Lifshitz Tails in Constant Magnetic Fields

  • Frédéric KloppEmail author
  • Georgi Raikov


We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] for the case of a vanishing magnetic field.


Landau Level Random Potential Spectral Shift Function Minimax Principle Magnetic Periodicity 
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  1. 1.
    Avron J., Herbst I., Simon B. (1978) Schrödinger operators with magnetic fields. I. General interactions. Duke. Math. J. 45, 847–883zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Broderix K., Hundertmark D., Kirsch W., Leschke H. (1995) The fate of Lifshits tails in magnetic fields. J. Stat. Phys. 80, 1–22zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bruneau V., Pushnitski A., Raikov G.D. (2004) Spectral shift function in strong magnetic fields. Alg. i Analiz 16, 207–238MathSciNetGoogle Scholar
  4. 4.
    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Notice Series 268, Cambridge: Cambridge University Press, 1999Google Scholar
  5. 5.
    Dubrovin B.A., Novikov S.P. (1980) Ground states in a periodic field. Magnetic Bloch functions and vector bundles. Sov. Math., Dokl. 22, 240–244zbMATHGoogle Scholar
  6. 6.
    Erdős L. (1998) Lifschitz tail in a magnetic field: the nonclassical regime. Probab. Th. Related Fields 112, 321–371CrossRefGoogle Scholar
  7. 7.
    Erdős L. (2001) Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case. Probab. Theory Related Fields 121, 219–236CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fock V. (1928) Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld, Z. Physik 47, 446–448CrossRefADSGoogle Scholar
  9. 9.
    Gradshteyn I.S., Ryzhik I.M., (1965). Table of Integrals, Series, and Products. New York San Francisco London, Academic PressGoogle Scholar
  10. 10.
    Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper, In: H. Holden, A. Jensen (eds.), Schrödinger operators, Proceedings, Sonderborg, Denmark 1988, Lect. Notes in Physics 345 Berlin: Springer (1981), pp. 118–197Google Scholar
  11. 11.
    Hupfer T., Leschke H., Müller P., Warzel S. (2001) Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials. Rev. Math. Phys. 13, 1547–1581zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hupfer T., Leschke H., Warzel S. (1999) Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshits tailing in magnetic fields. J. Stat. Phys. 97, 725–750zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hupfer T., Leschke H., Warzel S. The multiformity of Lifshits tails caused by random Landau Hamiltonians with repulsive impurity potentials of different decay at infinity. In: Differential equations and mathematical physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., 16, Providence, RI: Amer. Math. Soc., 2000, pp. 233–247Google Scholar
  14. 14.
    Hupfer T., Leschke H., Warzel S. (2001) Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials. J. Math. Phys. 42, 5626–5641zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Kirsch, W.: Random Schrödinger operators: a course. In: Schrödinger operators, Proc. Nord. Summer Sch. Math., Sandbjerg Slot, Soenderborg/Denmark 1988, Lect. Notes Phys. 345, Berlin: Springer, (1989), pp. 264–370Google Scholar
  16. 16.
    Kirsch W., Martinelli F. (1982) On the spectrum of Schrödinger operators with a random potential. Commun. Math. Phys. 85, 329–350zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Kirsch W., Martinelli F. (1983) Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians. Commun. Math. Phys. 89, 27–40zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kirsch W., Simon B. (1986) Lifshitz tails for periodic plus random potentials. J. Statist. Phys. 42(5-6): 799–808zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Kirsch W., Simon B. (1987) Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75, 396–410zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Klopp F. (1995) An asymptotic expansion for the density of states of a random Schrödinger operator with Bernoulli disorder. Random Oper. Stochastic Equations 3, 315–331zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Klopp F. (1999) Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J. 98, 335–396CrossRefMathSciNetGoogle Scholar
  22. 22.
    Klopp F. (2002) Lifshitz tails for random perturbations of periodic Schrödinger operators. In: Spectral and inverse spectral theory (Goa, 2000). Proc. Indian Acad. Sci. Math. Sci. 112, 147–162zbMATHMathSciNetGoogle Scholar
  23. 23.
    Klopp F., Pastur L. (1999) Lifshitz tails for random Schrödinger operators with negative singular Poisson potential. Commun. Math. Phys. 206, 57–103zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Klopp, F., Ralston, J.: Endpoints of the spectrum of periodic operators are generically simple. In: Cathleen Morawetz: a great mathematician. Methods Appl. Anal. 7, 459–463 (2000)Google Scholar
  25. 25.
    Klopp F., Wolff T. (2002) Lifshitz tails for 2-dimensional random Schrödinger operators. Dedicated to the memory of Tom Wolff. J. Anal. Math. 88, 63–147zbMATHMathSciNetGoogle Scholar
  26. 26.
    Landau L. (1930) Diamagnetismus der Metalle. Z. Physik 64, 629-637CrossRefADSGoogle Scholar
  27. 27.
    Mather, J.N.: On Nirenberg’s proof of Malgrange’s preparation theorem. In: Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, 192, Berlin: Springer 1971, pp. 116–120Google Scholar
  28. 28.
    Mezincescu G. (1987) Lifschitz singularities for periodic operators plus random potentials. J. Statist. Phys. 49, 1181–1190zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Mezincescu G. (1993) Internal Lifshitz singularities for one-dimensional Schrödinger operators. Commun. Math. Phys. 158, 315-325zbMATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Mohamed, A., Raikov, G.: On the spectral theory of the Schrödinger operator with electromagnetic potential. In: Pseudo-differential calculus and mathematical physics, Math. Top., 5 Berlin: Akademie Verlag, 1994, pp. 298–390Google Scholar
  31. 31.
    Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Grundlehren der Mathematischen Wissenschaften 297 Berlin: Springer-Verlag, 1992Google Scholar
  32. 32.
    Raikov G.D., Warzel S. (2002) Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials. Rev. Math. Phys. 14, 1051–1072zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Reed M., Simon B., (1978) Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York, Academic PresszbMATHGoogle Scholar
  34. 34.
    Shubin M.A., (2001) Pseudodifferential Operators and Spectral Theory Second Edition. Berlin, Springer- VerlagGoogle Scholar
  35. 35.
    Sjöstrand, J.: Microlocal analysis for the periodic magnetic Schrödinger equation and related questions. In: Microlocal analysis and applications (Montecatini Terme, 1989), Lecture Notes in Math., 1495, Berlin: Springer, 1991, pp. 237–332Google Scholar
  36. 36.
    Veselić, I.: Integrated density of states and Wegner estimates for random Schrödinger operators. In: Spectral Theory of Schrödinger Operators, Contemp. Math. 340, Providence, RI: AMS, 2004, pp. 97–183Google Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité de Paris Nord et Institut Universitaire de FranceVilletaneuseFrance
  2. 2.Departamento de Matemáticas, Facultad de CienciasUniversidad de ChileSantiagoChile

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