  We study the quantum mechanical motion in the $x^2y^2$ potentials with
$n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM)
equations. These systems show strong stochasticity in the classical limit
($\hbar = 0$) and exhibit a quantum mechanical confinement feature. We
calculate the partition function $Z(t)$ going beyond the Thomas-Fermi (TF)
approximation by means of the semiclassical expansion using the Wigner-Kirkwood
(WK) method. We derive a novel compact form of the differential equation for
the WK function. After separating the motion in the channels of the
equipotential surface from the motion in the central region, we show that the
leading higher-order corrections to the TF term vanish up to eighth order in
$\hbar$, if we treat the quantum motion in the hyperbolic channels correctly by
adiabatic separation of the degrees of freedom. Finally, we obtain an
asymptotic expansion of the partition function in terms of the parameter
$g^2\hbar^4t^3$.
