# American Institue of Mathematical Sciences

## Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations

 Department of Mathematics and Statistics, University of Nebraska Kearney, Kearney, Nebraska 68849, USA

* Corresponding author: Katherine A. Kime

Received  May 2017 Published  January 2018

We consider discrete potentials as controls in systems of finite difference equations which are discretizations of a 1-D Schrödinger equation. We give examples of palindromic potentials which have corresponding steerable initial-terminal pairs which are not mirror-symmetric. For a set of palindromic potentials, we show that the corresponding steerable pairs that satisfy a localization property are mirror-symmetric. We express the initial and terminal states in these pairs explicitly as scalar multiples of vector-valued functions of a parameter in the control.

Citation: Katherine A. Kime. Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018063
##### References:
 [1] G. D. Akrivis, V. A. Dougalis, Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain, Bulletin Greek Mathematical Society, 31 (1990), 19-28. [2] K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl., 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005. [3] D. Bohm, Quantum Theory, Dover Publications Inc., New York, 1989. [4] U. Boscain, J.-P. Gauthier, F. Rossi, M. Sigalotti, Approximate controllability, exact controllability and conical eigenvalue intersectons for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239. doi: 10.1007/s00220-014-2195-6. [5] T. Boykin, G. Klimeck, The discretized Schrödinger equation and simple models for semiconductor quantum wells, Eur. J. Phys., 25 (2004), 503-514. doi: 10.1088/0143-0807/25/4/006. [6] M. Buttiker, R. Landauer, Traversal time for tunneling, Advances in Solid State Physics, 25 (2007), 711-717. doi: 10.1007/BFb0108208. [7] R. Burden and J. Faires, Numerical Analysis, 5th edition, PWS, Boston, 1993. [8] T. Chan, L. Shen, Stability analysis of difference schemes for variable coefficient Schrödinger type equations, SIAM. J. Numer. Anal., 24 (1987), 336-349. doi: 10.1137/0724025. [9] K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, Journal of Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021. [10] A. Goldberg, H. Schey, J. Schwartz, Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena, American Journal of Physics, 35 (1967), 177-186. doi: 10.1119/1.1973991. [11] A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Communications in Mathematical Physics, 174 (1995), 149-159. doi: 10.1007/BF02099468. [12] A. Kacar, O. Terzioglu, Symbolic computation of the potential in a nonlinear Schrödinger Equation, Numer. Methods Partial Differential Equations, 23 (2007), 475-483. doi: 10.1002/num.20192. [13] K. Kime, Finite difference approximation of control via the potential in a 1-D Schrodinger equation, Electronic Journal of Differential Equations, 2000 (2000), 1-10. [14] I. Lasiecka, R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33. doi: 10.1016/0022-247X(90)90330-I. [15] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [16] M. Morancey, V. Nersesyan, Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations, J. Math. Pures Appl., 103 (2015), 228-254. doi: 10.1016/j.matpur.2014.04.002. [17] A. Nissen, G. Kreiss, M. Gerritsen, High order stable finite difference methods for the Schrödinger equation, J. Sci. Comput., 55 (2013), 173-199. doi: 10.1007/s10915-012-9628-1. [18] K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition, McGraw Hill, New York, 2007. [19] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Applied Mathematics, 52 (1973), 189-211. doi: 10.1002/sapm1973523189. [20] L. I. Schiff, Quantum Mechanics, McGraw Hill, New York, 1968. [21] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243. doi: 10.1137/S0036144503432862.

show all references

##### References:
 [1] G. D. Akrivis, V. A. Dougalis, Finite difference discretization with variable mesh of the Schrödinger equation in a variable domain, Bulletin Greek Mathematical Society, 31 (1990), 19-28. [2] K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl., 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005. [3] D. Bohm, Quantum Theory, Dover Publications Inc., New York, 1989. [4] U. Boscain, J.-P. Gauthier, F. Rossi, M. Sigalotti, Approximate controllability, exact controllability and conical eigenvalue intersectons for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239. doi: 10.1007/s00220-014-2195-6. [5] T. Boykin, G. Klimeck, The discretized Schrödinger equation and simple models for semiconductor quantum wells, Eur. J. Phys., 25 (2004), 503-514. doi: 10.1088/0143-0807/25/4/006. [6] M. Buttiker, R. Landauer, Traversal time for tunneling, Advances in Solid State Physics, 25 (2007), 711-717. doi: 10.1007/BFb0108208. [7] R. Burden and J. Faires, Numerical Analysis, 5th edition, PWS, Boston, 1993. [8] T. Chan, L. Shen, Stability analysis of difference schemes for variable coefficient Schrödinger type equations, SIAM. J. Numer. Anal., 24 (1987), 336-349. doi: 10.1137/0724025. [9] K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, Journal of Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021. [10] A. Goldberg, H. Schey, J. Schwartz, Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena, American Journal of Physics, 35 (1967), 177-186. doi: 10.1119/1.1973991. [11] A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Communications in Mathematical Physics, 174 (1995), 149-159. doi: 10.1007/BF02099468. [12] A. Kacar, O. Terzioglu, Symbolic computation of the potential in a nonlinear Schrödinger Equation, Numer. Methods Partial Differential Equations, 23 (2007), 475-483. doi: 10.1002/num.20192. [13] K. Kime, Finite difference approximation of control via the potential in a 1-D Schrodinger equation, Electronic Journal of Differential Equations, 2000 (2000), 1-10. [14] I. Lasiecka, R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33. doi: 10.1016/0022-247X(90)90330-I. [15] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [16] M. Morancey, V. Nersesyan, Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations, J. Math. Pures Appl., 103 (2015), 228-254. doi: 10.1016/j.matpur.2014.04.002. [17] A. Nissen, G. Kreiss, M. Gerritsen, High order stable finite difference methods for the Schrödinger equation, J. Sci. Comput., 55 (2013), 173-199. doi: 10.1007/s10915-012-9628-1. [18] K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition, McGraw Hill, New York, 2007. [19] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Applied Mathematics, 52 (1973), 189-211. doi: 10.1002/sapm1973523189. [20] L. I. Schiff, Quantum Mechanics, McGraw Hill, New York, 1968. [21] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243. doi: 10.1137/S0036144503432862.
Example 1. $\alpha$-Localized, Mirror-Symmetric
Example 2. Not Localized, Not Mirror-Symmetric
Example 3. Localized with Equal Degree of Restriction Equal to 1, Not $\alpha$-Localized, Not Mirror-Symmetric
 [1] Lu Gan Liu, ming Wei. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096 [2] Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124 [3] Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11/12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 [4] Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173 [5] Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems & Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033 [6] Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 [7] Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 [8] Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161 [9] M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473 [10] Bo Wang, Jiguang Bao. Mirror symmetry for a Hessian over-determined problem and its generalization. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2305-2316. doi: 10.3934/cpaa.2014.13.2305 [11] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [12] M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982 [13] Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033 [14] Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 [15] Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631 [16] Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure & Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 [17] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [18] Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827 [19] Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 [20] Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831

2016 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables