标题： 势能的通道对称性分解 显示英文标题

标题： Symmetry Decomposition of Potentials with Channels

摘要： 我们讨论二维势能$V=x^2y^2$的平均态密度的对称性分解及其三维推广$V=x^2y^2+y^2z^2+z^2x^2$。在这两个问题中，由于沿坐标轴存在无限通道，可获得的能量相空间是非紧致的。 已知在二维情况下，这些通道中的相空间体积是无限的，从而导致了平均态密度的非标准形式。 这里我们展示，这些通道也使得对称性分解的效果比在没有通道的势能中更强，从而产生了本质上是主导阶次的项。 我们通过数值验证了这些结果，并观察到一个与通道相关的奇特数值效应。 在三维情况下，相空间的体积是有限的，对称性分解更接近于典型势能的情况——然而，仍然存在一些与群元素相关的非典型效应。 显示英文摘要

摘要： We discuss the symmetry decomposition of the average density of states for the two dimensional potential $V=x^2y^2$ and its three dimensional generalisation $V=x^2y^2+y^2z^2+z^2x^2$. In both problems, the energetically accessible phase space is non-compact due to the existence of infinite channels along the axes. It is known that in two dimensions the phase space volume is infinite in these channels thus yielding non-standard forms for the average density of states. Here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels, leading to terms which are essentially leading order. We verify these results numerically and also observe a peculiar numerical effect which we associate with the channels. In three dimensions, the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials --- however there are still non-generic effects related to some of the group elements.