  We consider a surface M with constant curvature cusp ends and its Eisenstein
functions E_j(\lambda). These are the plane waves associated to the j-th cusp
and the spectral parameter \lambda, (\Delta - 1/4 - \lambda^2)E_j = 0. We prove
that as Re\lambda \to \infty and Im\lambda \to \nu > 0, E_j converges
microlocally to a certain naturally defined measure decaying exponentially
along the geodesic flow. In particular, for a sequence of \lambda's
corresponding to scattering resonances, we find the microlocal limit of
resonant states with energies away from the real line. This statement is
similar to quantum unique ergodicity (QUE), which holds in certain other
situations; however, the proof uses only the structure of the infinite ends,
not the global properties of the geodesic flow. As an application, we also show
that the scattering matrix tends to zero in strips separated from the real
line.
