Off-Shell Correction to the High-Energy Optical Potential

We obtain the leading off-shell correction to the high-energy optical potential: $v=v_0\times [1-(\frac{\gamma }{2k^2}\left)v_0\right]$, where $\gamma =k{\left[\right(\frac{\partial }{\partial q}\left)\ln f\right(q;k\left)\right]}_{q=k}$ for $f(q;k)=f(\overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{q}};k)$ the off-shell projectile-nucleon scattering amplitude. A slow energy variation of the amplitude at very high energy gives $\gamma =k\left(\frac{\partial }{\partial k}\right)\ln f_k\mathrm{}\left(0\right)\mathrm{}$, where $f_k\mathrm{}\left(0\right)\mathrm{}$ is the on-shell amplitude, and the correction is small. In the region of projectile-nucleon resonances, we describe the two-body interaction by a separable potential and relate $\gamma$ to the scattering phase shifts. The leading off-shell correction is substantial for an absorptive resonance but "anomalously" small at an elastic resonance.