EXACT C = 1 BOUNDARY CONFORMAL FIELD-THEORIES

We present a solution of the problem of a free massless scalar field o n the half line interacting through a sinusoidal potential on the boun dary. For a critical value of the period, this system is a conformal f ield theory with a nontrivial and explicitly calculable S matrix for s cattering from the boundary. It describes the critical behavior of a n umber of condensed matter systems, including dissipative quantum mecha nics and of barriers in ''quantum wires.''