiteness of the Morse index of \(H\). More precisely, the main result can be stated as Theorem 1.3. Let \(M\) be a complete Riemannian manifold. Let \(V\in L^\infty_{\text{loc}}\) and denote \(H= \Delta+ V\) the corresponding Schrödinger operator, which we assume to be bounded from below. The following assertions are equivalent: 1) \(H\) has finite Morse index; 2) There exists a positive smooth function \(\phi\) in \(W^{1,2}_{\text{loc}}\) which satisfies \(H\phi=0\) outside a compact set. A corollary concerning minimal surfaces is obtained. All these results are related with the so-called Bochner technique and the notion of parabolicity of a manifold.