Diffraction and boundary conditions in semi-classical open billiards

The conductance through open quantum dots or quantum billiards shows fluctuations, that can be explained as interference between waves following different paths between the leads of the billiard. We examine such systems by the use of a semi-classical Green's functions. In this paper we examine how the choice of boundary conditions at the lead mouths affect the diffraction. We derive a new formula for the S-matrix element. Finally we compare semi-classical simulations to quantum mechanical ones, and show that this new formula yield superior results.