Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects

Abstract

A nonrelativistic scalar particle that is constrained to move on an asymptotically flat curved surface undergoes a geometric scattering that is sensitive to the mean and Gaussian curvatures of the surface. A careful study of possible realizations of this phenomenon in typical condensed matter systems requires dealing with the presence of defects. We examine the effect of delta-function point defects residing on a curved surface ${S}$. In particular, we solve the scattering problem for a multi-delta-function potential in plane, which requires a proper regularization of divergent terms entering its scattering amplitude, and include the effects of nontrivial geometry of ${S}$ by treating it as a perturbation of the plane. This allows us to obtain analytic expressions for the geometric scattering amplitude for a surface consisting of one or more Gaussian bumps. In general the presence of the delta-function defects enhances the geometric scattering effects.