Microlocal limits of Eisenstein functions away from the unitarity axis

TL;DR

This paper analyzes the microlocal behavior of Eisenstein functions on surfaces with cusp ends as the spectral parameter grows large, revealing their limits and implications for scattering resonances and the scattering matrix.

Contribution

It establishes the microlocal limits of Eisenstein functions away from the unitarity axis, extending understanding of resonant states and scattering phenomena on cusp surfaces.

Findings

Eisenstein functions converge microlocally to a natural measure as spectral parameter tends to infinity.

Resonant states have specific microlocal limits away from the real line.

The scattering matrix tends to zero in certain spectral strips.

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 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.