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On resonant energy sets for Hamiltonian systems with reflections

We study two uncoupled oscillators, horizontal and vertical, residing in rectilinear polygons (with only vertical and horizontal sides) and impacting elastically from their boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytical potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs more often than rarely. We focus on unimodal analytic potentials with the minimum 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 mostly empty or is one-element or is large, i.e. non-empty and open. In this 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.