Vortices in an Imperfect Bose Gas. II. Single-Particle Excitations

Abstract

The single-particle excitations of an imperfect Bose gas with a vortex in the condensate are studied using the Bogoliubov approximation. The problem is reduced to an eigenvalue equation, whose solution yields both the single-particle energies and the eigenfunction expansion of the single-particle Green's function. The spectrum of the eigenvalues contains a discrete portion (bound states) and a continuous portion (scattering states). The lowest eigenvalue corresponding to angular momentum $l=0\mathrm{}$ and $l=\pm 1$ is determined variationally. In the long-wavelength limit, the eigenvalue for $l=\pm 1$ agrees with the frequency of normal modes calculated by Kelvin and Pitaevskii. The density of noncondensed particles is expressed in terms of the single-particle Green's function and is shown to be finite at the center of the vortex.