Martinez A., Nakamura S. , Sordoni V. Phase Space Tunneling in Multispace Scattering (50K, LATeX) ABSTRACT. We prove exponentially small estimates on the off-diagonal terms of the scattering matrix associated to two-state semiclassical Schr\"odinger Hamiltonians. Our method is based on phase space tunneling estimates and a splitting of the operator by means of Toeplitz-type phase space cut-off operators.