Vortices in an Imperfect Bose Gas. III. Scattering of Single-Particle Excitations

The scattering of single-particle excitatins by a vortex in the condensate of an imperfect Bose gas is studied using the Bogoliubov approximation. The $S$ matrix is calculated as in conventional scattering theory, and the phase shifts are determined with the Schwinger variational principle. The differential cross section in the long-wavelength limit is found to be $\frac{d\sigma }{d\chi }=\frac{1}{2}\pi {\left(\frac{\hslash }{\mathrm{mc}}\right)}^{2}k{cot}^2\left(\frac{1}{2}\chi \right)$, where $m$ is the mass of one atom and $c$ is the speed of sound. This cross section agrees precisely with that found previously for scattering of sound by a classical vortex with circulation $\frac{h}{m}$.