Scattering due to geometry: Case of a spinless particle moving on an asymptotically flat embedded surface

Abstract

A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.

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

“…12, we see that the presence of the point defects enhances the contribution of the nontrivial geometry of the surface to the differential cross-section. We can extend the above results to cases where the surface S is made of several Gaussian bumps that are sufficiently far from one another [7]. To do this, first we recall that under a translation, x → x − c, the scattering amplitude transforms according to f (k ′ , k) → e i(k−k ′ )·c f (k ′ , k).…”

“…In the following we drop the superscript + and label the the scattering amplitude by f(k ′ , k) for brevity. We can use (7) to obtain a series expansion for the scattering solutions of (6) in powers of the perturbation parameter ζ. To do this we substitute the ansatz…”

“…in both sides of (7) and demand that it holds at each order of ζ separately. This yields a set of equations for the unknowns |ψ + n (k) that admit the following solution.…”

Scattering by a collection of $δ$-function point and parallel line defects in two dimensions

“…∇ 2 is the Laplacian in two dimensions, ∆ S , K, and M are respectively the Laplace-Beltrami operator, the Gaussian curvature, and the mean curvature of the surface, and λ 1 and λ 2 are a pair of real coupling constants whose values depends on the details of the confining forces that keeps the particle on the surface [23].…”

“…The present investigation is motivated by our interest in the study of the consequences of the presence of point and line defects on the geometric scattering of a particle moving on a curved surface [22,23]. Refs.…”