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.

Adventures of the coupled Yang–Mills oscillators: II. YM–Higgs quantum mechanics

“…This approach is motivated by the need to take the quantum fluctuations inside the hyperbolic channels into account even in the lowest order approximation. We show that the new approach reproduces the TF result obtained with the method of [8,9] without requiring an artificial subdivision of the phase space into different regions. A modified WK expansion can be derived to systematically improve on this result.…”

“…This is due to the negligible time spent by the classical trajectory in the depth of the hyperbolic channels [9]. Higher-order corrections change this situation essentially, as we shall see below.…”

“…These generate a confining potential, which does not disappear in the limit v → 0, closes the flat direction, and eliminates the mentioned singularities, which are essentially classical. Taking these fluctuations into account, we are able to compare the expression for Z(t) obtained by our method with the result for Z(t) obtained by the method of [8,9] in the TF approximation.…”

The partition function and level density for Yang–Mills–Higgs quantum mechanics

“…(9 ′ ), and Eq. (18) are in accordance with the well-known relation of the trace of the n-harmonic-oscillator heat kernel, viz., Z(t) = 2 sinhh vt 2 −n in the quasiclassical approximation (t → 0) [7,8]. From Eq.…”

“…Some time ago, there appeared two important papers [7,8] devoted to the calculation of the partition function Z(t) and the asymptotic integrated level density N(E) for the x 2 y 2 model. Their method was based on an adiabatic separation in the partition function's dependence on x and y out in the narrow channels of the equipotential surface |xy| = constant.…”