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We construct the one-dimensional analogous of von-Neumann Wigner potential tothe relativistic Klein-Gordon operator, in which is defined taking asymptoticmathematical rules in order to obtain existence conditions of eigenvaluesembedded in the continuous spectrum. Using our constructed potential, weprovide an explicit and analytical example of the Klein-Gordon operator withpositive eigenvalues embedded in the so called relativistic "continuum region".Even so in this not standard example, we present the region of the "continuum"where those eigenvalues cannot occur. Besides, the absence of eigenvalues inthe continuous spectrum for Klein-Gordon operators is proven to a broad generalpotential classes, including the minimally coupled electric Coulomb potential.Considering known techniques available in literature for Schrodinger operators,we demonstrate an expression for Klein-Gordon operator written in Schrodinger'sform, whereby is determined the mathematical spectrum region of absence ofeigenvalues.

Absence of eigenvalues in the continuous spectrum for Klein-Gordon operators