Title: On resonant energy sets for Hamiltonian systems with reflections Show Chinese title

Title: 关于具有反射的哈密顿系统中的共振能量集

Abstract: 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. We also indicate which classes of potentials each of the three possibilities can occur in. From this point of view, the last case (strongly resonant) is the most interesting. Then, the potentials belong to a special class of potentials, denoted by $\mathcal{SP}$, which seems unknown in the literature. The presented results appear to be new, even in the simplest case, when the uncoupled oscillators are not trapped in any set. Show Chinese abstract

Abstract: 我们研究两个不耦合的振子，水平和垂直，位于矩形多边形（只有垂直和水平边）中，并从其边界弹性碰撞。 文章的主要目的是分析这种系统中共振的发生情况，这取决于决定振子的解析势的形状。 我们定义了共振能级；粗略地说，这些是共振现象比罕见更常发生的能级。 我们关注在零处有最小值的单峰解析势。 这项工作的最重要结果以以下三分法形式描述了共振能级集合的大小：它主要为空集，或为单元素集，或为较大的集合，即非空且开集。 我们还指出了三种可能性中每一种可能出现的势类。 从这个角度来看，最后一种情况（强共振）是最有趣的。 然后，这些势属于一个特殊的势类，记为$\mathcal{SP}$，这在文献中似乎尚未被提及。 所呈现的结果似乎是新的，即使在最简单的情况下也是如此，当不耦合的振子未被任何集合所束缚。