Electromagnetic gauge choice for scattering of Schr\"odinger particle

Abstract

We consider a Schr\"{o}dinger particle placed in an external electromagnetic field of the form typical for scattering settings in the field theory: $F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}$, where the current producing $F^{\mathrm{ret}/\mathrm{adv}}$ has the past and future asymptotes homogeneous of degree $-3$, and the free fields $F^{\mathrm{in}/\mathrm{out}}$ are radiation fields produced by currents with similar asymptotic behavior. We show that with appropriate choice of electromagnetic gauge the particle has 'in' and 'out' states reached with no further modification of the asymptotic dynamics. We use a special quantum-mechanical evolution 'picture' in which the free evolution operator has well-defined limits for $t\to\pm\infty$, thus the scattering wave operators do not need the free evolution counteraction. The existence of wave operators in this setting is established, but the proof of asymptotic completeness is not complete: more precise characterization of the asymptotic behavior of the particle for $|\mathbf{x}|=|t|$ would be needed.