In this paper we investigate an abstract Klein-Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space K induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the 'size' of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space K. Finally, the conditions on the potential required in the paper are illustrated for the Klein-Gordon equation in R n ; they include potentials consisting of a Coulomb part and an L p -part with n ≤ p < ∞.

In this paper the spectral properties of the abstract Klein-Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V , two operators are associated with the Klein-Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein-Gordon equation in R n .

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Consider the Klein-Gordon equation (1) where q, Aj are real valued functions on R", m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.AssumeDenote by K the operator of multiplication by eq in ^ -L 2 (R n ); K is a bounded selfadjoint operator. H is the natural selfadjoint realisation of The equation (1)

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