On the spectral singularities of Klein-Gordon equation under interface conditions

Abstract

In this study, we begin by deriving the differential operator associated with the Klein-Gordon  equation under interface conditions. Next, we introduce the transfer matrix and the resolvent operator for the Klein-Gordon operator with interface conditions, observing that the zeros of a component of the transfer matrix coincide with the poles of the resolvent operator. Building on this observation, we use the transfer matrix to characterize the eigenvalues and spectral singularities of mentioned operator through an alternative approach. Finally, we establish the finiteness of eigenvalues and spectral singularities, with finite multiplicities, under certain specific conditions.