Exponentially small scattering amplitude in high energy potential scattering

The high energy scattering amplitude of spherical symmetric potentials, which are expandable in ascending even powers of r and are nonsingular in coordinate space, is calculated. The Gaussian potential, which serves as a prototype of these potentials, is dealt with extensively. The first Born approximation is completely inadequate to estimate the amplitude off the forward direction. We show that the amplitude decreases faster than any power law as function of the momentum transfer q, and thus in that respect is similar in behavior to that of elastic scattering of elementary particles. Essentially, the whole range of the scattering angle is covered. Some speculations concerning shrinkage and antishrinkage effects are briefly mentioned.