Multidimensional WKB approach to high-energy elastic scattering at fixed angle

We discuss here the application of a multidimensional WKB method to the Schroedinger equation and the Klein-Gordon equation. This method enables us to calculate scattering amplitudes in the limit of large incident momentum p and fixed scattering angle theta. As an application, we calculate the scattering amplitude for the Klein-Gordon equation with the potential αe/sup -lambdar//r, and find that in the limit of large p and fixed theta, it is equal to the scattering amplitude for a Coulomb potential multiplied by (lambdae/sup γ//2p)/sup 2ialpha/, where γ = 0.57721... . Our result differs from the eikonal formula, in the high-energy fixed-angle region. We also show that our formula agrees with the eikonal formula for small scattering angles satisfying 1>>theta>>lambda/p. Thus our formula, together with the eikonal formula, give complete information on the high-energy amplitude for all angles. A side result of the application of the WKB method is a generalized eikonal formula for the case of the Klein-Gordon equation with a four-potential A/sub μ/(x) which depends not only on space but also on time. This formula holds for the scattering amplitude in the region p→infinity with the momentum transfer fixed