  We calculate the partition function $Z(t)$ and the asymptotic integrated
level density $N(E)$ for Yang-Mills-Higgs Quantum Mechanics for two and three
dimensions ($n = 2, 3$). Due to the infinite volume of the phase space $\Gamma$
on energy shell for $n= 2$, it is not possible to disentangle completely the
coupled oscillators ($x^2 y^2$-model) from the Higgs sector. The situation is
different for $n = 3$ for which $\Gamma$ is finite. The transition from order
to chaos in these systems is expressed by the corresponding transitions in
$Z(t)$ and $N(E)$, analogous to the transitions in adjacent level spacing
distribution from Poisson distribution to Wigner-Dyson distribution. We also
discuss a related system with quartic coupled oscillators and two dimensional
quartic free oscillators for which, contrary to YMHQM, both coupling constants
are dimensionless.
