Non-real eigenvalues of singular indefinite Sturm-Liouville operators

Abstract

We study a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on \mathbb{R} and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at \pm \infty we prove that there are no real eigenvalues and the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to −d^2/dx^2 + V in L^2(\mathbb{R}). The general results are illustrated with examples.