Let \(X\) be a noncompact manifold that is Euclidean outside of a compact set. Let us consider a symmetric semiclassical pseudodifferential operator \(P(h)\rho^w (x,hD)+hp_1^w(x,hD;h)\), where \(p\in S^2(T^* x)\) is elliptic, \(p=0\Rightarrow dp\neq 0\) and \(p_1\in S^{0,2}(T^*x)\) is a usual semiclassical symbol on \(X:| \Omega^\alpha_x\partial^\beta_\xi p_1(x,\xi;h)|\leq C_{\alpha\beta}\langle\xi \rangle^{2-|\beta|}\), \((n,\xi) \in T^*X\), \(h\in(0,1]\). Outside of a compact set, \(P(h)\) is a differential operator of second-order which tends to \(-\Delta-1\) uniformly with respect to \(h\) as \(|x|\to\infty\). The simplest exemple is \(X= \mathbb{R}^n\), \(P(h)=-h^2\Delta+V(x)-1\), \(V\in{\mathcal C}^\infty_c(\mathbb{R}^n;\mathbb{R})\).
The main result of this article is the following. Suppose that the flow of the Hamilton vector field \(H_p\) near zero energy is hyperbolic in some sense. If we write the Minkowski dimension of the trapped set at zero energy as \(m_0=r\nu_0+ 1\), then for any \(\nu>\nu_0\) and \(C_0>0\), there exists \(C_1\) such that the number of resonances of \(P(h)\) in the disc \(D(0,C_0h)\) is bounded by \(C_1h^{-\nu}\). When the trapped set is of pure dimension, \(\nu\) can be replaced by \(\nu_0\). To prove the main result, the authors first develop methods for proving the natural results on teh absence of resonances and on general upper bounds at nondegenerate energies.