Spectral Properties, Including Spectral Singularities, of a Quadratic Pencil of Schrödinger Operators on the Whole Real Axis

Abstract

We consider the Quadratic Pencil of Schr¨ odinger Operator L generated in L 2 (R) by the differential expression l(y) = − y'' + q [q(x) + 2 λp (x) − λ 2]y, x ∈ R = (−∞, ∞), where p and q are complex valued functions. Using the uniqueness theorems of analytic functions, we investigate the dependence of the structure of eigenvalues and spectral singularities of L on the behavior of p and q at infinity. We also obtain the conditions on p and q under which the operator L has a finite number of eigenvalues and spectral singularities with finite multiplicities. The results about the discrete spectrum of L are applied to non-selfadjoint Sturm-Liouville and Klein-Gordon operators on the whole real axis.