International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 27, Pages 1437-1445
doi:10.1155/S0161171204203088
Spectral properties of the Klein-Gordon s-wave equation with spectral parameter-dependent boundary condition
Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara 06100, Turkey
Received 12 March 2002
Copyright © 2004 Gülen Başcanbaz-Tunca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.