Abstract:
The spectrum of Schrödinger operators H with periodic point potentials in dimensions d= 2, 3 is studied. In the general case of N points in the Wigner–Seitz cell it is proven that H has a band structure with at most a finite number of gaps (Bethe–Sommerfeld conjecture). It is also proven that in the case of a generic local point perturbation no singular continuous components are present; in the non-local case a fractal component like the Cantor set is exhibited, this component can either consist of a singular continuous or a dense point spectrum.
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Received: 4 November 1998 / Accepted: 30 August 1999
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Albeverio, S., Geyler, V. The Band Structure of the General Periodic¶ Schrödinger Operator with Point Interactions. Comm Math Phys 210, 29–48 (2000). https://doi.org/10.1007/s002200050771
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DOI: https://doi.org/10.1007/s002200050771