Figure 1.
Example 1. $\alpha$ -Localized, Mirror-Symmetric
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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
Department of Mathematics and Statistics, University of Nebraska Kearney, Kearney, Nebraska 68849, USA |
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.
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 and Continuous Dynamical Systems - B,
2018, 23
(4)
: 1601-1621.
doi: 10.3934/dcdsb.2018063
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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.
|
| [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 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. |
| [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. |
| [6] |
M. Buttiker and 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 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 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. |
| [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 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. |
| [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. |
| [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 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.
|
| [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 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. |
| [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. |
| [6] |
M. Buttiker and 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 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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 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. |
| [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 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. |
| [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. |
| [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. |
Figure 2.
Example 2. Not Localized, Not Mirror-Symmetric
Figure 3.
Example 3. Localized with Equal Degree of Restriction Equal to 1, Not $\alpha$ -Localized, Not Mirror-Symmetric
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