SLAC-PUB-747 May 1970 (TH) METHODS I?OR THE I~ETlIE-SRLP~~TER EQUATION I: SPECIAL FUNCTIONS AND EXPANSIONS IN SPHERICAL HARMONICS*

Elements of the structure of the Bethe-Salpeter equation are studied. Properties of useful special functions are obtained and free particle solutions to the truncated expansion of the equation in four-dimensional spherical harmonics are derived in terms of known special functions. Validity of the truncation approximation is examined in terms of a convenient representation of the Green’s function. In particular, it is shown that the method of truncating the differential Bethe-Salpeter equation cannot succeed for scattering. The development of alternative procedures is deferred to the paper following. As a by-product, a simply computational technique for the approximation of integrals by Gaussian quadrature is derived, r (Submitted to Phys. Rev.) * Work supported in part by the U. S. Atomic Energy Commission and by the Air Force Office of Scientific Research.