Quadratic Pencil of Schr¨odinger Operators with Spectral Singularities: Discrete Spectrum and Principal Functions

In this article we investigated the spectrum of the quadratic pencil of Schr¨o- dinger operators L l. generated in L2 Rq. by the equation yy0 qwV x.q2lU x.yl2xys0, xgRqs w0, `. and the boundary condition y 0. s0, where U, V are complex valued functions and U is absolutely continuous in each finite subinterval of Rq. Discussing the spectrum, we proved that L l. has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions sup exp eʹx.w