Completeness of trajectories associated to Appell hypergeometric functions

Abstract

We examine the linear completeness of trajectories of eigenfunctions associated to non-linear eigenvalue problems, subject to Dirichlet boundary conditions on a segment. We pursue two specific goals. On the one hand, we establish that linear completeness persists for the non-linear Schr\"odinger equation, even when the trajectories lie far from those of the linear equation where bifurcations occur. On the other hand, we show that this is also the case for a fully non-linear version of this equation which is naturally associated with Appell hypergeometric functions. Both models shed new light on a framework for completeness in the non-linear setting, considered by L.E. Fraenkel over 40 years ago, that may have significant potential but which does not seem to have received much attention.