Sum rule for metal surfaces

A sum rule relating the asymptotic phases of the wave functions of two complementary metal surfaces (which would "fit" together if joined) is derived. The sum rule is shown to be valid in the presence of all band-structure effects, all many-body effects, and even when the ions near the surface are allowed to relax to their new equilibrium positions. The sum rule essentially states that the integral of the sum of the two appropriately defined phases over a two-dimensional $\overrightarrow{\mathrm{k}}$-space surface (which in some cases is multisheeted) bounded by the "Fermi perimeter" vanishes. It is shown that the sum rule may be cast into a form where the techniques developed for proving the Friedel rule for point impurities may easily be applied; nevertheless there are numerous pitfalls, including the failure of Levinson's theorem, and these are given due attention.