# American Institute of Mathematical Sciences

May  2008, 9(3&4, May): 541-554. doi: 10.3934/dcdsb.2008.9.541

## On the spectrum of the one-dimensional Schrödinger operator

 1 Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam 2 Università di Firenze, Dipartimento di Sistemi e Informatica, Via Santa Marta 3, 50139 Firenze

Received  January 2007 Revised  July 2007 Published  February 2008

The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
Citation: Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
 [1] Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75 [2] Sijia Zhang, Shengfan Zhou. Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022056 [3] Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585 [4] Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101 [5] Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216 [6] Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047 [7] Fabien Durand, Alejandro Maass. A note on limit laws for minimal Cantor systems with infinite periodic spectrum. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 745-750. doi: 10.3934/dcds.2003.9.745 [8] Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467 [9] Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006 [10] Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537 [11] Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104 [12] Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169 [13] Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks and Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1 [14] Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171 [15] Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080 [16] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [17] Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019 [18] Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615 [19] Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125 [20] Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure and Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839

2021 Impact Factor: 1.497