笼形各向异性振子的超可积性
Superintegrability of the Caged Anisotropic Oscillator
摘要：
我们研究了“笼状各向异性谐振子”，这是一个新的超可积或偶然退化的哈密顿量的例子。潜在的是谐振子与合理的频率比（l：m：n），但另外与障碍项描述排斥力从主平面。这将经典运动限制在由主平面或笼形物限定的扇区内。在3度中，有五个孤立的运动积分，确保所有的约束轨迹是封闭的和严格的周期性。其中三个积分在动量上是二次的，其余两个是2（l+m-1）和2（l+n-1）阶的多项式。在量子问题中，本征态是多重简并的，表现出对称群SU（3）基本模式的多个副本。
英文摘要：
We study "the Caged Anisotropic Harmonic Oscillator", which is a new example of a superintegrable, or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n), but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or cage. In 3 degrees, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m-1) and 2(l+n-1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting multiple copies of the fundamental pattern of the symmetry group SU(3).
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