Abstract
LetG n ,N∞N, denote the set of gaps of the Hill operator. We solve the following problems: 1) find the effective massesM ± n , 2) compare the effective massM ± n with the length of the gapG n , and with the height of the corresponding slit on the quasimomentum plane (both with fixed numbern and their sums), 3) consider the problems 1), 2) for more general cases (the Dirac operator with periodic coefficients, the Schrödinger operator with a limit periodic potential). To obtain 1)–3) we use a conformal mapping corresponding to the quasimomentum of the Hill operator or the Dirac operator.
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Communicated by B. Simon
Partially supported by Russian Fund of Fundamental Research (93-011-1697)
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Kargaev, P., Korotyaev, E. Effective masses and conformal mappings. Commun.Math. Phys. 169, 597–625 (1995). https://doi.org/10.1007/BF02099314
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DOI: https://doi.org/10.1007/BF02099314