Mathematical Physics, Analysis and Geometry

, Volume 8, Issue 3, pp 257–285 | Cite as

Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

Article

Abstract

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ℝd→[0,∞[ decay at infinity in an anisotropic way, for example, \(f(x_{1},x_{2})\sim (|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha_{2}})^{-1}\) as |(x1,x2)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α1 and α2 are big enough, and there is a so-called classical regime with Lifshits exponent depending on α1 and α2 if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α1 and α2 simultaneously.

Keywords

random Schrödinger operators integrated density of states Lifshits tails 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billingsley, P.: Convergence of Probability Measures, Wiley, 1968. Google Scholar
  2. 2.
    Broderix, K., Hundertmark, D., Kirsch, W. and Leschke, H.: The fate of Lifshits tails in magnetic fields, J. Statist. Phys. 80 (1995), 1–22. CrossRefGoogle Scholar
  3. 3.
    Carmona, R. and Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990. Google Scholar
  4. 4.
    Durrett, R.: Probability: Theory and Examples, Duxbury, Belmont, 1996. Google Scholar
  5. 5.
    Donsker, M. D. and Varadhan, S. R. S.: Asymptotics of the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), 525–565. Errata: ibid, p. 677. Google Scholar
  6. 6.
    Daley, D. J. and Vere-Jones, D.: An Introduction to the Theory of Point Processes, Springer, New York, 1988. Google Scholar
  7. 7.
    Dembo, A. and Zeitouni, O.: Large Deviations Techniques and Applications, Springer, New York, 1998. Google Scholar
  8. 8.
    Erdős, L.: Lifschitz tail in a magnetic field: The nonclassical regime, Probab. Theory Related Fields 112 (1998), 321–371. CrossRefGoogle Scholar
  9. 9.
    Erdős, L.: Lifschitz tail in a magnetic field: Coexistence of the classical and quantum behavior in the borderline case, Probab. Theory Related Fields 121 (2001), 219–236. Google Scholar
  10. 10.
    Hundertmark, D., Kirsch, W. and Warzel, S.: Lifshits tails in three space dimensions: Impurity potentials with slow anisotropic decay, Markov Process. Related Fields 9 (2003), 651–660. Google Scholar
  11. 11.
    Hupfer, T., Leschke, H. and Warzel, S.: Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshits tailing in magnetic fields, J. Statist. Phys. 97 (1999), 725–750. CrossRefGoogle Scholar
  12. 12.
    Hupfer, T., Leschke, H. and Warzel, S.: The multiformity of Lifshits tails caused by random Landau Hamiltonians with repulsive impurity potentials of different decay at infinity, AMS/IP Stud. Adv. Math. 16 (2000), 233–247. Google Scholar
  13. 13.
    Kallenberg, O.: Random Measures, Akademie-Verlag, Berlin, 1983. Google Scholar
  14. 14.
    Kirsch, W.: Random Schrödinger operators: A course, In: H. Holden and A. Jensen (eds), Schrödinger Operators, Lecture Notes in Phys. 345, Springer, Berlin, 1989, pp. 264–370. Google Scholar
  15. 15.
    Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators, Duke Math. J. 98 (1999), 335–369. Erratum: mp_arc 00-389. CrossRefGoogle Scholar
  16. 16.
    Klopp, F.: Une remarque á propos des asymptotiques de Lifshitz internes, C.R. Acad. Sci. Paris Ser. I 335 (2002), 87–92. Google Scholar
  17. 17.
    Kirsch, W., Kotani, S. and Simon, B.: Absence of absolutely continuous spectrum for some one-dimensional random but deterministic Schrödinger operators, Ann. Inst. H. Poincare Phys. Théor. 42 (1985), 383–406. Google Scholar
  18. 18.
    Kirsch, W. and Martinelli, F.: On the ergodic properties of the spectrum of general random operators, J. Reine Angew. Math. 334 (1982), 141–156. Google Scholar
  19. 19.
    Kirsch, W. and Martinelli, F.: Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians, Comm. Math. Phys. 89 (1983), 27–40. CrossRefGoogle Scholar
  20. 20.
