Behavior of Scattering Amplitudes at High Energies, Bound States, and Resonances

An exact Fourier-Bessel representation of the scattering amplitude is introduced and discussed for potential scattering and field theory. It is shown to contain the Mandelstam representation as a special case. The representation automatically satisfies unitarity exactly in the high-energy limit even in the many-channel situation. The behavior of the scattering amplitude for large momentum transfers is discussed and it is demonstrated that this limit is directly connected with the formation of bound states and resonances. A selection rule governing the ordering of resonances is derived. A variational principle for calculating the asymptotic dependence on momentum transfer in potential scattering is formulated. Some interesting relations between the asymptotic behaviors of $\pi -\pi$, $\pi -N$, and $N-N$ are developed, and related to the single-particle poles and low-energy resonances.