Revivals and fractalisation in the linear free space Schrödinger equation
Authors:
Peter J. Olver, Natalie E. Sheils and David A. Smith
Journal:
Quart. Appl. Math. 78 (2020), 161-192
MSC (2010):
Primary 35Q41; Secondary 35B30, 35B65
DOI:
https://doi.org/10.1090/qam/1547
Published electronically:
July 8, 2019
MathSciNet review:
4077460
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Additional Information
Abstract: We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (“rational”) times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other (“irrational”) times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogenous linear boundary conditions in terms of numerically computable eigenfunctions. Alternative solution formulas are derived using the Unified Transform Method (UTM), that can prove useful in more general situations. We then investigate the effects of general linear boundary conditions, including Robin, and find novel “dissipative” revivals in the case of energy decreasing conditions.
References
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References
- Mark J. Ablowitz and Athanassios S. Fokas, Complex variables: introduction and applications, 2nd ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003. MR 1989049
- M. V. Berry, Quantum fractals in boxes, J. Phys. A 29 (1996), no. 20, 6617–6629. MR 1421017, DOI https://doi.org/10.1088/0305-4470/29/20/016
- M. V. Berry and S. Klein, Integer, fractional and fractal Talbot effects, J. Modern Opt. 43 (1996), no. 10, 2139–2164. MR 1410175, DOI https://doi.org/10.1080/095003496154761
- M.V. Berry, I. Marzoli, and Schleich W., Quantum carpets, carpets of light, Physics Wolrd 14 (2001), no. 6, 39–44.
- G. Chen and P.J. Olver, Dispersion of discontinuous periodic waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2149, 21pp. MR 3001377
- Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI https://doi.org/10.1137/110821871
- P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322
- Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953
- A. S. Fokas and B. Pelloni, A transform method for linear evolution PDEs on a finite interval, IMA J. Appl. Math. 70 (2005), no. 4, 564–587. MR 2156459, DOI https://doi.org/10.1093/imamat/hxh047
- A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion, Appl. Phys. Lett. 23 (1973), no. 3, 142–144.
- A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. normal dispersion, Appl. Phys. Lett. 23 (1973), no. 4, 171–172.
- Emine Kesici, Beatrice Pelloni, Tristan Pryer, and David Smith, A numerical implementation of the unified Fokas transform for evolution problems on a finite interval, European J. Appl. Math. 29 (2018), no. 3, 543–567. MR 3788455, DOI https://doi.org/10.1017/S0956792517000316
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- P.J. Olver, Dispersive quantization, Amer. Math. Monthly 117 (2010), no. 7, 599–610. MR 2681520
- Peter J. Olver, Introduction to partial differential equations, Undergraduate Texts in Mathematics, Springer, Cham, 2014. MR 3136142
- E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, The Physical Review 28 (1926December), no. 6, 1049–1070.
- D.A. Smith, Spectral theory of ordinary and partial linear differential operators on finite intervals, Ph.D., 2011.
- David A. Smith, Well-posed two-point initial-boundary value problems with arbitrary boundary conditions, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 3, 473–496. MR 2911141, DOI https://doi.org/10.1017/S030500411100082X
- H.F. Talbot, Facts related to optical science. no. IV, Philos. Mag. 9 (1836), 401–407.
- Bernd Thaller, Visual quantum mechanics, Springer-Verlag–TELOS, New York, 2000. Selected topics with computer-generated animations of quantum-mechanical phenomena; With 1 CD-ROM (Windows and Macintosh). MR 1764328
- M.J.J. Vrakking, D.M. Villeneuve, and A. Stolow, Observation of fractional revivals of a molecular wave packet, Phys. Rev. A 54 (1996July), R37–R40.
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- J.A. Yeazell and C.R. Stroud, Observation of fractional revivals in the evolution of a Rydberg atomic wave packet, Phys. Rev. A 43 (1991May), 5153–5156.
- V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics 9 (1968), no. 2, 190–194 (English).
- V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP. 35 (1972), 908–914.
- A.K. Zvezdin and A.F. Popkov, Contribution to the nonlinear theory of magnetostatic spin waves, Sov. Phys. JETP. 57 (1983February), no. 2, 350–355.
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Additional Information
Peter J. Olver
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID:
133815
Natalie E. Sheils
Affiliation:
UnitedHealth Group R&D, Minnetonka, Minnesota 55343
MR Author ID:
929715
David A. Smith
Affiliation:
Division of Science, Yale-NUS College, 16 College Avenue West, #01-220 138527, Singapore
MR Author ID:
975701
Received by editor(s):
December 19, 2018
Received by editor(s) in revised form:
May 27, 2019
Published electronically:
July 8, 2019
Article copyright:
© Copyright 2019
Brown University
