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Hofstadter’s butterfly for a periodic array of quantum dots. (English) Zbl 0898.35082
Constanda, C. (ed.) et al., Integral methods in science and engineering. Vol. I: Analytic methods. Proceedings of the 4th international conference, IMSE ’96, Oulu, Finland, June 17–20, 1996. Harlow: Longman. Pitman Res. Notes Math. Ser. 374, 74-78 (1997).
The main results here are concerned with theoretical explanation of the quantum Hall effect. From the mathematical point of view this subject is of considerable interest because of its relations to a number of modern areas of mathematics: theory of characteristic classes, non-commutative geometry, operator extension theory, fractal geometry, etc. The most interesting properties of periodic systems with a magnetic field are conditioned by the presence of two natural geometric scales, namely, the magnetic length and the size of an elementary cell of the period lattice. The commensurability (or incommensurability) of the scales leads to such a peculiarity of the systems as the transition from a band structure of the spectrum of a fractal one. This transition is described by the flux-energy diagram known as the “Hofstadter butterfly”.
For the entire collection see [Zbl 0883.00026].

35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis