  The electronic structure of the superconducting surface sheath in a type-II
superconductor in magnetic fields $H_{c2}<H<H_{c3}$ is calculated
self-consistently using the Bogoliubov-de Gennes equations. We find that the
pair potential $\Delta(x)$ exhibits pronounced Friedel oscillations near the
surface, in marked contrast with the results of Ginzburg-Landau theory. The
role of magnetic edge states is emphasized. The local density of states near
the surface shows a significant depletion near the Fermi energy due to the
development of local superconducting order. We suggest that this structure
could be unveiled by scanning-tunneling microscopy studies performed near the
edge of a superconducting sample.
