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

• Main Author: Gülen Başcanbaz-Tunca
• Format: Article
• Language: English
• Published: Hindawi Limited 2004-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/S0161171204203088

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.