Osaka Journal of Mathematics

Infinite divisibility of random measures associated to some random Schrödinger operators

Fumihiko Nakano

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Abstract

We study a random measure which describes distribution of eigenvalues and corresponding eigenfunctions of random Schrödinger operators on $L^{2}(\mathbf{R}^{d})$. We show that in the natural scaling every limiting point is infinitely divisible.

Article information

Source
Osaka J. Math., Volume 46, Number 3 (2009), 845-862.

Dates
First available in Project Euclid: 26 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1256564209

Mathematical Reviews number (MathSciNet)
MR2583332

Zentralblatt MATH identifier
1180.82106

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis

Citation

Nakano, Fumihiko. Infinite divisibility of random measures associated to some random Schrödinger operators. Osaka J. Math. 46 (2009), no. 3, 845--862. https://projecteuclid.org/euclid.ojm/1256564209


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References

  • M. Aizenman, A. Elgart, S. Naboko, J. Schenker and G. Stolz: Moment analysis for localization in random Schrödinger operators, Invent. Math. 163 (2006), 343–413.
  • J.-M. Combes and P.D. Hislop: Localization for some continuous, random Hamiltonians in $d$-dimensions, J. Funct. Anal. 124 (1994), 149–180.
  • R. Carmona and J. Lacroix: Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, MA, 1990.
  • E. Giere: Spektrale Mittelung und Lokale, Spektrale Eigenschaften Eindimensionaler, Zufälliger Schrödinger-Operatoren, Dissertion, Bochum, 1998.
  • O. Kallenberg: Random measures, fourth edition, Akademie-Verlag, Berlin; Academic Press, Inc., London, 1986.
  • R. Killip and F. Nakano: Eigenfunction statistics in the localized Anderson model, Ann. Henri Poincaré 8 (2007), 27–36.
  • W. Kirsch and F. Martinelli: On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys. 85 (1982), 329–350.
  • H. Makino and S. Tasaki: Level spacing statistics of classically integrable systems: Investigation along the lines of the Berry-Robnik approach, Phys. Rev. E 67 (2003), 066205.
  • N. Minami: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), 709–725.
  • S.A. Molčanov: The local structure of the spectrum of the one-dimensional Schrödinger operator, Comm. Math. Phys. 78 (1980/81), 429–446.
  • S. Nakamura: Lectures on Schrödinger Operators, Lectures in Mathematical Sciences, University of Tokyo, 1994.
  • F. Nakano: The repulsion between localization centers in the Anderson model, J. Stat. Phys. 123 (2006), 803–810.
  • F. Nakano: Distribution of localization centers in some discrete random systems, Rev. Math. Phys. 19 (2007), 941–965.
  • B. Simon: Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447–526.
  • P. Stollmann: Caught by Disorder, Birkhäuser, Boston, MA, 2001.