Sum rules for surface plasmon frequencies

Surfaces exhibit spatially localized collective electronic excitations. The Mie modes of a particulate is one well-known example. We derive sum rules which (1) relate the surface mode frequencies of complementary systems like a sphere and a void and (2) relate the surface mode frequencies of a system with several interfaces to the ones for the individual surfaces. Our approach is of general nature since the results are based on properties of the Poisson equation and should be of interest for spectroscopies of matter in different fields of physics, in particular for scanning transmission electron microscopy.