The Wigner caustic on shell and singularities of odd functions

TL;DR

This paper investigates the singularities of the Wigner caustic on shell associated with Lagrangian submanifolds, classifying them through odd function singularities and their deformations, with implications for symplectic geometry.

Contribution

It introduces a mathematical framework for analyzing singularities of the Wigner caustic on shell using odd function theory and classifies these singularities via odd versal deformations.

Findings

Classification of singularities of the Wigner caustic on shell.

Application of odd function singularity theory to symplectic geometry.

Interpretation of singularities in terms of local geometry of L.

Abstract

We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.