Abstract
We consider a single-band approximation to the random Schrödinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the entire spectrum of this Hamiltonian when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level in the absence of a random potential being infinitely degenerate, while the eigenfunctions corresponding to energies in the rest of the spectrum are localized.
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References
B. Huckestein,Rev. Mod. Phys. 67:357 (1995).
J. M. Combes and P. D. Hislop,Commun. Math. Phys. 177:603 (1996).
W.-M. Wang, Microlocalization, percolation and Anderson localization for the magnetic Schrödinger operator with a random potential,J. Funct. Anal., to appear.
T. C. Dorlas, N. Macris, and J. V. Pulé,Helv. Phys. Acta 68:330 (1995).
T. C. Dorlas, N. Macris, and J. V. Pulé,J. Math. Phys. 37:1574 (1996).
J. M. Barbaroux, J. M. Combes, and P. D. Hislop, Landau Hamiltonians with unbounded random potentials: Localization in unperturbed gaps, preprint (1996).
Y. Avishai, R. M. Redheffer, and Y. B. Band,J. Phys. A 25:3883 (1992); see also Y. Avishai and R. M. Redheffer,Phys. Rev. B 47:2089 (1993); Y. Avishai, M. Ya. Azbel', and S. A. Gredeskul,Phys. Rev. B 48:17280 (1993).
H. Kunz,Commun. Math. Phys. 112:121 (1987).
J. Bellisard, A. Van Elst, and H. Schulz-Baldes,J. Math. Phys. 35:5373 (1994).
D. J. Thouless,J. Phys. C 14:3475 (1981).
H. Aoki and T. Ando,J. Phys. Soc. Jpn. 54:2238 (1985).
H. Aoki and T. Ando,Phys. Rev. Lett. 54:831 (1985).
H. von Dreifus and A. Klein,Commun. Math. Phys. 124:285 (1989).
R. Carmona and J. Lacroix,Spectral Theory of Random Schrödinger Operators (Birkhauser, Boston, 1990).
R. Ph. Boas,Entire Functions (Academic Press, New York, 1954).
D. J. Thouless,J. Phys. C 17:L325 (1984).
I. M. Gel'fand and N. Ya. Vilenkin,Generalized Functions, Vol. 4 (Academic Press, New York, 1964).
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Dorlas, T.C., Macris, N. & Pulé, J.V. The nature of the spectrum for a Landau Hamiltonian with delta impurities. J Stat Phys 87, 847–875 (1997). https://doi.org/10.1007/BF02181247
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DOI: https://doi.org/10.1007/BF02181247