An impulsive problem for quadratic pencil of Schrödinger operators on semi-axis

The present article examines a quadratic pencil of Schrödinger operator given with an impulsive condition. The novelty of this study lies in its focus on the half-line and inclusion of an arbitrary single discontinuity point, distinguishing it from previous works in the literature. First, some solutions to the impulsive equation associated with so-called operator $\mathcal{L}$ are presented. Next, the Jost solutions and resolvent operator of $\mathcal{L}$ are defined. The primary work includes deriving an asymptotic equation for the function related to the Wronskian of the Jost solutions, providing sufficient conditions to ensure the finiteness of eigenvalues and spectral singularities, and demonstrating that their multiplicities are finite.