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On integral separation of bounded linear random differential equations
Characterizations of uniform hyperbolicity and spectra of CMV matrices
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 |
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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