The Geometry of the Wigner Caustic and a Decomposition of a Curve Into Parallel Arcs

TL;DR

This paper explores the geometry and singularities of the Wigner caustic for closed planar curves, providing a decomposition method into parallel arcs to analyze its branches and properties, with applications to quantum particle dynamics.

Contribution

It introduces a new decomposition of curves into parallel arcs to analyze the Wigner caustic's branches and their geometric features, advancing understanding of its singularities and global properties.

Findings

Decomposition method for smooth branches of the Wigner caustic

Determination of the number of branches and their properties

Application to quantum particle dynamics in optical lattices

Abstract

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls - the important object to study the dynamics of a quantum particle in the optical lattice potential.