Bound states of edge dislocations: The quantum dipole problem in two dimensions

Abstract

We investigate bound-state solutions of the two-dimensional Schr\"odinger equation with a dipole potential originating from the elastic effects of a single edge dislocation. The knowledge of these states could be useful for understanding a wide variety of physical systems, including superfluid behavior along dislocations in solid $^{4}\text{H}\text{e}$. We present a review of the results obtained by previous workers together with an improved variational estimate of the ground-state energy. We then numerically solve the eigenvalue problem and calculate the energy spectrum. In our dimensionless units, we find a ground-state energy of $\ensuremath{-}0.139$, which is lower than any previous estimate. We also make successful contact with the behavior of the energy spectrum as derived from semiclassical considerations.