Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with Boundary Conditions Dependent on the Eigenparameter.

In this paper, we consider the following quadratic pencil of Schrödinger operators L(λ) generated in L 2 (ℝ +) by the equation with the boundary condition y ′ (0) y (0) = β 1 λ + β 0 α 1 λ + α 0 , where p(x)and q(x) are complex valued functions and α0, α1, β0, β1 are complex numbers with α 0 β 1 − α 1 β 0 ≠ 0 . It is proved that L(λ) has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions and sup 0 ≤ x < + ∞ { e ε x [ | p ′ (x) | + | q ′ ′ (x) | ] } < + ∞ hold, where ε> 0. [ABSTRACT FROM AUTHOR]