Let be given a Schrödinger operator \(A_\hbar = - \hbar^2 \Delta + V\) with \(\Delta\) the Laplace-Beltrami operator associated with a Riemannian metric on a compact manifold \(M\), and \(V \in C^\infty (M)\) is strictly positive. \(A_\hbar\) then has a discrete spectrum, consisting of eigenvalues \(\{E_j (\hbar)\}\) with finite multiplicities, and the trace formula states that, under certain assumptions, the sums \[ \Upsilon_{\hbar, E} (\varphi) = \sum_j \varphi \left( {E_j (\hbar) - E \over \hbar} \right), \tag{1} \] where \(\varphi\) is a test function with compactly supported Fourier transform, have an asymptotic expansion in integer powers of \(\hbar\) as \(\hbar \to 0\), and the leading coefficients of these expansion are computed. The behavior of (1) in case there are equilibria with energy \(E\) is studied.