Images and nonlocal vortex pinning in thin superfluid films

For thin films of superfluid adsorbed on a disordered substrate, we derive within a mean field (Hartree) description of the condensate the equation of motion for a vortex in the presence of a random potential. The compressible nature of the condensate leads to an effective pinning potential experienced by the vortex which is nonlocal, with a long-range tail that smooths out the random potential coupling the condensate to the substrate. We interpret this nonlocality in terms of images, and relate the effective potential governing the dynamics to the pinning energy arising from the expectation value of the Hamiltonian with respect to the vortex wave function.