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Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials
School of Mathematical Sciences, Yangzhou University, Yangzhou 225009, China |
In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential $ u(\omega t) $. We show that if the frequency vector $ \omega $ is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy $ E $. In contrast with previous results, the conditions of small potential $ u $ or large energy $ E $ are no longer needed.
References:
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D. Bambusi,
Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbation.I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.
doi: 10.1090/tran/7135. |
[2] |
N. N. Bogoljubov, J. A. Mitropoliskii and A. M. Samo$\breve{\mathrm{i}}$lenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-verlag, New York, 1976. |
[3] |
L. Corsi and G. Genovese,
Periodic driving at high frequencies of an impurity in the isotropic XY chain, Comm. Math. Phys., 354 (2017), 1173-1203.
doi: 10.1007/s00220-017-2917-7. |
[4] |
E. Dinaburg and Y. Sinai,
The one dimensional Schrödinger equation with a quasiperiodic potential, Funct. Anal. Appl., 9 (1975), 279-289.
|
[5] |
L. H. Eliasson,
Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.
doi: 10.1007/BF02097013. |
[6] |
L. Franzoi and A. Maspero,
Reducibility for a fast-driven linear Klein-Gordon equation, Annali. di. Matematica. Pura. ed. Applicata., 198 (2019), 1407-1439.
doi: 10.1007/s10231-019-00823-2. |
[7] |
J.-M. Fokam,
Forced Vibrations via Nash-Moser Iteration, Commun. Math. Phys., 283 (2008), 285-304.
doi: 10.1007/s00220-008-0509-2. |
[8] |
A. Fedotov and F. Klopp,
Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys., 227 (2002), 1-92.
doi: 10.1007/s002200200612. |
[9] |
J. Moser and J. Pöschel,
An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85.
|
[10] |
J. Liang and J. Xu,
A note on the extension of the Dinaburg-Sinai theorem to higher dimension, Ergod. Th. Dynam. Sys., 25 (2005), 1539-1549.
doi: 10.1017/S0143385705000118. |
[11] |
Y. Shi, Absence of eigenvalues of analytic quasi-periodic Schrödinger operators on $\mathbb{R}^d$, preprint, arXiv: 2006.11925. |
[12] |
Y. Sun, Reducibility of Schrödinger equation at high frequencies, J. Math. Phys., 61 (2020), 062701, 16 pp.
doi: 10.1063/1.5125120. |
show all references
References:
[1] |
D. Bambusi,
Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbation.I, Trans. Amer. Math. Soc., 370 (2018), 1823-1865.
doi: 10.1090/tran/7135. |
[2] |
N. N. Bogoljubov, J. A. Mitropoliskii and A. M. Samo$\breve{\mathrm{i}}$lenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-verlag, New York, 1976. |
[3] |
L. Corsi and G. Genovese,
Periodic driving at high frequencies of an impurity in the isotropic XY chain, Comm. Math. Phys., 354 (2017), 1173-1203.
doi: 10.1007/s00220-017-2917-7. |
[4] |
E. Dinaburg and Y. Sinai,
The one dimensional Schrödinger equation with a quasiperiodic potential, Funct. Anal. Appl., 9 (1975), 279-289.
|
[5] |
L. H. Eliasson,
Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.
doi: 10.1007/BF02097013. |
[6] |
L. Franzoi and A. Maspero,
Reducibility for a fast-driven linear Klein-Gordon equation, Annali. di. Matematica. Pura. ed. Applicata., 198 (2019), 1407-1439.
doi: 10.1007/s10231-019-00823-2. |
[7] |
J.-M. Fokam,
Forced Vibrations via Nash-Moser Iteration, Commun. Math. Phys., 283 (2008), 285-304.
doi: 10.1007/s00220-008-0509-2. |
[8] |
A. Fedotov and F. Klopp,
Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys., 227 (2002), 1-92.
doi: 10.1007/s002200200612. |
[9] |
J. Moser and J. Pöschel,
An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85.
|
[10] |
J. Liang and J. Xu,
A note on the extension of the Dinaburg-Sinai theorem to higher dimension, Ergod. Th. Dynam. Sys., 25 (2005), 1539-1549.
doi: 10.1017/S0143385705000118. |
[11] |
Y. Shi, Absence of eigenvalues of analytic quasi-periodic Schrödinger operators on $\mathbb{R}^d$, preprint, arXiv: 2006.11925. |
[12] |
Y. Sun, Reducibility of Schrödinger equation at high frequencies, J. Math. Phys., 61 (2020), 062701, 16 pp.
doi: 10.1063/1.5125120. |
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