Quadratic Pencil of Schrödinger Operators with Spectral Singularities: Discrete Spectrum and Principal Functions

Abstract

In this article we investigated the spectrum of the quadratic pencil of Schrödinger operatorsL(λ) generated inL2(R+) by the equation[formula]and the boundary condition[formula] whereU,Vare complex valued functions andUis absolutely continuous in each finite subinterval of R+. Discussing the spectrum, we proved thatL(λ) has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions [formula]and[formula] hold. It is shown that the principal functions corresponding to eigenvalues ofL(λ) are in the spaceL2(R+), and the principal functions corresponding to spectral singularities are in another Hilbert space, which containsL2(R+). Some results about the spectrum ofL(λ) have also been applied to radial Klein–Gordon and one-dimensional Schrödinger equations.