Spectral properties of singular Sturm—Liouville operators with indefinite weight sgn x

Abstract

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d 2 /dx 2 + q) with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.

Accumulation of complex eigenvalues of indefinite Sturm–Liouville operators

“…Our first theorem, which also follows from the considerations in [3] and [15], summarizes the spectral properties of the indefinite Sturm-Liouville operator A. Recall that for a selfadjoint operator in a Krein space the essential spectrum σ ess consists of all spectral points which are no isolated eigenvalues with finite-dimensional algebraic eigenspaces.…”

“…We mention that an existence result on such potentials was already proved in [15]. The models considered here arise from completely solvable models on the half axes, so that eigenvalues and eigenfunctions can be computed by using standard software packages, e.g.…”

Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications

“…Darboux Transformation for Hamiltonian Systems 549 equations, non-real eigenvalues are of interest (see [6,7,11,26,27,41] and references therein). Thus, the insertion (and the removal as an operation which is inverse to the insertion) of non-real eigenvalues is of interest as well.…”

“…(9.1) (Assumptions (9.1) were made, for instance, in [26].) First, we consider Shin-Zettl system (1.1), (5.4) and assume that the equalities (9.1) and…”

“…We note that it is often required that u (from D) is absolutely continuous everywhere on (−∞, ∞) (see [26]), although the requirements in [7] are somewhat weaker. Proposition 8.3 yields the following corollary.…”