Generalized Pareto optimum and semi-classical spinors

TL;DR

This paper extends Morse theory to multiple functions using Smale's critical Pareto set concept, applying it to analyze singularities in 2x2 Hamiltonians relevant to elasticity theory.

Contribution

It introduces the Pareto property for Hamiltonians, enabling their decomposition near singular points, and applies this framework to elasticity matrix Hamiltonians.

Findings

Decomposition of Hamiltonians near singular points using Pareto property

Application to elasticity theory Hamiltonians

Extension of Morse theory to multiple functions

Abstract

In 1971, S.Smale presented a generalization of Pareto optimum he called the critical Pareto set. The underlying motivation was to extend Morse theory to several functions, i.e. to find a Morse theory for $m$ differentiable functions defined on a manifold $M$ of dimension $\ell$. We use this framework to take a $2\times2$ Hamiltonian ${\cal H}={\cal H}(p)\in C^\infty(T^*{\bf R}^2)$ to its normal form near a singular point of the Fresnel surface. Namely we say that ${\cal H}$ has the Pareto property if it decomposes, locally, up to a conjugation with regular matrices, as ${\cal H}(p)=u'(p)C(p)(u'(p))^*$, where $u:{\bf R}^2\to{\bf R}^2$ has singularities of codimension 1 or 2, and $C(p)$ is a regular Hermitian matrix ("integrating factor"). In particular this applies in certain cases to the matrix Hamiltonian of Elasticity theory and its (relative) perturbations of order 3 in momentum at the origin.