Annales Henri Poincaré

, Volume 17, Issue 9, pp 2341–2377 | Cite as

Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate



We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e., the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baker J., Loss M., Stolz G.: Minimizing the ground state energy of an electron in a randomly deformed lattice. Commun. Math. Phys. 283(2), 397–415 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borisov D.: Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Sb.: Math. 197(4), 475–504 (2006)MathSciNetMATHGoogle Scholar
  3. 3.
    Borisov D., Veselić I.: Low lying spectrum of weak-disorder quantum waveguides. J. Stat. Phys. 142(1), 58–77 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Borisov D., Veselić I.: Low lying eigenvalues of randomly curved quantum waveguides. J. Funct. Anal. 265(11), 2877–2909 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borisov D.I.: On spectrum of a two-dimensional periodic operator with small localized perturbation. Izv. Math. 75(3), 471–505 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Borisov D.I., Gadyl’shin R.R.: Discrete spectrum of an asymmetric pair of waveguides coupled through a window. Izv. Math. 72(4), 659–688 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bourgain J.: An approach to Wegner’s estimate using subharmonicity. J. Stat. Phys. 134(5-6), 969–978 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Elgart A., Krüger H., Tautenhahn M., Veselić I.: Discrete Schrödinger operators with random alloy-type potential. Oper. Theory: Adv. Appl. 224, 107–131 (2012)MATHGoogle Scholar
  9. 9.
    Erdös L., Hasler D.: Anderson localization at band edges for random magnetic fields. J. Stat. Phys. 146, 900–923 (2012). doi: 10.1007/s10955-012-0445-6 ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Erdös L., Hasler D.: Wegner estimate and anderson localization for random magnetic fields. Commun. Math. Phys. 309, 507–542 (2012). doi: 10.1007/s00220-011-1373-z ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gadyl’shin R.R.: On local perturbations of the Schrödinger operator on the axis. Theor. Math. Phys. 132(1), 976–982 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ghribi F., Hislop P.D., Klopp F.: Localization for Schrödinger operators with random vector potentials. In: Adventures in Mathematical Physics, Contemporary Mathematics, vol. 447, pp. 123–138. American Mathematical Society, Providence (2007)Google Scholar
  13. 13.
    Ghribi F., Klopp F.: Localization for the random displacement model at weak disorder. Ann. Henri Poincaré 11(1–2), 127–149 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hislop P.D., Klopp F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kleespies F., Stollmann P.: Lifshitz asymptotics and localization for random quantum waveguides. Rev. Math. Phys. 12(10), 1345–1365 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Klopp F.: Localization for semiclassical continuous random Schrödinger operators II: The random displacement model. Helv. Phys. Acta 66, 810–841 (1993)MathSciNetMATHGoogle Scholar
  17. 17.
    Klopp F.: Localisation pour des opérateurs de Schrödinger aléatoires dans L 2(R d): un modéle semi-classique. Ann. Inst. Fourier (Grenoble) 45(1), 265–316 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Klopp F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167, 553–569 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Klopp F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. Henri Poincaré 3(4), 711–737 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Klopp F., Loss M., Nakamura S., Stolz G.: Localization for the random displacement model. Duke Math. J. 161(4), 587–621 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Klopp F., Nakamura S.: Spectral extrema and Lifshitz tails for non-monotonous alloy type models. Commun. Math. Phys. 287(3), 1133–1143 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Klopp F., Nakamura S., Nakano F., Nomura Y.: Anderson localization for 2D discrete Schrödinger operators with random magnetic fields. Ann. Henri Poincaré 4(4), 795–811 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kostrykin V., Veselić I.: On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials. Math. Z. 252(2), 367–392 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lenz D., Peyerimhoff N., Post O., Veselić I.: Continuity properties of the integrated density of states on manifolds. Jpn. J. Math. 3(1), 121–161 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lenz D., Peyerimhoff N., Post O., Veselić I.: Continuity of the integrated density of states on random length metric graphs. Math. Phys. Anal. Geom. 12(3), 219–254 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lenz D., Peyerimhoff N., Veselić I.: Integrated density of states for random metrics on manifolds. Proc. Lond. Math. Soc. (3) 88(3), 733–752 (2004)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Leonhardt, K., Peyerimhoff, N., Tautenhahn, M., Veselić, I.: Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials. Rev. Math. Phys. 27, 1550007 (2015)Google Scholar
  28. 28.
    Martinelli F., Holden H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on \({ L^{2}(R^{\nu}) }\). Commun. Math. Phys. 93, 197–217 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Stolz G.: Non-monotonic random Schrödinger operators: the Anderson model. J. Math. Anal. Appl. 248(1), 173–183 (2000)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ueki N.: On spectra of random Schrödinger operators with magnetic fields. Osaka J. Math. 31(1), 177–187 (1994)MathSciNetMATHGoogle Scholar
  31. 31.
    Ueki N.: Simple examples of Lifschitz tails in Gaussian random magnetic fields. Ann. Henri Poincaré 1(3), 473–498 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ueki N.: Wegner estimate and localization for random magnetic fields. Osaka J. Math. 45(3), 565–608 (2008)MathSciNetMATHGoogle Scholar
  33. 33.
    Veselić I.: Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators. Lett. Math. Phys. 59(3), 199–214 (2002)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Veselić, I.: Existence and regularity properties of the integrated density of states of random Schrödinger operators. Lecture Notes in Mathematics, vol. 1917. Springer (2008)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Denis Borisov
    • 1
    • 2
    • 3
  • Anastasia Golovina
    • 4
  • Ivan Veselić
    • 5
  1. 1.Department of Differential Equations, Institute of Mathematics with Computer Center, Ufa Scientific CenterRussian Academy of SciencesUfaRussia
  2. 2.Faculty of Physics and MathematicsBashkir State Pedagogical UniversityUfaRussia
  3. 3.Faculty of ScienceUniversity of Hradec KrálovéHradec KrálovéCzech Republic
  4. 4.Department of Fundamental SciencesBauman Moscow State Technical UniversityMoscowRussia
  5. 5.Department of MathematicsTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations