  Quantum mechanical boundary conditions along a timelike line, corresponding
to the origin in radial coordinates, in two-dimensional dilaton gravity coupled
to $N$ matter fields, are considered. Conformal invariance and vacuum stability
severely constrain the possibilities. The simplest choice found corresponds to
a nonlinear Liouville-type boundary interaction. The scattering of low-energy
matter off the boundary can be computed perturbatively. It is found that weak
incident pulses induce damped oscillations at the boundary while large incident
pulses produce black holes. The response of the boundary to such pulses is
semi-classically characterized by a second order, nonlinear ordinary
differential equation which is analyzed numerically.
