Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with a General Boundary Condition

Abstract

In this article we investigate the spectrum and the spectral singularities of the Quadratic Pencil of Schrödinger OperatorLgenerated inL2(R+) by the differential expressionℓ(y)=−y″+[q(x)+2λp(x)−λ2]y,x∈R+=[0,∞)and the boundary condition∫∞0K(x)y(x)dx+αy′(0)−βy(0)=0,wherep,q, and K are complex valued functions, p is continuously differentiable onR+,K∈L2(R+), andα,β∈C, with |α|+|β|≠0. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditionslimx→∞p(x)=0,supx∈R+{eεx[|q(x)|+|p′(x)|+|K(x)|]}<∞,ε>0.Later we investigate the properties of the principal functions corresponding to the spectral singularities. Moreover, some results about the spectrum ofLare applied to non-selfadjoint Sturm–Liouville and Klein–Gordons-wave operators.