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

TL;DR

This paper studies bound states of a 2D Schrödinger equation with a dipole potential from an edge dislocation, providing improved estimates and numerical solutions relevant for physical systems like superfluidity in solid helium.

Contribution

It offers an improved variational estimate and numerical solutions for the ground state energy of the 2D dipole problem, advancing understanding of dislocation-related quantum states.

Findings

Ground state energy estimated at -0.139, lower than previous results.

Numerical solutions confirm semiclassical energy spectrum behavior.

Review of prior results and methodological improvements.

Abstract

We investigate bound state solutions of the 2D 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 $^4$He. 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 -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.