Characterizations of uniform hyperbolicity and spectra of CMV matrices

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August 2016, 9(4): 1009-1023. doi: 10.3934/dcdss.2016039

Characterizations of uniform hyperbolicity and spectra of CMV matrices

David Damanik ^1, , Jake Fillman ^2, , Milivoje Lukic ^3, and William Yessen ^1,

1.	Department of Mathematics, Rice University, Houston, TX 77005, United States
2.	Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, United States
3.	Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Received September 2014 Revised July 2015 Published August 2016

Abstract
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We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.

Keywords: Linear cocycles, generalized eigenfunctions, CMV matrices, uniform hyperbolicity, orthogonal polynomials..

Mathematics Subject Classification: Primary: 37D20, 42C05; Secondary: 37A2.

Citation: David Damanik, Jake Fillman, Milivoje Lukic, William Yessen. Characterizations of uniform hyperbolicity and spectra of CMV matrices. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1009-1023. doi: 10.3934/dcdss.2016039

References:

[1]	Ju. M. Berezanskii, Expansions in Eigenfuncions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.
[2]	J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. doi: 10.1007/s00209-009-0494-y.
[3]	D. Damanik, J. Fillman, M. Lukic and W. Yessen, Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model, Int. Math. Res. Not., 2015 (2015), 7110-7129. doi: 10.1093/imrn/rnu158.
[4]	D. Damanik, J. Fillman and D. C. Ong, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices, J. Math. Pures Appl., 105 (2016), 293-341. doi: 10.1016/j.matpur.2015.11.002.
[5]	J. Geronimo and R. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations, 132 (1996), 140-178. doi: 10.1006/jdeq.1996.0175.
[6]	F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139 (2006), 172-213. doi: 10.1016/j.jat.2005.08.002.
[7]	R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Diff. Eq., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.
[8]	Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.
[9]	M. Lukic and D. Ong, Generalized Prüfer variables for perturbations of Jacobi and CMV matrices, J. Math. Anal. Appl., in press. DOI:10.1016/j.jmaa.2016.07.036. (arXiv:1409.7116).
[10]	P. Munger and D. Ong, The Hölder continuity of spectral measures of an extended CMV matrix, J. Math. Phys., 55 (2014), 093507, 10 pp. doi: 10.1063/1.4895762.
[11]	D. Ong, Purely singular continuous spectrum for CMV operators generated by subshifts, J. Stat. Phys., 155 (2014), 763-776. doi: 10.1007/s10955-014-0974-2.
[12]	M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
[13]	R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems I., J. Diff. Eq., 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.
[14]	R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Diff. Eq., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.
[15]	J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.
[16]	B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, American Mathematical Society, Providence, RI, 2005.
[17]	J.-C. Yoccoz, Some questions and remarks about SL$(2,\mathbbR)$ cocycles, Modern Dynamical Systems and Applications, 447-458, Cambridge Univ. Press, Cambridge, 2004.
[18]	Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, preprint, (arXiv:1305.4226).

show all references

References:

[1]	Ju. M. Berezanskii, Expansions in Eigenfuncions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.
[2]	J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231. doi: 10.1007/s00209-009-0494-y.
[3]	D. Damanik, J. Fillman, M. Lukic and W. Yessen, Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model, Int. Math. Res. Not., 2015 (2015), 7110-7129. doi: 10.1093/imrn/rnu158.
[4]	D. Damanik, J. Fillman and D. C. Ong, Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices, J. Math. Pures Appl., 105 (2016), 293-341. doi: 10.1016/j.matpur.2015.11.002.
[5]	J. Geronimo and R. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations, 132 (1996), 140-178. doi: 10.1006/jdeq.1996.0175.
[6]	F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139 (2006), 172-213. doi: 10.1016/j.jat.2005.08.002.
[7]	R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Diff. Eq., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.
[8]	Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.
[9]	M. Lukic and D. Ong, Generalized Prüfer variables for perturbations of Jacobi and CMV matrices, J. Math. Anal. Appl., in press. DOI:10.1016/j.jmaa.2016.07.036. (arXiv:1409.7116).
[10]	P. Munger and D. Ong, The Hölder continuity of spectral measures of an extended CMV matrix, J. Math. Phys., 55 (2014), 093507, 10 pp. doi: 10.1063/1.4895762.
[11]	D. Ong, Purely singular continuous spectrum for CMV operators generated by subshifts, J. Stat. Phys., 155 (2014), 763-776. doi: 10.1007/s10955-014-0974-2.
[12]	M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
[13]	R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems I., J. Diff. Eq., 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.
[14]	R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Diff. Eq., 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.
[15]	J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), 359-390. doi: 10.1090/S0002-9947-1975-0368080-X.
[16]	B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, American Mathematical Society, Providence, RI, 2005.
[17]	J.-C. Yoccoz, Some questions and remarks about SL$(2,\mathbbR)$ cocycles, Modern Dynamical Systems and Applications, 447-458, Cambridge Univ. Press, Cambridge, 2004.
[18]	Z. Zhang, Resolvent set of Schrödinger operators and uniform hyperbolicity, preprint, (arXiv:1305.4226).

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