Arkiv för Matematik

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  • Volume 51, Number 1 (2013), 157-183.

Simplicity of eigenvalues in Anderson-type models

Sergey Naboko, Roger Nichols, and Günter Stolz

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We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.


S. N. was supported by Russian research grant RFBR 09-01-00515a.


G. S. was supported in part by NSF grant DMS-0653374.

Article information

Ark. Mat., Volume 51, Number 1 (2013), 157-183.

Received: 22 October 2010
First available in Project Euclid: 31 January 2017

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2011 © Institut Mittag-Leffler


Naboko, Sergey; Nichols, Roger; Stolz, Günter. Simplicity of eigenvalues in Anderson-type models. Ark. Mat. 51 (2013), no. 1, 157--183. doi:10.1007/s11512-011-0155-3.

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  • Bellissard, J., Hislop, P. D. and Stolz, G., Correlation estimates in the Anderson model, J. Stat. Phys. 129 (2007), 649–662.
  • Carmona, R. and Lacroix, J., Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
  • Combes, J.-M., Germinet, F. and Klein, A., Poisson statistics for eigenvalues of continuum random Schrödinger operators, Anal. PDE 3 (2010), 49–80.
  • Combes, J.-M. and Hislop, P. D., Localization for some continuous, random Hamiltonians in d-dimensions, J. Funct. Anal. 124 (1994), 149–180.
  • Garnett, J. B., Bounded Analytic Functions, Pure and Applied Mathematics 96, Academic Press, Orlando, FL, 1981.
  • Graf, G.-M. and Vaghi, A., A remark on an estimate by Minami, Lett. Math. Phys. 79 (2007), 17–22.
  • Jakšić, V. and Last, Y., Simplicity of singular spectrum in Anderson-type Hamiltonians, Duke Math. J. 133 (2006), 185–204.
  • Kirsch, W., An invitation to random Schrödinger operators, in Random Schrödinger Operators, Panoramas et Synthèses 25, pp. 1–119, Société Mathématique de France, Paris, 2008.
  • Klein, A. and Molchanov, S., Simplicity of eigenvalues in the Anderson model, J. Stat. Phys. 167 (2006), 95–99.
  • Minami, N., Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), 709–725.
  • Naboko, S. N., Nontangential boundary values of operator R-functions in a half-plane, Algebra i Analiz 1 (1989), 197–222 (Russian). English transl.: Leningrad Math. J. 1 (1990), 1255–1278.
  • Privalov, I. I., Boundary Properties of Analytic Functions, 2nd ed., Gosudarstv. Izdat. Tekhn.-Teor. Lit., Moscow, 1960 (Russian).
  • del Río, R., Jitomirskaya, S., Last, Y. and Simon, B., Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153–200.
  • Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.
  • Simon, B., Cyclic vectors in the Anderson model, Rev. Math. Phys. 6 (1994), 1183–1185.