Symmetry Decomposition of Potentials with Channels

Abstract

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.

References in corpus (4)

• Statistical properties of energy levels of chaotic systems: Wigner or non-Wigner
• The semiclassical resonance spectrum of Hydrogen in a constant magnetic field
• Weyl Expansion for Symmetric Potentials
• Discrete Symmetries in the Weyl Expansion for Quantum Billiards

Cited by in corpus (6)

• Progress in Classical and Quantum Variational Principles
• Spectral fluctuations and 1/f noise in the order-chaos transition regime
• Semiclassical Trace Formulas for Noninteracting Identical Particles
• The Partition Function and Level Density for Yang-Mills-Higgs Quantum Mechanics
• Adventures of the Coupled Yang-Mills Oscillators: I. Semiclassical Expansion
• Adventures of the Coupled Yang-Mills Oscillators: II. YM-Higgs Quantum Mechanics