The Exact $S$-matrix of the Deformed $c=1$ Matrix Model

Abstract

We consider the $c=1$ matrix model deformed by the operator ${1\over 2} M\Tr\Phi^{-2}$, which was conjectured by Jevicki and Yoneya to describe a two-dimensional black hole of mass $M$. We calculate the exact non-perturbative $S$-matrix and show that all the amplitudes involving an odd number of particles vanish at least to all orders of perturbation theory. We conjecture that these amplitudes vanish non-perturbatively and prove this for the $2n \to 1$ scattering. For the 2-- and 4--particle amplitudes we give some leading terms of the perturbative expansion.