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 $2\times 2$ Hamiltonian $\mathcal{H}=\mathcal{H}(p)\in C^{\infty }(T^{\ast }{\mathbf{R}}^2)$ to its normal form near a singular point of the Fresnel surface. Namely we say that $\mathcal{H}$ has the Pareto property if it decomposes, locally, up to a conjugation with regular matrices, as $\mathcal{H}(p)=u^{\prime }(p)C(p)(u^{\prime }(p){)}^{\ast }$, where $u:{\mathbf{R}}^2\to {\mathbf{R}}^2$ 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.

References in corpus (1)

• Semiclassical theory of quasiparticles in the superconducting state