  We construct the one-dimensional analogous of von-Neumann Wigner potential to
the relativistic Klein-Gordon operator, in which is defined taking asymptotic
mathematical rules in order to obtain existence conditions of eigenvalues
embedded in the continuous spectrum. Using our constructed potential, we
provide an explicit and analytical example of the Klein-Gordon operator with
positive 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 in
the continuous spectrum for Klein-Gordon operators is proven to a broad general
potential 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's
form, whereby is determined the mathematical spectrum region of absence of
eigenvalues.
