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

Abstract

We investigate the spectrum of the differential operator L λ defined by the Klein-Gordon s -wave equation y ″ + ( λ − q ( x ) ) 2 y = 0 , x ∈ ℝ + = [ 0 , ∞ ) , subject to the spectral parameter-dependent boundary condition y ′ ( 0 ) − ( a λ + b ) y ( 0 ) = 0 in the space L 2 ( ℝ + ) , where a ≠ ± i , b are complex constants, q is a complex-valued function. Discussing the spectrum, we prove that L λ has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions lim x → ∞ q ( x ) = 0 , sup x ∈ R + { exp ( ϵ x ) | q ′ ( x ) | } ∞ , 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> ϵ > 0 , hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.