Physics – Condensed Matter – Mesoscale and Nanoscale Physics
Scientific paper
1998-06-05
Phys. Rev. B 58, 9881 (1998)
Physics
Condensed Matter
Mesoscale and Nanoscale Physics
5 pages, 4 figures, to appear in Phys. Rev. B
Scientific paper
10.1103/PhysRevB.58.9881
Based on a thorough numerical analysis of the spectrum of Harper's operator, which describes, e.g., an electron on a two-dimensional lattice subjected to a magnetic field perpendicular to the lattice plane, we make the following conjecture: For any value of the incommensurability parameter sigma of the operator its spectrum can be covered by the bands of the spectrum for every rational approximant of sigma after stretching them by factors with a common upper bound. We show that this conjecture has the following important consequences: For all irrational values of sigma the spectrum is (i) a zero measure Cantor set and has (ii) a Hausdorff dimension less or equal to 1/2. We propose that our numerical approach may be a guide in finding a rigorous proof of these results.
Geisel Theo
Ketzmerick Roland
Kruse Karsten
Steinbach Frank
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