Phase Rule for Wave Functions near Metallic Surfaces

Abstract

It is shown that a weighted Fermi-surface average of the asymptotic phase of the electronic wave function in the presence of a metallic surface is equal to $-\frac{1}{4}\pi$. We thus give a rigorous proof of a theorem appearing in the work of Sugiyama. An asymptotic expression for the amplitude of the charge-density wave associated with a metallic surface is also derived.