Masakuni Ida

An Attempt at a General Theory of the Bethe-Salpeter Equation

The Bethe-Salpeter equation is investigated under some general assumptions, without using specific approximations. We assume 1) the validity of integral representations for the interaction kernel and eigenfunctions, 2) the permissibility of the Wick transform, 3) the absolute boundedness of the transformed kernel and 4) the kernel to be of Hilbert-Schmidt type. In the rest frame the kernel is found to be a real analytic function of P4 in a strip region of the complex P4 plane, where P4 is the total energy multiplied by i. The eigenvalue problem for the hermitian analytic kernel is discussed. As a consequence, the sign of the norm of bound states represented by eigenfunctions with real eigenvalues for total energies in a real interval, is shown to be determined by three factors. This is a generalization of a result known in the equal-mass ladder models.