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June  2022, 11(3): 869-894. doi: 10.3934/eect.2021029

Persistence of superoscillations under the Schrödinger equation

1. 

Department of Mathematics & Statistics, Saint Louis University, Saint Louis, MO 63103, USA

2. 

Department of Mathematics & Statistics, Washington University in Saint Louis, Saint Louis, MO 63130, USA

* Corresponding author: Elodie Pozzi

Received  December 2020 Revised  April 2021 Published  June 2022 Early access  July 2021

Fund Project: The second author is supported in part by a National Science Foundation DMS grant # 1800057 and Australian Research Council – DP 190100970

The goal of this paper is to provide new proofs of the persistence of superoscillations under the Schrödinger equation. Superoscillations were first put forward by Aharonov and have since received much study because of connections to physics, engineering, signal processing and information theory. An interesting mathematical question is to understand the time evolution of superoscillations under certain Schrödinger equations arising in physics. This paper provides an alternative proof of the persistence of superoscillations by some elementary convergence facts for sequence and series and some connections with certain polynomials and identities in combinatorics. The approach given opens new perspectives to establish persistence of superoscillations for the Schrödinger equation with more general potentials.

Citation: Elodie Pozzi, Brett D. Wick. Persistence of superoscillations under the Schrödinger equation. Evolution Equations and Control Theory, 2022, 11 (3) : 869-894. doi: 10.3934/eect.2021029
References:
[1]

Y. AharonovJ. BehrndtF. Colombo and P. Schlosser, Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations, J. Differential Equations, 277 (2021), 153-190.  doi: 10.1016/j.jde.2020.12.029.

[2]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, How superoscillating tunneling waves can overcome the step potential, Ann. Physics, 414 (2020), 168088, 19 pp. doi: 10.1016/j.aop.2020.168088.

[3]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Schrödinger evolution of superoscillations under different potentials, Quantum Stud. Math. Found., 5 (2018), 485-504.  doi: 10.1007/s40509-018-0161-2.

[4]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, Evolution of superoscillatory initial data in several variables in uniform electric field, J. Phys. A, 50 (2017), 185201, 19 pp. doi: 10.1088/1751-8121/aa66d9.

[5]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, The mathematics of superoscillations, Mem. Amer. Math. Soc., 247 (2017). doi: 10.1090/memo/1174.

[6]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Superoscillating sequences in several variables, J. Fourier Anal. Appl., 22 (2016), 751-767.  doi: 10.1007/s00041-015-9436-8.

[7]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Superoscillating sequences as solutions of generalized Schrödinger equations, J. Math. Pures Appl., 103 (2015), 522-534.  doi: 10.1016/j.matpur.2014.07.001.

[8]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, Evolution of superoscillatory data, J. Phys. A, 47 (2014), 20531, 18 pp. doi: 10.1088/1751-8113/47/20/205301.

[9]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, On some operators associated to superoscillations, Complex Anal. Oper. Theory, 7 (2013), 1299-1310.  doi: 10.1007/s11785-012-0227-9.

[10]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, On the Cauchy problem for the Schrödinger equation with superoscillatory initial data, J. Math. Pures Appl., 99 (2013), 165-173.  doi: 10.1016/j.matpur.2012.06.008.

[11]

T. AokiF. ColomboI. Sabadini and D. C. Struppa, Continuity theorems for a class of convolution operators and applications to superoscillations, Ann. Mat. Pura Appl., 197 (2018), 1533-1545.  doi: 10.1007/s10231-018-0736-x.

[12]

T. AokiF. ColomboI. Sabadini and D. C. Struppa, Continuity of some operators arising in the theory of superoscillations, Quantum Stud. Math. Found., 5 (2018), 463-476.  doi: 10.1007/s40509-018-0159-9.

[13]

T. O. de Carvalho, Exact space-time propagator for the step potential, Phys. Rev. A, 47 (1993), 2562-2573.  doi: 10.1103/PhysRevA.47.2562.

[14]

F. Colombo, J. Gantner and D. C. Struppa, Evolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potential, Proc. A., 475 (2019), 20180390, 17 pp. doi: 10.1098/rspa.2018.0390.

[15]

F. Colombo, J. Gantner and D. C. Struppa, Evolution of superoscillations for Schrödinger equation in a uniform magnetic field, J. Math. Phys., 58 (2017), 092103, 17 pp. doi: 10.1063/1.4991489.

[16]

F. Colombo, I. Sabadini and D. C. Struppa, An introduction to superoscillatory sequences, in Noncommutative Analysis, Operator Theory and Applications, Oper. Theory Adv. Appl., Vol. 252, Birkhäuser/Springer, 2016,105–121. doi: 10.1007/978-3-319-29116-1_7.

[17]

F. ColomboI. SabadiniD. C. Struppa and A. Yger, Gauss sums, superoscillations and the Talbot carpet, J. Math. Pures Appl., 147 (2021), 163-178.  doi: 10.1016/j.matpur.2020.07.011.

[18]

F. ColomboD. C. Struppa and A. Yger, Superoscillating sequences towards approximation in $S$ or $S'$-type spaces and extrapolation, J. Fourier Anal. Appl., 25 (2019), 242-266.  doi: 10.1007/s00041-018-9592-8.

