Spectral properties of the Klein-Gordon s-wave equation with spectral parameter-dependent boundary condition

We investigate the spectrum of the differential operator Lλ defined by the Klein-Gordon s-wave equation y′′ +(λ−q(x))2y = 0, x ∈R+ = [0,∞), subject to the spectral parameterdependent boundary condition y′(0)−(aλ+b)y(0)= 0 in the space L(R+), where a≠±i, b are complex constants, q is a complex-valued function. Discussing the spectrum, we prove that Lλ has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions limx→∞q(x) = 0, supx∈R+{exp(ε √ x)|q′(x)|} < ∞, ε > 0, hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.