Scattering om two Aharonov-Bohm vortices

TL;DR

This paper develops an explicit, integrable solution for the scattering problem involving two Aharonov-Bohm vortices by generalizing a diffraction method and linking the problem to Painleve equations.

Contribution

It extends a diffraction method to solve the two AB vortices scattering problem and connects it to Painleve equations, demonstrating integrability.

Findings

Explicit solutions for large and small vortex distances.

The scattering problem is shown to be integrable.

Green function and scattering amplitude satisfy nonlinear differential equations.

Abstract

The problem of two Aharonov-Bohm (AB) vortices for the Helmholtz equation is examined in detail. It is demonstrated that the method proposed in [J. M. Myers, J. Math. Phys. \textbf{6}, 1839 (1963)] for diffraction on a slit can be generalized to get an explicit solution for AB vortices. Due to singular nature of AB interaction the Green function and the scattering amplitude for two AB vortices obey a series of partial differential equations. Coefficients entering these equations, in their turn, fulfill ordinary non-linear differential equations whose solutions can be obtained from a solution of the Painleve V (or III) equation. The asymptotics of necessary functions for very large and very small distances between two vortices are calculated explicitly. Taken together, it means that the problem of two AB vortices is integrable.