Quantum Ergodicity of Eisenstein Functions at Complex Energies

We consider a surface M with constant curvature cusp ends and its Eisenstein functions Ej(λ; z), z ∈ M . These are the plane waves associated to the jth cusp and the spectral parameter λ, (∆ − 1/4 − λ)Ej = 0. We prove quantum unique ergodicity (QUE) of Ej ’s for Reλ → ∞ and Imλ → ν > 0; the limiting measure is naturally defined and decays exponentially along the geodesic flow. In particular, taking a sequence of λ’s corresponding to scattering resonances, we obtain QUE of resonant states with energies away from the real line. As an application, we also show that the scattering matrix tends to 0 in strips separated from the real line.