On resonant energy sets for Hamiltonian systems with reflections

Abstract

We study two uncoupled oscillators, one horizontal and one vertical, moving in a rectilinear polygon (with only vertical and horizontal sides) and undergoing elastic reflections at its boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytic potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs for a large set of values of the parameter. We focus on unimodal analytic potentials whose unique minimum is at zero. The most important result of the work describes the size of the set of resonance levels in the form of the following trichotomy: it is either empty, a singleton, or large, namely non-empty and open. In the latter case, we show that an abundance of resonant orbits occurs only when the potentials are of a special type; we denote this family by $\mathcal{SP}$. This result can be regarded as a distant analogue of the classical Bertrand's theorem (1873), which characterizes centrally symmetric potentials in the presence of an abundance of periodic orbits.