Abstract
The approximation of fixed points by numerical fixed points was presented in the elegant monograph of Krasnosel’skii et al. (1972). The theory, both in its formulation and implementation, requires a differential operator calculus, so that its actual application has been selective. The writer and Kerkhoven demonstrated this for the semiconductor drift-diffusion model in 1991. In this article, we show that the theory can be applied to time dependent quantum systems on bounded domains, via evolution operators. In addition to the kinetic operator term, the Hamiltonian includes both an external time dependent potential and the classical nonlinear Hartree potential. Our result can be paraphrased as follows: For a sequence of Galerkin subspaces, and the Hamiltonian just described, a uniquely defined sequence of Faedo–Galerkin solutions exists; it converges in Sobolev space, uniformly in time, at the maximal rate given by the projection operators.
A Notation and norms
We employ complex Hilbert spaces in this article.
However, ∫Ωfg dx is interpreted as
For f ∈ L2, as just defined, if each component fj satisfies fj ∈
∫Ω∇f ⋅ ∇g dx is interpreted as
Finally, H–1 is defined as the dual of
References
[1] R. Adams and J. Fournier, Sobolev Spaces. 2nd. ed., Elsevier / Academic Press, Amsterdam–Boston, 2003.Search in Google Scholar
[2] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994.10.1007/978-1-4757-4338-8Search in Google Scholar
[3] A. Castro and M. A. L. Marques, Propagators for the time dependent Kohn–Sham equations. In: Time Dependent Density Functional Theory, Lect. Notes in Phys., Springer, 706 (2006), 197–210.10.1007/3-540-35426-3_12Search in Google Scholar
[4] T. Cazenave, Semilinear Schrödinger Equations. Courant Inst. Lecture Notes, Vol. 10, Amer. Math. Society, 2003.10.1090/cln/010Search in Google Scholar
[5] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations. Oxford Science Publ., Oxford, 1998.Search in Google Scholar
[6] Z. Chen and E. Polizzi, Spectral-based propagation schemes for time-dependent quantum systems with applications to carbon nanotubes. Phys. Rev. B82 (2010), 205410.10.1103/PhysRevB.82.205410Search in Google Scholar
[7] J. W. Jerome, Approximation of Nonlinear Evolution Systems. Academic Press, 1983.Search in Google Scholar
[8] J. W. Jerome, An asymptotically linear fixed point extension of the inf–sup theory of Galerkin approximation. Numer. Funct. Anal. Optim. 16 (1995), No. 3-4, 345–361.10.1080/01630569508816622Search in Google Scholar
[9] J. W. Jerome, Analysis of Charge Transport: A mathematical study of semiconductor devices. Springer-Verlag, Berlin, 1996.10.1007/978-3-642-79987-7Search in Google Scholar
[10] J. W. Jerome, Time-dependent closed quantum systems: Nonlinear Kohn–Sham potential operators and weak solutions. J. Math. Anal. Appl. 429 (2015), 995–1006.10.1016/j.jmaa.2015.04.047Search in Google Scholar
[11] J. W. Jerome, The quantum Faedo–Galerkin equation: Evolution operator and time discretization. Numer. Funct. Anal. Optim. 38 (2017), No. 5, 590–601.10.1080/01630563.2016.1252393Search in Google Scholar
[12] J. W. Jerome, Convergent iteration in Sobolev space for time dependent closed quantum systems. Nonlinear Analysis: Real World Appl. 40 (2018), 130–147.10.1016/j.nonrwa.2017.08.016Search in Google Scholar
[13] J. W. Jerome and T. Kerkhoven, A finite element approximation theory for the drift-diffusion semiconductor model. SIAM J. Numer. Anal. 28 (1991), No. 2, 403–422.10.1137/0728023Search in Google Scholar
[14] J. W. Jerome and E. Polizzi, Discretization of time dependent quantum systems: Real time propagation of the evolution operators. Appl. Anal. 93 (2014), 2574–2597.10.1080/00036811.2013.878863Search in Google Scholar
[15] T. Kato, Linear equations of hyperbolic type. J. Fac. Sci. Univ. Tokyo17 (1970), 241–258.Search in Google Scholar
[16] T. Kato, Linear equations of hyperbolic type II. J. Math. Soc. Japan25 (1973), 648–666.10.2969/jmsj/02540648Search in Google Scholar
[17] A. N. Kolmogorov, Über die beste Annähergung von Funktionen einer gegebenen Funktionklasse. Ann. Math. 37 (1936), No. 2, 107–111.10.2307/1968691Search in Google Scholar
[18] M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rititskii, and V. Ya. Stetsenko, Approximate Solution of Operator Equations. Wolters-Noordhoff, Groningen, 1972.10.1007/978-94-010-2715-1Search in Google Scholar
[19] E. Runge and E. K. U. Gross, Density functional theory for time dependent systems. Phys. Rev. Lett. 52 (1984), 997–1000.10.1103/PhysRevLett.52.997Search in Google Scholar
[20] M. Sprengel, G. Ciarmella, and A. Borzi, A theoretical investigation of time-dependent Kohn–Sham equations, SIAM J. Math. Anal. 49 (2017), 1681–1704.10.1137/15M1053517Search in Google Scholar
[21] C. A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications. Oxford University Press, Oxford, 2012.10.1093/acprof:oso/9780199563029.001.0001Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
