Authors’ summary: This paper is devoted to certain semi-classical asymptotics of a Schrödinger type operator \(A(h)\) in the vicinity of a regular value \(E\) of its principal symbol \(a_0(x,\xi)\). We investigate the semi-classical behaviour of the number \(N_{E+rh,c}(h)\) of all eigenvalues \(\lambda_{j}(h)\) of \(A(h)\) situated in the interval \([E+rh-ch,E+rh+ch]\), where the energy shift parameter \(r\) and the size constant \(c>0\) are both bounded. The behaviour of \(N_{E+rh,c}(h)\) for small \(h\) depends on an oscillating term \(Q(h,r)\) which is related to the periodic trajectories of the Hamiltonian vector field \(H_{a_0}\) on the energy hypersurface \(\Sigma=\{(x,\xi):a_0(x,\xi)=E\}\). If \(Q(h,r)\) is uniformly continuous in \(r\) for any \(0<h\leq h_0\), we obtain asymptotics of the counting function \(N_{E+rh,c}(h)\) as \(h\) tends to zero. On the other hand, the points of discontinuity of \(Q(h,r)\) in \(r\) may give rise to a clustering of eigenvalues of \(A(h)\) near the energy level \(E\). Such jumps of the function \(Q\) in \(r\) are described in terms of a suitable quantization condition. In particular, if \(a_0\) is analytic in a neighborhood of \(\Sigma\) and the energy surface is connected and of contact type we obtain a complete description of the asymptotics of \(N_{E+rh,c}(h)\). Moreover, we obtain a new semi-classical trace formula giving for any \(\rho\) with Fourier transform \(\hat{\rho}\in C_0^{\infty}({\mathbb R})\) the asymptotics of \[ \sum_{\lambda_{j}(h)\leq\lambda}\rho\left(\frac{E-\lambda_{j}(h)}{h}\right) \] in terms of certain dynamical and topological characteristics of the periodic trajectories of \(H_{a_0}\) on \(\Sigma\) without any additional clean intersection assumptions.