High Degeneracy of Effective Hamiltonian in Two Dimensions

TL;DR

This paper proves that for generic potentials, the effective Hamiltonian in two dimensions exhibits a piecewise one-dimensional structure on a dense open set, using Aubry-Mather theory.

Contribution

It demonstrates the high degeneracy of the effective Hamiltonian in two dimensions for generic potentials, revealing a piecewise 1D structure.

Findings

Effective Hamiltonian is piecewise 1D in a dense open set.

High degeneracy of the effective Hamiltonian in 2D.

Use of Aubry-Mather theory to establish structural properties.

Abstract

Consider the effective Hamiltonian $\overline H(p)$ associated with the mechanical Hamiltonian $H(p,x)={1\over 2}|p|^2+V(x)$. We prove that for generic $V$, $\overline H$ is piecewise 1d in a dense open set in two dimensions using Aubry-Mather theory.