Linear response functions for an inhomogeneous electron gas in the hydrodynamic approximation

The current-current response functions for one-component inhomogeneous plasmas are evaluated in two geometries: the infinite half-space and the sphere. Retardation effects and dispersion to order k² are included by constructing Green's functions at the plasma-vacuum interface for the hydrodynamic plasma equations coupled to the Maxwell equations. The surface-plasmon dispersion relation is found to appear as a pole of the current-current response, and the residue at this pole is identified as the appropriate coupling constant for interactions of surface plasmons with weak external probes. Contact is made with Feibelman's theory of linear response for the infinite half-space.