Analysis & PDE

  • Anal. PDE
  • Volume 3, Number 4 (2010), 409-426.

Lifshitz tails for generalized alloy-type random Schrödinger operators

Frédéric Klopp and Shu Nakamura

Full-text: Open access


We study Lifshitz tails for random Schrödinger operators where the random potential is alloy-type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have a Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models.

Article information

Anal. PDE, Volume 3, Number 4 (2010), 409-426.

Received: 30 March 2009
Revised: 18 February 2010
Accepted: 4 April 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47B80: Random operators [See also 47H40, 60H25] 47N55 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

random Schrödinger operators sign-indefinite potentials Lifshitz tail


Klopp, Frédéric; Nakamura, Shu. Lifshitz tails for generalized alloy-type random Schrödinger operators. Anal. PDE 3 (2010), no. 4, 409--426. doi:10.2140/apde.2010.3.409.

Export citation


  • J. Baker, M. Loss, and G. Stolz, “Minimizing the ground state energy of an electron in a randomly deformed lattice”, Comm. Math. Phys. 283:2 (2008), 397–415.
  • J. Baker, M. Loss, and G. Stolz, “Low energy properties of the random displacement model”, J. Funct. Anal. 256:8 (2009), 2725–2740.
  • R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990.
  • G. B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995.
  • F. Germinet and A. Klein, “Bootstrap multiscale analysis and localization in random media”, Comm. Math. Phys. 222:2 (2001), 415–448.
  • P. D. Hislop and F. Klopp, “The integrated density of states for some random operators with nonsign definite potentials”, J. Funct. Anal. 195:1 (2002), 12–47.
  • W. Kirsch, “Random Schrödinger operators and the density of states”, pp. 68–102 in Stochastic aspects of classical and quantum systems (Marseille, 1983), edited by S. Albeverio et al., Lecture Notes in Math. 1109, Springer, Berlin, 1985.
  • W. Kirsch, “Random Schrödinger operators: A course”, pp. 264–370 in Schrödinger operators (Sønderborg, 1988), edited by H. Holden and A. Jensen, Lecture Notes in Phys. 345, Springer, Berlin, 1989.
  • F. Klopp, “Localization for some continuous random Schrödinger operators”, Comm. Math. Phys. 167:3 (1995), 553–569.
  • F. Klopp and S. Nakamura, “Spectral extrema and Lifshitz tails for non-monotonous alloy type models”, Comm. Math. Phys. 287:3 (2009), 1133–1143.
  • L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Grundlehren der Math. Wissenschaften 297, Springer, Berlin, 1992.
  • P. Stollmann, Caught by disorder: Bound states in random media, Progress in Mathematical Physics 20, Birkhäuser, Boston, 2001.
  • I. Veselić, “Integrated density of states and Wegner estimates for random Schrödinger operators”, pp. 97–183 in Spectral theory of Schrödinger operators (Mexico City, 2001), edited by R. del Río and C. Villegas-Blas, Contemp. Math. 340, Amer. Math. Soc., Providence, 2004.
  • I. Veselić, Existence and regularity properties of the integrated density of states of random Schrödinger operators, Lecture Notes in Mathematics 1917, Springer, Berlin, 2008.