International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 25-28, Pages 1437-1445
doi:10.1155/S0161171204203088

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

Abstract

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 parameter-dependent boundary condition y′(0)−(aλ+b)y(0)=0 in the space L2(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(ϵx)|q′(x)|}<∞, ϵ>0, hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.