Microlocal limits of Eisenstein functions away from the unitarity axis

Abstract

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 $\mathrm {Re} \lambda\to \infty$ and $\mathrm {Im} \lambda\to\nu>0$, $E_j$ converges microlocally to a certain naturally defined measure decaying exponentially along the geodesic flow. In particular, for a surface with one cusp and 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.