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June  2018, 23(4): 1601-1621. doi: 10.3934/dcdsb.2018063

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

Katherine A. Kime ,

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

* Corresponding author: Katherine A. Kime

Received  May 2017 Published  June 2018 Early access  February 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.

Keywords: Mirror, symmetry, palindromic, control, potential, Schrödinger, complex-valued matrix.
Mathematics Subject Classification: Primary: 93B03, 93B40, 93C20; Secondary: 81Q05, 81Q93.
Citation: Katherine A. Kime. Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1601-1621. doi: 10.3934/dcdsb.2018063
References:
[1]

G. D. Akrivis and 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.   Google Scholar

[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.  Google Scholar

[3]

D. Bohm, Quantum Theory, Dover Publications Inc., New York, 1989. Google Scholar

[4]

U. Boscain, J.-P. Gauthier, F. Rossi and 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.  Google Scholar

[5]

T. Boykin and 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.  Google Scholar

[6]

M. Buttiker and R. Landauer, Traversal time for tunneling, Advances in Solid State Physics, 25 (2007), 711-717.  doi: 10.1007/BFb0108208.  Google Scholar

[7]

R. Burden and J. Faires, Numerical Analysis, 5th edition, PWS, Boston, 1993. Google Scholar

[8]

T. Chan and 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.  Google Scholar

[9]

K. Beauchard and 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.  Google Scholar

[10]

A. Goldberg, H. Schey and 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.  Google Scholar

[11]

A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Communications in Mathematical Physics, 174 (1995), 149-159.  doi: 10.1007/BF02099468.  Google Scholar

[12]

A. Kacar and 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.  Google Scholar

[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.   Google Scholar

[14]

I. Lasiecka and 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.  Google Scholar

[15]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[16]

M. Morancey and 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.  Google Scholar

[17]

A. Nissen, G. Kreiss and 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.  Google Scholar

[18]

K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition, McGraw Hill, New York, 2007. Google Scholar

[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.  Google Scholar

[20]

L. I. Schiff, Quantum Mechanics, McGraw Hill, New York, 1968. Google Scholar

[21]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

show all references

References:
[1]

G. D. Akrivis and 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.   Google Scholar

[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.  Google Scholar

[3]

D. Bohm, Quantum Theory, Dover Publications Inc., New York, 1989. Google Scholar

[4]

U. Boscain, J.-P. Gauthier, F. Rossi and 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.  Google Scholar

[5]

T. Boykin and 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.  Google Scholar

[6]

M. Buttiker and R. Landauer, Traversal time for tunneling, Advances in Solid State Physics, 25 (2007), 711-717.  doi: 10.1007/BFb0108208.  Google Scholar

[7]

R. Burden and J. Faires, Numerical Analysis, 5th edition, PWS, Boston, 1993. Google Scholar

[8]

T. Chan and 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.  Google Scholar

[9]

K. Beauchard and 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.  Google Scholar

[10]

A. Goldberg, H. Schey and 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.  Google Scholar

[11]

A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Communications in Mathematical Physics, 174 (1995), 149-159.  doi: 10.1007/BF02099468.  Google Scholar

[12]

A. Kacar and 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.  Google Scholar

[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.   Google Scholar

[14]

I. Lasiecka and 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.  Google Scholar

[15]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[16]

M. Morancey and 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.  Google Scholar

[17]

A. Nissen, G. Kreiss and 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.  Google Scholar

[18]

K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition, McGraw Hill, New York, 2007. Google Scholar

[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.  Google Scholar

[20]

L. I. Schiff, Quantum Mechanics, McGraw Hill, New York, 1968. Google Scholar

[21]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Review, 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

Figure 1.  Example 1. $\alpha$-Localized, Mirror-Symmetric
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Figure 2.  Example 2. Not Localized, Not Mirror-Symmetric
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Figure 3.  Example 3. Localized with Equal Degree of Restriction Equal to 1, Not $\alpha$-Localized, Not Mirror-Symmetric
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