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

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Abstract

We discuss here the application of a multidimensional WKB method to the Schr\"odinger 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 $\ensuremath{\theta}$. As an application, we calculate the scattering amplitude for the Klein-Gordon equation with the potential $\frac{\ensuremath{\alpha}{e}^{\ensuremath{-}\ensuremath{\lambda}r}}{r}$, and find that in the limit of large $p$ and fixed $\ensuremath{\theta}$, it is equal to the scattering amplitude for a Coulomb potential multiplied by ${(\frac{\ensuremath{\lambda}{e}^{\ensuremath{\gamma}}}{2p})}^{2i\ensuremath{\alpha}}$, where $\ensuremath{\gamma}=0.57721\dots{}$. 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 $1\ensuremath{\gg}\ensuremath{\theta}\ensuremath{\gg}\frac{\ensuremath{\lambda}}{p}$. 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 ${A}_{\ensuremath{\mu}}(x)$ which depends not only on space but also on time. This formula holds for the scattering amplitude in the region $p\ensuremath{\rightarrow}\ensuremath{\infty}$ with the momentum transfer fixed.

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