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

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.

33 pages, 27 figures. We change the title, add physical motivations to study the Wigner caustic, and add Section 6 where we describe the geometry of the Wigner caustic of whorls

References in corpus (3)

• The improved isoperimetric inequality and the Wigner caustic of planar ovals
• A note on Hurwitz's inequality
• The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals

Cited by in corpus (4)

• Singular points of the Wigner caustic and affine equidistants of planar curves
• The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
• The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals
• The Geometry of the Secant Caustic of a Planar Curve