The author considers Schrödinger equations with variable coefficients, which are long-range type perturbations of the flat Laplacian on \(\mathbb{R}^n\). Let \(H\) be a Schrödinger operator \[ H= -{1\over 2} \sum^n_{j,k=1} \partial_{x_j} a_{jk}(x) \partial_{x_k}+ V(x), \] on \(L^2(\mathbb{R}^n)\), where \(n\geq 1\) and \(a_{jk}(x)\) and \(V(x)\) are real-valued \(C^\infty\)-class functions and satisfy \[ |\partial^\alpha_x(a(x)- \delta_{jk})|\leq C_\alpha\langle x\rangle^{-\mu-|\alpha|}, \]
\[ |\partial^\alpha_x V(x)|\leq C_\alpha\langle x\rangle^{2-\mu-|\alpha|}, \] for \(x\in\mathbb{R}^n\), \(\alpha\in Z^n_+\). Denote \[ k(x,\xi)= {1\over 2}\sum^n_{j,k= 1} a_{jk}(x)\xi_j\xi_k,\;p(x,\xi)= k(x,\xi)+ V(x),\;x,\xi\in\mathbb{R}^n. \] Let \(\exp t_Hp(x_0,\xi_0)\) denote the Hamilton flow generated by the symbol \(p\) with the initial data \((x_0,\xi_0)\). For \((x_0,\xi_0)\in \mathbb{R}^{2n}\) denote \((\widetilde y(t),\widetilde\eta(t))= \exp tH_p(x_0,\xi_0)\). \((x_0,\xi_0)\) is called backward nontrapping if \(|\widetilde\eta(t)|\to \infty\) as \(t\to-\infty\). Let \(u(t)= e^{-tH}u_0\), \(u_0\in L^2(\mathbb{R}^n)\) and \(t_0> 0\). Assume that \((x_0,\xi_0)\) is backward nontrapping. The author proved that \((x_0,\xi_0)\) is not in \(WF(u(t_0))\) if and only if there is \(a(x,\xi)\in C^\infty_0(\mathbb{R}^{2n})\) such that \((a(x_0,\xi_0)\neq 0\) and \[ \| (a_h\circ\exp t_0 H_p)(x, D)u_0\|_{L^2(\mathbb{R}^n)}= 0(h^\infty),\quad h\to 0, \] where \(a_h(x, D)= a(x,hD)\).