Scattering by the two-dimensional potential sin /spl phi//r

Abstract

The solution of the Schro/spl uml/dinger equation with potential sin /spl phi//r ((r,/spl phi/)-polar coordinates) presents serious mathematical difficulties which so far have prevented the reliable calculation of the electrical resistivity of edge dislocations in metals. The nature of the difficulties is analyzed by studying related, explicitly solvable problems and by physical reasoning. It is argued that a complete solution will show a logarithmic dependence on an external cut-off radius and that this may account for the experimental results. A recursion-formula approach based on Hankel transformations and a mathematical technique originally developed for Mathieu functions promise to permit a full solution of the scattering problem.