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The time-dependent Schrödinger equation with piecewise constant potentials

Part of: Representations of solutions Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 September 2018

NATALIE E. SHEILS  and
BERNARD DECONINCK
NATALIE E. SHEILS
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: nesheils@umn.edu
BERNARD DECONINCK
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA email: deconinc@uw.edu
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Abstract

The linear Schrödinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with physically important models. In examples such as ‘particle in a box’ and tunnelling, attention is restricted to the time-independent Schrödinger equation. This paper combines the unified transform method and recent insights for interface problems to present fully explicit solutions for the time-dependent problem.

Keywords

Unified Transform MethodFokas MethodSchrödinger equation

MSC classification

Primary: 35B30: Dependence of solutions on initial and boundary data, parameters
Secondary: 35B40: Asymptotic behavior of solutions 35C05: Solutions in closed form 35C15: Integral representations of solutions 35Q40: PDEs in connection with quantum mechanics 35Q41: Time-dependent Schrödinger equations, Dirac equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 57 - 83
Copyright
© Cambridge University Press 2018 

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References

Ablowitz, M. & Fokas, A. (2003) Complex Variables: Introduction and Applications, 2nd ed. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Ablowitz, M. & Segur, H. (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.CrossRefGoogle Scholar
Agranovich, Z. & Marchenko, V. (1963) The Inverse Problem of Scattering Theory, Gordon and Breach, New York.Google Scholar
Asvestas, M., Sifalakis, A., Papadopoulou, E. & Saridakis, Y. (2014) Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity. J. Phys. Conf. Ser. 490(1), 012143.CrossRefGoogle Scholar
Bender, C. & Orszag, S. (1978) Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Co., New York, NY.Google Scholar
Biondini, G. & Trogdon, T. (2015) Gibbs phenomenon for dispersive PDEs, arXiv preprint arXiv:1411.6142.Google Scholar
Deconinck, B., Pelloni, B. & Sheils, N. (2014) Non-steady state heat conduction in composite walls. Proc. R. Soc. A 470(2165), 1–22.CrossRefGoogle ScholarPubMed
Deconinck, B., Sheils, N. & Smith, D. (2016) The linear KdV equation with an interface. Commun. Math. Phys. 347(2), 489–509.CrossRefGoogle Scholar
Deconinck, B., Trogdon, T. & Vasan, V. (2014) The method of Fokas for solving linear partial differential equations. SIAM Rev. 56(1), 159–186.CrossRefGoogle Scholar
Deift, P. & Trubowitz, E. (1979) Inverse scattering on the line. Commun. Pure Appl. Math. 32(2), 121–251.CrossRefGoogle Scholar
Faddeyev, L. D. & Seckler, B. (1963) The inverse problem in the quantum theory of scattering. J. Math. Phys. 4(1), 72–104.CrossRefGoogle Scholar
Feynman, R., Leighton, R. & Sands, M. (1964) The Feynman Lectures on Physics, vol. II, Addison-Wesley, Reading, MA.CrossRefGoogle Scholar
Fokas, A. (2005) The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Comm. Pure Appl. Math. 58(5), 639–670.CrossRefGoogle Scholar
Fokas, A. (2008) 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.CrossRefGoogle Scholar
Fokas, A. & Pelloni, B. (2005) A transform method for linear evolution PDEs on a finite interval. IMA J. Appl. Math. 70(4), 564–587.CrossRefGoogle Scholar
Griffiths, D. (2004) Introduction to Quantum Mechanics, 2nd ed., Pearson Prentice Hall, New York, New York.Google Scholar
Janowicz, M. (2003) Method of multiple scales in quantum optics. Phys. Rep. 375(5), 327–410.CrossRefGoogle Scholar
Kevorkian, J. & Cole, J. (1996) Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, Springer, New York.CrossRefGoogle Scholar
Landau, L. & Lifshitz, E. (1981) Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Elsevier Science, Burlington, MA.Google Scholar
Langer, R. (1929) The asymptotic location of the roots of a certain transcendental equation. Trans. Amer. Math. Soc. 31(4), 837–844.CrossRefGoogle Scholar
Langer, R. (1931) The zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37, 213–239.CrossRefGoogle Scholar
Levin, D. (1996) Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67(1), 95–101.CrossRefGoogle Scholar
Levitan, B. & Sargsjan, I. (1975) Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators. American Mathematical Society, Providence, RI. Translated from the Russian by Amiel Feinstein, Translations of Mathematical Monographs, Vol. 39.CrossRefGoogle Scholar
Levitan, B. & Sargsjan, I. (1991) Sturm–Liouville and Dirac Operators. Mathematics and Its Applications, Vol. 59. Kluwer Academic Publishers, Dordrecht, Netherlands.CrossRefGoogle Scholar
Mantzavinos, D., Papadomanolaki, M., Saridakis, Y. & Sifalakis, A. (2016) Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in 1 + 1 dimensions. Appl. Numer. Math. 104, 47–61.CrossRefGoogle Scholar
Merzbacher, E. (1970) Quantum Mechanics, 2nd ed., John Wiley & Sons, New York, NY.Google Scholar
Miller, P. D. (2006) Applied Asymptotic Analysis. Graduate Studies in Mathematics, Vol. 75, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Razavy, M. (2003) Quantum Theory of Tunneling, World Scientific, Singapore.CrossRefGoogle Scholar
Rybalko, Y. (2016) Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the uniform transform method. arXiv preprint arXiv:1601.08008, p. 12.Google Scholar
Sheils, N. & Deconinck, B. (2014) Heat conduction on the ring: Interface problems with periodic boundary conditions. Appl. Math. Lett. 37, 107–111.CrossRefGoogle Scholar
Sheils, N. & Deconinck, B. (2015) Interface problems for dispersive equations. Stud. Appl. Math. 134(3), 253–275.CrossRefGoogle Scholar
Sheils, N. & Deconinck, B. (2016) Initial-to-interface maps for the heat equation on composite domains. Stud. Appl. Math. 137(1), 140–154.CrossRefGoogle Scholar
Sheils, N. & Smith, D. (2015) Heat equation on a network using the Fokas method. J. Phys. A: Math. Theor. 48(33), 1–21.CrossRefGoogle Scholar
Sylvester, J. & Uhlmann, G. (1990) The Dirichlet to Neumann map and applications. In: Colton, D., Ewing, R. & Rundell, W. (editors) Inverse Problems in Partial Differential Equations. Proceedings in Applied Mathematics Series, SIAM, Philadelphia, PA, pp. 101–139.Google Scholar
Trogdon, T. (2012) Riemann–Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions. PhD thesis, University of Washington.Google Scholar
Trogdon, T. (2015) A unified numerical approach for the nonlinear Schrödinger equations. In: Fokas, A. & Pelloni, B. (editors) Unified Transform for Boundary Value Problems: Applications and Advances, SIAM, Philadelphia, PA.Google Scholar