THE LOCALIZATION OF SURFACE-STATES - AN EXACTLY SOLVABLE MODEL

We discuss the discrete Schrodinger operator H = -Delta + V with surfa ce potential: V as a function of lattice point vanishes outside a surf ace. In general, the operator has surface eigenstates, i.e. eigenfunct ions decreasing with distance away from the surface. We show that for a particular case of strongly incommensurate surface potential, all su rface states with energies in the exterior of the spectrum of the free operator -Delta are exponentially localized in all directions. The co rresponding centers of localization are uniformly distributed on the s urface and the set of surface energies is everywhere dense in the exte rior of the free spectrum. We find explicitly these surface energies a nd their density (the density of surface states). We also discuss Lifs hitz's approach to studying the low-dimensional perturbations which is an important ingredient of our calculation.