Dispersion relations and the causality concept

Abstract

It is well known that the scattering functionS(x) associated with a cut potential has certain analytic properties that make it satisfy dispersion relations. It is of interest to see how these analytic properties are modified when the potentials are not cut-off at a certain point, but continue to infinity, going asymptotically to zero there. The discussion is first carried using a causality condition enunciated as follows : The wave function associated with any initial wave packet remains bounded for all time. As a consequence of the causality condition, we obtained that it is no longer theS(x), but a new function, which we call the dispersion function, that satisfies the analytic properties that imply dispersion relations. We also check these analytic properties directly from the Schrodinger equation. Finally, to discuss the significance of the poles of the dispersion and scattering functions, we analize in detail the scattering by the Eckart potential, obtaining the time dependent Green function in terms of basic interaction Green (BIG) functions associated with the poles of the dispersion function. From the behaviour of the BIG functions, as functions of time, we can also obtain restrictions on the analytic behaviour of the dispersion function.