Interactions of Collective Excitations with Vortices in Superfluid Systems

We investigate the interactions of collective excitations with vortices in superfluid systems, including $~^4$He and superconductors. The dynamical equations are obtained by the aid of the many-body wavefunction and the density-density correlation function. The scattering cross-section of collective excitations with a vortex is calculated in the Born approximation (valid at long wavelengths), and is expressed in terms of the collective excitation spectrum $\omega (q)$ by the simple and general formula $\sigma = \frac{\pi}{2} \frac{q}{[\omega'(q)]^2} \cot^2\frac{\theta}{2}$, where $q$ is the wave number of the excitation, and $\theta$ is the scattering angle. At short wave lengths, the classical equations of motion are derived.