    Kirsch, W. and Martinelli, F.: On the essential self adjointness of stochastic Schrödinger operators, Duke Math. J. 50 (1983), 1255–1260. CrossRefGoogle Scholar
  21. 21.
    Kirsch, W. and Simon, B.: Lifshits tails for periodic plus random potentials, J. Statist. Phys. 42 (1986), 799–808. CrossRefGoogle Scholar
  22. 22.
    Kirsch, W. and Simon, B.: Comparison theorems for the gap of Schrödinger operators, J. Funct. Anal. 75 (1987), 396–410. CrossRefGoogle Scholar
  23. 23.
    Klopp, F. and Wolff, T.: Lifshitz tails for 2-dimensional random Schrödinger operators, J. Anal. Math. 88 (2002), 63–147. Google Scholar
  24. 24.
    Lang, R.: Spectral Theory of Random Schrödinger Operators, Lecture Notes in Math. 1498, Springer, Berlin, 1991. Google Scholar
  25. 25.
    Leschke, H., Müller, P. and Warzel, S.: A survey of rigorous results on random Schrödinger operators for amorphous solids, Markov Process. Related Fields 9 (2003), 729–760. Google Scholar
  26. 26.
    Leschke, H. and Warzel, S.: Quantum-classical transitions in Lifshits tails with magnetic fields, Phys. Rev. Lett. 92 (2004), 086402 (1–4). CrossRefPubMedGoogle Scholar
  27. 27.
    Lifshitz, I. M.: Structure of the energy spectrum of the impurity bands in disordered solid solutions, Soviet Phys. JETP 17 (1963), 1159–1170. Russian original: Zh. Eksper. Teoret. Fiz. 44 (1963), 1723–1741. Google Scholar
  28. 28.
    Mezincescu, G. A.: Internal Lifshitz singularities for disordered finite-difference operators, Comm. Math. Phys. 103 (1986), 167–176. CrossRefGoogle Scholar
  29. 29.
    Mezincescu, G. A.: Lifschitz singularities for periodic operators plus random potential, J. Statist. Phys. 49 (1987), 1181–1190. CrossRefGoogle Scholar
  30. 30.
    Mezincescu, G. A.: Internal Lifshitz singularities for one-dimensional Schrödinger operators, Comm. Math. Phys. 158 (1993), 315–325. CrossRefGoogle Scholar
  31. 31.
    Mine, T.: The uniqueness of the integrated density of states for the Schrödinger operators for the Robin boundary conditions, Publ. Res. Inst. Math. Sci., Kyoto Univ. 38 (2002), 355–385. Google Scholar
  32. 32.
    Nakao, S.: On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math. 3 (1977), 111–139. Google Scholar
  33. 33.
    Pastur, L. A.: Behavior of some Wiener integrals as t→∞ and the density of states of Schrödinger equations with random potential, Theoret. Math. Phys. 32 (1977), 615–620. Russian original: Teoret. Mat. Fiz. 6 (1977), 88–95. CrossRefGoogle Scholar
  34. 34.
    Pastur, L. and Figotin, A.: Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992. Google Scholar
  35. 35.
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic, New York, 1978. Google Scholar
  36. 36.
    Simon, B.: Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526. Erratum: Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526. Google Scholar
  37. 37.
    Simon, B.: Lifshitz tails for the Anderson model, J. Statist. Phys. 38 (1985), 65–76. CrossRefGoogle Scholar
  38. 38.
    Simon, B.: Internal Lifshitz tails, J. Statist. Phys. 46 (1987), 911–918. CrossRefGoogle Scholar
  39. 39.
    Stoyan, D., Kendal, W. S. and Mecke, J.: Stochastic Geometry and Its Applications, Wiley, Chichester, 1987. Google Scholar
  40. 40.
    Stollmann, P.: Lifshitz asymptotics via linear coupling of disorder, Math. Phys. Anal. Geom. 2 (1999), 2679–2689. CrossRefGoogle Scholar
  41. 41.
    Stollmann, P.: Caught by Disorder: Bound States in Random Media, Birkhäuser, Boston, 2001. Google Scholar
  42. 42.
    Veselic, I.: Integrated density of states and Wegner estimates for random Schrödinger operators, Contemp. Math. 340 (2004), 97–183. Google Scholar
  43. 43.
    Warzel, S.: On Lifshits Tails in Magnetic Fields, Logos, Berlin, 2001. PhD thesis, University Erlangen–Nürnberg (2001). Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Institut für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für Theoretische PhysikUniversität Erlangen–NürnbergErlangenGermany

Personalised recommendations