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Spectral properties of limit-periodic Schrödinger operators
On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows
1. | Dipartimento di Sistemi e Informatica, Università di Firenze, 50139 Firenze |
2. | Department of Mathematics, Rutgers University, Camden NJ 08102, United States |
$ |\gamma^i - \frac{p^i_n}{q_n}| \leq Ce^{-q^{1+\delta}_n}, 1\leq i\leq d, n\in N,$
where $C > 0$ and $\delta > 0$ are some constants and $p^i_n, q_n$ are some sequences of integers with $q_n\to \infty$.
References:
[1] |
J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696.
doi: doi:10.1017/S0143385702001165. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.
doi: doi:10.4007/annals.2005.161.1423. |
[3] |
Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due," Ph.D. Thesis, Università di Firenze, 1997. |
[4] |
R. Fabbri and R. Johnson, On the Lyapunov exponent of certain $SL(2,R)$ valued cocycles, Differential Equations and Dynamical Systems, 7 (1999), 349-370. |
[5] |
R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Analysis: Real World Applications, 3 (2002), 37-59.
doi: doi:10.1016/S1468-1218(01)00012-8. |
[6] |
R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients, Jour. Diff. Equn., 61 (1986), 54-78.
doi: doi:10.1016/0022-0396(86)90125-7. |
[7] |
R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.
doi: doi:10.1007/BF01208484. |
[8] |
R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.
doi: doi:10.1137/0518001. |
[9] |
J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici, 56 (1981), 198-224.
doi: doi:10.1007/BF02566210. |
[10] |
M. Nerurkar, Positive exponents for a dense class of continuous $SL(2,R)$ valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math., 215 (1998), 265-278. |
[11] |
M. Nerurkar, Density of positive Lyapunov exponents in the smooth category, preprint (2008). |
show all references
References:
[1] |
J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems, 22 (2002), 1667-1696.
doi: doi:10.1017/S0143385702001165. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic maps, Ann. of Math., 161 (2005), 1423-1485.
doi: doi:10.4007/annals.2005.161.1423. |
[3] |
Roberta Fabbri, "Genericità dell'iperbolicità nei sistemi differenziali lineari di dimensione due," Ph.D. Thesis, Università di Firenze, 1997. |
[4] |
R. Fabbri and R. Johnson, On the Lyapunov exponent of certain $SL(2,R)$ valued cocycles, Differential Equations and Dynamical Systems, 7 (1999), 349-370. |
[5] |
R. Fabbri, R. Johnson and R. Pavani, On the nature of the spectrum of the quasi-periodic Schrödinger operator, Nonlinear Analysis: Real World Applications, 3 (2002), 37-59.
doi: doi:10.1016/S1468-1218(01)00012-8. |
[6] |
R. Johnson, Exponential dichotomy, rotation number and linear differential operatorss with bounded coefficients, Jour. Diff. Equn., 61 (1986), 54-78.
doi: doi:10.1016/0022-0396(86)90125-7. |
[7] |
R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.
doi: doi:10.1007/BF01208484. |
[8] |
R. Johnson, K. Palmer and G. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.
doi: doi:10.1137/0518001. |
[9] |
J. Moser, An example of a Schrodinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici, 56 (1981), 198-224.
doi: doi:10.1007/BF02566210. |
[10] |
M. Nerurkar, Positive exponents for a dense class of continuous $SL(2,R)$ valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math., 215 (1998), 265-278. |
[11] |
M. Nerurkar, Density of positive Lyapunov exponents in the smooth category, preprint (2008). |
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