Symmetry Decomposition of Potentials with Channels

We discuss the symmetry decomposition of the average density of states for the two dimensional potential $V=x^2{y}^2$ and its three dimensional generalisation $V=x^2{y}^2+y^2{z}^2+z^2{x}^2$. 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.