Publication Cover
Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 12
133
Views
8
CrossRef citations to date
0
Altmetric
Articles

Discretization of time-dependent quantum systems: real-time propagation of the evolution operator

&
Pages 2574-2597 | Received 13 Sep 2013, Accepted 18 Dec 2013, Published online: 06 Feb 2014

References

  • Castro A, Marques MAL. Time dependent density functional theory. Lec. Notes in Phys. 2006;706:197–210.
  • Mikhailova TY, Pupyshev VI. Symmetric approximations for the evolution operator. Phys. Lett. A. 1999;257:1–6.
  • Polizzi E. Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B. 2009;79:p115112.
  • FEAST eigenvalue solver. Available from: http://www.feast-solver.org.
  • Crank J, Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Adv. Comput. Math. 1996;6:207–226.
  • Yajima K. Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 1987;10:415–426.
  • Chen Z, Polizzi E. Spectral-based propagation schemes for time-dependent quantum systems with applications to carbon nanotubes. Phys. Rev. B. 2010;82:205410, 8 p.
  • Magnus W. On the exponential solutions of differential equations for a linear operator. Commun. Pure Appl. Math. 1954;VII:649–673.
  • Alvermann A, Fehske H. High-order commutator-free exponential time propagation of driven quantum systems. J. Comp. Phys. 2011;230:5930–5956.
  • Cancès E, Le Bris C. On the time-dependent Hartree-Fock equations coupled with a classical nonlinear dynamics. Math. Models Meth. Appl. Sc. 1999;93:963–990.
  • Cazenave T. Semilinear Schrödinger equations. Vol. 10. Courant Institute Lecture Notes. Providence, RI:American Mathematical Society; 2003.
  • Elgart A, Erdös L, Schlein B, Yau H-T. Nonlinear Hartree equation as the mean field limit of weakly coupled Fermions. J. Math. Pures. Appl. 2004;83:1241–1273.
  • Kato T. Linear equations of hyperbolic type. J. Fac. Sc. Univ. Tokyo. 1970;17:241–258.
  • Kato T. Linear equations of hyperbolic type II. J. Math. Soc. Japan. 1973;25:648–666.
  • Dorroh JR. A simplified proof of a theorem of Kato on linear evolution equations. J. Math. Soc. Japan. 1975;27:474–478.
  • Jerome JW. Approximation of nonlinear evolution equations. New York (NY): Academic Press; 1983.
  • Kohn W, Vashista P. in Lunqvbist S, March NH[eds] Theory of the inhomogeneous electron gas. New York (NY): Plenum Press; 1983. Chapter 2, General density functional theory. pp. 79-147
  • Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965;140:A1133–A1138.
  • Prodan E, Nordlander P. On the Kohn-Sham equations with periodic background potentials. J. Stat. Phys. 2003;111:967–992.
  • Freeman AJ, Wimmer E. Density functional theory as a major tool in computational materials science. Annu. Rev. Mater. Sci. 1995;25:7–36.
  • Runge E, Gross EKU. Density functional theory for time dependent systems. Phys. Rev. Lett. 1984;52:997–1000.
  • Le Bris C, Lions P-L. From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. (N.S.). 2005;42:291–363.
  • Castro A, Marques MAL, Rubio A. Propagators for the time-dependent Kohn-Sham equations. J. Chem. Phys. 2004;121:3425–3433.
  • Stone M. Linear transformations in Hilbert space. IV. Proc. Nat. Acad. Sci. USA. 1929;15:198–200.
  • Lax PD. Functional analysis. New York (NY): Wiley-Interscience; 2002.
  • Kato T. Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Amer. Math. Soc. 1951;70:195–211.
  • Kato T. On the existence of solutions of the helium wave equation. Trans. Amer. Math. Soc. 1951;70:212–218.
  • Caspers W, Sweers G. Point interactions on bounded domains. Proc. Roy. Soc. Edinburgh Sect. A. 1994;124:917–926.
  • Calderón AP. Commutators of singular integral operators. Proc. Nat. Acad. Sci. 1965;53:1092–1099.
  • Sauer T. Numerical analysis. Pearson Addison Wesley; 2006.
  • Lehtovaara L, Havu V, Puska M. All-electron time-dependent density functional theory with finite elements: Time-propagation approach. J. Chemical Physics. 2011;135:154104.
  • Stoer J, Bulirsch R. Introduction to numerical analysis. 3rd ed. Vol. 12. Texts in applied mathematics. Springer Verlag; 2002.
  • Bramble JH, Hilbert SR. Estimation of linear functionals on Sobolev spaces with applications to Fourier analysis and spline interpolation. SIAM J. Numer. Anal. 1970;7:112–124.
  • Munkres, JR. Topology: a first course. Upper Saddle River, NJ:Prentice-Hall; 1975.
  • Gilbarg D, Trudinger N. Elliptic partial differential equations of second order. Reprint of the 1998 ed. Classics in mathematics. Berlin: Springer-Verlag; 2001.
  • Polizzi E, Ben Abdallah N, Vanbésien O, Lippens D. Space lateral transfer and negative differential conductance regimes in quantum waveguide junctions. J. Appl. Phys. 2000;87:8700–8706.
  • Chen Z, Polizzi E. Spectral modeling and propagation schemes for time-dependent quantum systems. 14th International Workshop on Computational Electronics, IWCE proceedings; 2010; p. 295–298.
  • Yabana K, Nakatsukasa T, Iwata J-I, Bertsch GF. Real-time, real-space implementation of the linear response time-dependent density-functional theory. Phys. Stat. Sol. (b). 2006;243:1121–1138.
  • Ullrich CA. Time-dependent density-functional theory: concepts and applications. Oxford University Press; 2012.
  • Addagarla T, Polizzi E. Forthcoming.
  • Hughes T, Kato T, Marsden J. Well-posed quasi-linear, second order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 1977;63:273–294.
  • Aubin J-P. Applied functional analysis. Wiley Interscience; 1979.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.