  We study the semi-classical trace formula at a critical energy level for a
Schr\"odinger operator on $\mathbb{R}^{n}$. We assume here that the potential
has a totally degenerate critical point associated to a local minimum. The main
result, which computes the contribution of this equilibrium, is valid for all
time in a compact and establishes the existence of a total asymptotic expansion
whose top order coefficient depends only on the germ of the potential at the
critical point.
