Scattering matrices and scattering geodesics of locally symmetric spaces (Q2730736)

The authors study \( \mathbb{Q}\)-rank locally symmetric spaces and the scattering matrices on these spaces. The scattering matrices measure the density of the continuous spectrum and their analytic properties are important. Scattering geodesics are the geodesics that move to infinity in both directions. They are distance minimizing near both infinities. The sojourn time of a scattering geodesic can be described as the time this geodesic spends in a fixed compact region. The authors describe the singularities of the Fourier transformations, i.e. the frequencies of oscillation (with respect to the energy parameter) of the scattering matrices of a locally symmetric space. They prove that the singularities occur at sojourn time of scattering geodesics. A corresponding quantum property of this classical statement is given as well. NEWLINENEWLINENEWLINEIt is known [\textit{S. Zelditch}, Commun. Partial Differ. Equations 17, 221-260 (1992; Zbl 0749.58062)] that the real part of the scattering matrix can be given as an integral of the wave group over horocycles. This result makes it easier to understand the singularities of the Fourier transformation of the real part of the scattering matrix. Ji and Zworski realize that, instead of constructing the wave operators or studying Einstein series explicitly, the singularities of the scattering matrix itself can be obtained that way too. NEWLINENEWLINENEWLINEThe paper contributes a lot to a better understanding of the geometry and the spectral theory of locally symmetric spaces as well as of the scattering matrices and their frequencies of oscillation.