Integro-differential Wave Equation Research Articles

Bound-state solutions, invariant scalar products, and conserved currents for a class of two-body relativistic systems.

The simplest two-body relativistic system with direct interaction, described by two first-class constraints, is investigated. After a description of the multitime approach (canonical quantization without gauge fixings), the two coupled integro-differential wave equations are solved. The elementary solutions for the bound states are found and are shown to transform as irreducible representations of the Poincar\'e group. Invariant scalar products are introduced assuring the unitarity of the representations and some of the associated conserved currents are discussed. The initial data problem has been solved by means of a quantum canonical transformation, which transforms the integro-differential equations into differential equations.

Scattering by non-local potentials

The kernels of a class of non-local potentials are related to the Green functions of the second order differential equations. For these potentials, the integro-differential wave equation may be transformed into a set of coupled ordinary differential equations, which can be solved by a factorization method. A solvable example of the non-separable kernel is given and the variation of the exact wave function with the non-locality of the interaction is studied. The effective mass approximation of Frahn and Lemmer for the optical potential is derived and the connection between the wave function in this approximation and the wave function for the equivalent local potential is discussed.