Exact c=1 boundary conformal field theories.

We present a solution of the problem of a free massless scalar field on the half line interacting through a sinusoidal potential on the boundary. For a critical value of the period, this system is a conformal field theory with a nontrivial and explicitly calculable [ital S] matrix for scattering from the boundary. It describes the critical behavior of a number of condensed matter systems, including dissipative quantum mechanics and of barriers in quantum wires.''