An Eigenfunction Expansion for a Quadratic Pencil of a Schrödinger Operator with Spectral Singularities

In this paper, we consider the operatorLgenerated inL2(R+) by the differential expressionℓ(y)=−y″+[q(x)+2λp(x)−λ2] y, x R+=[0, ∞),and the boundary conditiony(0)=0, wherepandqare complex-valued functions andpis continuously differentiable onR+. We derive a two-fold spectral expansion ofL(in the sense of Keldysh, 1951,Soviet Math. Dokl.77, 11–14 [1971,Russian Math. Survey26, 15–44 (Engl. transl.)]) in terms of the principal functions under the conditions taking into account the spectral singularities. Also we investigate the convergence of the spectral expansion.