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Localization in a periodic system of the Aharonov-Bohm rings. (English) Zbl 1008.81104
Summary: A model for the periodic system of the Aharonov-Bohm rings is constructed by means of operator extension theory. When the uniform component of the field has a rational flux through an elementary cell of the Bravais lattice of the system, the dispersion equation is found in an explicit form. The band structure of the spectrum is studied. It is proved that under some commensurability condition the spectrum of the system consists of three parts: (1) the levels of a single ring; (2) the extended states; (3) the bound states satisfying the dispersion equation. A physical interpretation of this spectrum structure is discussed.

81V70 Many-body theory; quantum Hall effect
82D20 Statistical mechanical studies of solids
Full Text: DOI
[1] Büttiker, M.; Imry, Y.; Azbel, M.Y., Phys. rev., A30, 1982, (1984)
[2] Büttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S., Phys. rev., B31, 6207, (1985)
[3] Büttiker, M., J. math. phys., 37, 4793, (1996)
[4] Avron, J.E.; Raveh, A.; Zur, B., Rev. mod. phys., 60, 873, (1988)
[5] Ford, C.J.B.; Fowler, A.B.; Hong, J.M.; Knoedler, C.M.; Laux, S.E.; Wainer, J.J.; Washburn, S., Surf. sci., 229, 307, (1990)
[6] Beenakker, C.W.J.; van Houten, H., Solid state phys. advances in res. and appl., 44, 1, (1991)
[7] Yacoby, A.; Heiblum, M.; Mahalu, D.; Shtrikman, M., Phys. rev. lett., 74, 4047, (1995)
[8] Yejati, A.Levy; Büttiker, M., Phys. rev., B52, 14360, (1995)
[9] Carini, J.P.; Browne, D.A.; Nagel, S.R., Localization and metal-insulator transitions, (), 281, New York, London
[10] Shi, J.-R.; Gu, B.-Y., Phys. rev., B55, 4703, (1997)
[11] Li, J.; Zhang, Z.-Q.; Liu, Y., Phys. rev., B55, 5337, (1997)
[12] V. A. Geyler and I. Yu. Popov: paper submitted to Phys. Lett.
[13] Geyler, V.A.; Pavlov, B.S.; Popov, I.Yu., Atti sem. mat. fis. univ. modena, 45, 1, (1997)
[14] Hofstadter, D.R., Phys. rev., B14, 2239, (1976)
[15] Geyler, V.A.; Pavlov, B.S.; Popov, I.Yu., J. math. phys., 37, 5171, (1996)
[16] Geyler, V.A.; Popov, I.Yu., Rep. math. phys., 39, 275, (1997)
[17] Exner, P., J. phys., A29, 87, (1996)
[18] Exner, P.; Gawlista, R., Phys. rev., B53, 4275, (1996)
[19] Exner, P.; Šeba, P., (), 227
[20] Pavlov, B.S., Russ. math. surv., 42, 127-168, (1987)
[21] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable models in quantum mechanics, (1988), Springer Berlin · Zbl 0679.46057
[22] Krein, M.G.; Langer, H., Funkt. anal. pril., 5, 54, (1971)
[23] Geyler, V.A.; Popov, I.Yu., Zeitschr. phys., B93, 437, (1994)
[24] Zak, J., Phys. rev., 133, A1602, (1964)
[25] Geyler, V.A., Saint |St. Petersburg math. J., 3, 489, (1992)
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