[19]

E. HightT. OrabyJ. Palacio and E. Suazo, On persistence of superoscillations for the Schrödinger equation with time-dependent quadratic Hamiltonians, Math. Methods Appl. Sci., 43 (2020), 1660-1674.  doi: 10.1002/mma.5992.

[20]

G. Tsaur and J. Wang, Constructing Green functions of the Schrödinger equation by elementary transformations, Amer. J. Phys., 74 (2006), 600-606.  doi: 10.1119/1.2186688.

show all references

References:
[1]

Y. AharonovJ. BehrndtF. Colombo and P. Schlosser, Green's function for the Schrödinger equation with a generalized point interaction and stability of superoscillations, J. Differential Equations, 277 (2021), 153-190.  doi: 10.1016/j.jde.2020.12.029.

[2]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, How superoscillating tunneling waves can overcome the step potential, Ann. Physics, 414 (2020), 168088, 19 pp. doi: 10.1016/j.aop.2020.168088.

[3]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Schrödinger evolution of superoscillations under different potentials, Quantum Stud. Math. Found., 5 (2018), 485-504.  doi: 10.1007/s40509-018-0161-2.

[4]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, Evolution of superoscillatory initial data in several variables in uniform electric field, J. Phys. A, 50 (2017), 185201, 19 pp. doi: 10.1088/1751-8121/aa66d9.

[5]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, The mathematics of superoscillations, Mem. Amer. Math. Soc., 247 (2017). doi: 10.1090/memo/1174.

[6]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Superoscillating sequences in several variables, J. Fourier Anal. Appl., 22 (2016), 751-767.  doi: 10.1007/s00041-015-9436-8.

[7]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, Superoscillating sequences as solutions of generalized Schrödinger equations, J. Math. Pures Appl., 103 (2015), 522-534.  doi: 10.1016/j.matpur.2014.07.001.

[8]

Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa and J. Tollaksen, Evolution of superoscillatory data, J. Phys. A, 47 (2014), 20531, 18 pp. doi: 10.1088/1751-8113/47/20/205301.

[9]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, On some operators associated to superoscillations, Complex Anal. Oper. Theory, 7 (2013), 1299-1310.  doi: 10.1007/s11785-012-0227-9.

[10]

Y. AharonovF. ColomboI. SabadiniD. C. Struppa and J. Tollaksen, On the Cauchy problem for the Schrödinger equation with superoscillatory initial data, J. Math. Pures Appl., 99 (2013), 165-173.  doi: 10.1016/j.matpur.2012.06.008.

[11]

T. AokiF. ColomboI. Sabadini and D. C. Struppa, Continuity theorems for a class of convolution operators and applications to superoscillations, Ann. Mat. Pura Appl., 197 (2018), 1533-1545.  doi: 10.1007/s10231-018-0736-x.

[12]

T. AokiF. ColomboI. Sabadini and D. C. Struppa, Continuity of some operators arising in the theory of superoscillations, Quantum Stud. Math. Found., 5 (2018), 463-476.  doi: 10.1007/s40509-018-0159-9.

[13]

T. O. de Carvalho, Exact space-time propagator for the step potential, Phys. Rev. A, 47 (1993), 2562-2573.  doi: 10.1103/PhysRevA.47.2562.

[14]

F. Colombo, J. Gantner and D. C. Struppa, Evolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potential, Proc. A., 475 (2019), 20180390, 17 pp. doi: 10.1098/rspa.2018.0390.

[15]

F. Colombo, J. Gantner and D. C. Struppa, Evolution of superoscillations for Schrödinger equation in a uniform magnetic field, J. Math. Phys., 58 (2017), 092103, 17 pp. doi: 10.1063/1.4991489.

[16]

F. Colombo, I. Sabadini and D. C. Struppa, An introduction to superoscillatory sequences, in Noncommutative Analysis, Operator Theory and Applications, Oper. Theory Adv. Appl., Vol. 252, Birkhäuser/Springer, 2016,105–121. doi: 10.1007/978-3-319-29116-1_7.

[17]

F. ColomboI. SabadiniD. C. Struppa and A. Yger, Gauss sums, superoscillations and the Talbot carpet, J. Math. Pures Appl., 147 (2021), 163-178.  doi: 10.1016/j.matpur.2020.07.011.

[18]

F. ColomboD. C. Struppa and A. Yger, Superoscillating sequences towards approximation in $S$ or $S'$-type spaces and extrapolation, J. Fourier Anal. Appl., 25 (2019), 242-266.  doi: 10.1007/s00041-018-9592-8.

[19]

E. HightT. OrabyJ. Palacio and E. Suazo, On persistence of superoscillations for the Schrödinger equation with time-dependent quadratic Hamiltonians, Math. Methods Appl. Sci., 43 (2020), 1660-1674.  doi: 10.1002/mma.5992.

[20]

G. Tsaur and J. Wang, Constructing Green functions of the Schrödinger equation by elementary transformations, Amer. J. Phys., 74 (2006), 600-606.  doi: 10.1119/1.2186688.

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