Generalized Pareto optimum and semi-classical spinors

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.