Grushin Reduction and Finite-Dimensional Effective Hamiltonians in Semiclassical Resonance Theory

We present a rigorous formulation of the Grushin reduction method for semiclassical operators with hyperbolic trapped sets. The spectral problem for the operator P(h)-z is reduced to the study of the invertibility of a finite-dimensional effective Hamiltonian E₋₊(z), following the framework established by Sjöstrand and Zworski. We prove that z is a resonance if and only if E₋₊(z) = 0, providing a block-resolvent representation that facilitates both theoretical analysis and numerical computation of scattering resonances.