Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T17:01:54.270Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Properties of Random and Deterministic CMVMatrices

Published online by Cambridge University Press:  17 July 2014

M. Stoiciu
Get access

Abstract

The CMV matrices are unitary analogues of the discrete one-dimensional Schrödingeroperators. We review spectral properties of a few classes of CMV matrices and describefamilies of random and deterministic CMV matrices which exhibit a transition in thedistribution of their eigenvalues.

Keywords

spectral theorydistribution of eigenvaluesCMV matrices
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 270 - 281
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M., Molchanov, S.. Localization at large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys., 157 (1993), 245–278. CrossRefGoogle Scholar
Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D.. Finite-volume fractional moment criteria for Anderson localization. Commun. Math. Phys., 224 (2001), 219–253. CrossRefGoogle Scholar
Anderson, P.W.. Absence of diffusion in certain random lattices. Phys. Rev., 109 (1958), 1492–1505. CrossRefGoogle Scholar
Bellissard, J., Hislop, P., Stolz, G.. Correlation estimates in the Anderson model. J. Stat. Phys., 129 (2007), no. 4, 649–662. CrossRefGoogle Scholar
Bourgain, J., Kenig, C.. On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math., 161 (2005), 389–426. CrossRefGoogle Scholar
Combes, J-M., Germinet, F., Klein, A.. Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys., 135 (2009), no. 2, 201–216. CrossRefGoogle Scholar
Cantero, M.J., Moral, L., Velázquez, L.. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl., 362 (2003), 29–56. CrossRefGoogle Scholar
del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.. What is localization? Phys. Rev. Lett., 75 (1995), 117–119. CrossRefGoogle ScholarPubMed
Dyson, F.J.. Statistical theory of the energy levels of complex systems. I, II, and III. J. Math. Phys., 3 (1962), 140–156, 157–165, and 166–175. CrossRefGoogle Scholar
P.J. Forrester. Log-gases and Random matrices. London Mathematical Society Monographs Series, 34. Princeton University Press, Princeton, NJ, 2010.
Fröhlich, J., Spencer, T.. Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys., 88 (1983), no. 2, 151–184. CrossRefGoogle Scholar
Gittins, K., Peyerimhoff, N., Stoiciu, M., Wirosoetis, D.. Some spectral applications of McMullen’s Hausdorff dimension algorithm. Conform. Geom. Dyn., 16 (2012), 184–203. CrossRefGoogle Scholar
Goldsheid, I. Ja., Molchanov, S.A., Pastur, L.A.. A random homogeneous Schrödinger operator has a pure point spectrum. Funkcional. Anal. i Priložen., 11 (1977), no. 1, 1–10, 96. Google Scholar
Graf, G.M., Vaghi, A.. A remark on the estimate of a determinant by Minami. Lett. Math. Phys., 79 (2007), no. 1, 17–22. CrossRefGoogle Scholar
Klein, A., Lenoble, O., Müller, P.. On Mott’s formula for the ac-conductivity in the Anderson model. Ann. of Math., 166 (2007), no. 2, 549–577. CrossRefGoogle Scholar
Killip, R., Nenciu, I.. Matrix models for circular ensembles. Int. Math. Res. Not. (2004), no. 50, 2665–2701. CrossRefGoogle Scholar
Killip, R., Stoiciu, M.. Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Random Matrix Ensembles. Duke Math. J., 146 (2009), no. 3, 361–399. CrossRefGoogle Scholar
McMullen, C.T.. Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math., 120 (1998), no. 4, 691–721. CrossRefGoogle Scholar
M.L. Mehta. Random matrices. Third Edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004.
Minami, N.. Local fluctuation of the spectrum of a multidimensional Anderson tight-binding model. Comm. Math. Phys., 177 (1996), 709–725. CrossRefGoogle Scholar
Molchanov, S.. Structure of the eigenfunctions of one-dimensional unordered structures. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), no. 1, 70–103, 214. Google Scholar
Molchanov, S.. The local structure of the spectrum of the one-dimensional Schrödinger operator. Comm. Math. Phys., 78 (1981), 429–446. CrossRefGoogle Scholar
P.J. Nicholls. A measure on the limit set of a discrete group. In Ergodic theory, symbolic dynamics, and hyperbolic spaces. Edited by T. Bedford, M. Keane, C. Series. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.
Patterson, S.J.. The limit set of a Fuchsian group. Acta Math., 136 (1976), no. 3-4, 241–273. CrossRefGoogle Scholar
B. Simon. Orthogonal Polynomials on the Unit Circle, Vol. 1. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, Rhode Island, 2004.
B. Simon. Orthogonal Polynomials on the Unit Circle, Vol. 2. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, Rhode Island, 2004.
Simon, B.. Fine structure of the zeros of orthogonal polynomials. I. A tale of two pictures. Electron. Trans. Numer. Anal., 25 (2006), 328–368 (electronic). Google Scholar
Simon, B.. CMV matrices: five years after. J. Comput. Appl. Math., 208 (2007), no. 1, 120–154. CrossRefGoogle Scholar
Stoiciu, M.. The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle. J. Approx. Theory, 139 (2006), 29–64. CrossRefGoogle Scholar
M. Stoiciu. Poisson Statistics for Eigenvalues: From Random Schrödinger Operators to Random CMV Matrices. CRM Proceedings and Lecture Notes, volume 42 (2007), 465–475.
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. No., 50 (1979), 171–202. CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153 (1984), no. 3-4, 259–277. CrossRefGoogle Scholar