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The localization of surface states: an exactly solvable model. (English) Zbl 1001.82526

Summary: We discuss the discrete Schrödinger operator \(H=-\Delta+V\) with surface potential: \(V\) as a function of lattice point vanishes outside a surface. In general, the operator has surface eigenstates, i.e. eigenfunctions decreasing with distance away from the surface. We show that for a particular case of strongly incommensurate surface potential, all surface states with energies in the exterior of the spectrum of the free operator \(-\Delta\) are exponentially localized in all directions. The corresponding centers of localization are uniformly distributed on the surface and the set of surface energies is everywhere dense in the exterior of the free spectrum. We find explicitly these surface energies and their density (the density of surface states). We also discuss Lifshitz’s approach to studying the low-dimensional perturbations which is an important ingredient of our calculation.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
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