] 7.A [ Nuclear Physics A147 (1970) 215--224; ~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or rnierofilm without written permission from the publisher A SPECTRAL THEORY FOR THE KLEIN-GORDON EQUATION WITH AN EXTERNAL ELECTROSTATIC POTENTIAL K. VESELIC Institute "Ruder Bo§kovi{", Zagreb and Institute for Mathematics, University of Zagreb Received 22 January 1970 Abstract: It has been shown that the energy operator for the Klein-Gordon particle in the presence of an external electrostatic potential satisfying some suitable conditions possesses a spectral decomposition. The spectrum is shown to cover (--~, --mc 2] U [mc 2, ~), whereas the gap (--mc 2, mc 2) possibly contains isolated eigenvalues, each of them having a finite multiplicity. It has also been proved that in the non-relativistic limit the spectral projections of the Klein- Gordon Hamiltonian tend in a sense to those of the Schr6dinger operator with the same potential. 1. Introduction In this paper we shall consider the spectral properties of the Klein-Gordon equation describing the motion of a spinless relativistic particle in an external electrostatic field. In writing the Klein-Gordon equation in the Hamiltonian form we use the method of Feshbach-Villars 1), in which the Hamiltonian is symmetric with respect to an in- definite Hermitian form. The main problem we are concerned with is under what conditions of the potential the Hamiltonian possesses a spectral decomposition like an ordinary self-adjoint operator. It is physically plausible that such a condition should ensure the separation of a positive and a negative spectrum 2). In sect. 1, by applying the general theory developed in ref. 3) we prove that under a suitable condition on the potential the Klein-Gordon Hamiltonian is of scalar type with a real spectrum. In other words, we are able to construct a new, positive definite scalar product with respect to which the Hamiltonian is self-adjoint. Although this scalar product varies with the potential, the scalar products for all admissible poten- tials are topologically equivalent. On the other hand, if restricted to the space of positive energies of the Hamiltonian with a given potential, the definite scalar product and the indefinite form coincide (in the space of negative energies they are of opposite sign). The condition on the potential is rather strong and includes neither the square well potential nor the Yukawa potential. Taking the Gaussian potential as an example we show that this condition is fulfilled if the potential is not too deep and if its range is not too small in comparison with the Compton wavelength. 215