Boundary dynamics and multiple reflection expansion for Robin boundary conditions

In the presence of a boundary interaction, Neumann boundary conditions should be modified to contain a function S of the boundary fields: $({\nabla }_N+S)\phi =0.\mathrm{}$ Information on quantum boundary dynamics is then encoded in the S-dependent part of the effective action. In the present paper we extend the multiple reflection expansion method to the Robin boundary conditions mentioned above, and calculate the heat kernel and the effective action (i) for constant S, (ii) to the order $S^2$ with an arbitrary number of tangential derivatives. Some applications to symmetry breaking effects, tachyon condensation and a brane world are briefly discussed.