hamiltonian formalism - Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact. The harmonic oscillator is exact and should be an element of this class - the problem is that I don't completely understand the theorem, therefore can't generalize it.

I would like to know if, for systems of the form

$$ H(q,p) = \left( \frac{q^2+p^2}{2} \right)^\gamma \, , \quad \gamma \in \mathbb{N} \, ,$$

the partition function is semiclassically exact, and if this exactness implies that the semiclassical propagator is also exact.