The spectrum of relativistic one-electron atoms according to Bethe and Salpeter

Bethe and Salpeter introduced a relativistic equation — different from the Bethe-Salpeter equation — which describes relativistic multi-particle systems. Here we will begin some basic work concerning its mathematical structure. In particular we show self-adjointness of the one-particle operator which will be a consequence of a sharp Sobolev type inequality yielding semi-boundedness of the corresponding sesquilinear form. Moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum.