A theoretical investigation of time-dependent Kohn-Sham equations: new proofs. (English) Zbl 1479.35769

Summary: In this paper, a new analysis for the existence, uniqueness, and regularity of solutions to a time-dependent Kohn-Sham equation is presented. The Kohn-Sham equation is a nonlinear integral Schrödinger equation that is of great importance in many applications in physics and computational chemistry. To deal with the time-dependent, nonlinear and non-local potentials of the Kohn-Sham equation, the analysis presented in this manuscript makes use of energy estimates, fixed-point arguments, regularization techniques, and direct estimates of the non-local potential terms. The assumptions considered for the time-dependent and nonlinear potentials make the obtained theoretical results suitable to be used also in an optimal control framework.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35K55 Nonlinear parabolic equations
82M36 Computational density functional analysis in statistical mechanics
81Q99 General mathematical topics and methods in quantum theory


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[1] Hohenberg, P.; Kohn, W., Inhomogeneous electron gas, Phys Rev, 136, B864-B871 (1964)
[2] Kohn, W.; Sham, LJ., Self-consistent equations including exchange and correlation effects, Phys Rev, 140, A1133-A1138 (1965)
[3] Runge, E.; Gross, EKU., Density-functional theory for time-dependent systems, Phys Rev Lett, 52, 997-1000 (1984)
[4] Marques, MAL, Ullrich, CA, Nogueira, F, Rubio, A, Burke, K, Gross, EKU.Time-dependent density functional theory, volume 706 of Lecture notes in physics. Berlin Heidelberg: Springer-Verlag; 2006.
[5] Ruggenthaler, M.; Penz, M.; van Leeuwen, R., Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory, J Phys Condens Matter, 27 (2015)
[6] van Leeuwen, R., Mapping from densities to potentials in time-dependent density-functional theory, Phys Rev Lett, 82, 3863-3866 (1999)
[7] Borzì, A.; Ciaramella, G.; Sprengel, M., Formulation and numerical solution of quantum control problems (2017), Philadelphia, PA: Society for Industrial and Applied Mathematics, Philadelphia, PA · Zbl 1376.93047
[8] Castro, A.; Appel, H.; Oliveira, M.; Rozzi, CA; Andrade, X.; Lorenzen, F.; Marques, MAL; Gross, EKU; Rubio, A., OCTOPUS: a tool for the application of time-dependent density functional theory, Phys Status Solidi (b), 243, 11, 2465-2488 (2006)
[9] Castro, A.; Werschnik, J.; Gross, EKU., Controlling the dynamics of many-electron systems from first principles: A combination of optimal control and time-dependent density-functional theory, Phys Rev Lett, 109 (2012)
[10] Sprengel, M.; Ciaramella, G.; Borzì, A., A COKOSNUT code for the control of the time-dependent Kohn-Sham model, Comput Phys Commun, 214, 231-238 (2017) · Zbl 1376.93047
[11] Sprengel, M.; Ciaramella, G.; Borzì, A., Investigation of optimal control problems governed by a time-dependent Kohn-Sham model, J Dyn Control Syst, 24, 4, 657-679 (2018) · Zbl 1407.35170
[12] Cancés, E.; Le Bris, C., On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics, Math Models Methods Appl Sci, 9, 7, 963-990 (1999) · Zbl 1011.81087
[13] Cazenave, T., Semilinear Schrödinger equations. Courant lecture notes in mathematics (2003), Providence, Rhode Island: American Mathematical Society, Providence, Rhode Island
[14] Jerome, JW., Approximation of nonlinear evolution systems. Mathematics in science and engineering (1983), New York: Elsevier Science, New York
[15] Maspero, A.; Robert, D., On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms, J Funct Anal, 273, 2, 721-781 (2017) · Zbl 1366.35153
[16] Jerome, JW., Time dependent closed quantum systems: nonlinear Kohn-Sham potential operators and weak solutions, J Math Anal Appl, 429, 2, 995-1006 (2015) · Zbl 1405.35174
[17] Jerome, JW., Consistency of local density approximations and quantum corrections for time-dependent quantum systems, Appl Anal, 0, 1-21 (2019) · Zbl 1451.35151
[18] Jerome, JW; Polizzi, E., Discretization of time-dependent quantum systems: real-time propagation of the evolution operator, Appl Anal, 93, 2574-2597 (2014) · Zbl 1304.35591
[19] Sprengel, M.; Ciaramella, G.; Borzì, A., A theoretical investigation of time-dependent Kohn-Sham equations, SIAM J Math Anal, 49, 3, 1681-1704 (2017) · Zbl 1365.35132
[20] Adams, RA., Sobolev spaces (1970), Oxford, London: Academic Press, Oxford, London
[21] Ciarlet, PG., Linear and nonlinear functional analysis with applications (2013), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia
[22] Evans, LC.Partial differential equations, volume 19 of Graduate studies in mathematics, 2nd ed. Providence, RI: American Mathematical Society; 2010.
[23] Grisvard, P., Elliptic problems in nonsmooth domains. Classics in applied mathematics (2011), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1231.35002
[24] Gazzola, HC, Grunau, F, Sweers, G.Polyharmonic boundary value problems. Lecture Notes in Mathematics. Springer; 2010. · Zbl 1239.35002
[25] Walter, W.Ordinary differential equations. Graduate texts in mathematics. New York: Springer; 1998.
[26] Boyer, F.; Fabrie, P., Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Applied mathematical sciences (2012), New York: Springer, New York
[27] Lions, JL.Quelques Méthodes de Resolution des Problèmes aux Limites non Linéaires. Dunod. Paris: Gauthier-Villars; 1969.
[28] Yserentant, H.Regularity and approximability of electronic wave functions. Lecture notes in mathematics. Berlin Heidelberg: Springer; 2010. · Zbl 1204.35003
[29] Lieb, EH, Loss, M.Analysis. CRM proceedings & lecture notes. American Mathematical Society; 2001.
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