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

TL;DR

This paper investigates how the geometry of a curved surface influences the scattering behavior of a quantum particle, analyzing the effects of curvature and potential variations on scattering amplitudes, with implications for experimental setups.

Contribution

It provides a detailed analysis of geometric scattering for a scalar particle on asymptotically flat surfaces with arbitrary curvature coefficients, including symmetry-breaking perturbations.

Findings

Derived scattering amplitude formulas for arbitrary curvature coefficients.

Analyzed effects of symmetry-breaking perturbations on scattering.

Explored implications for experimental realization with electron gases.

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.