×

zbMATH — the first resource for mathematics

Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. (English) Zbl 1084.65125
This paper is concerned with the approximate solution of the electronic Schrödinger equation that is used to model the molecular behaviour of some particle systems in quantum mechanics. The author presents some results that complement his recent publication [Numer. Math. 98, 731–759 (2004; Zbl 1062.35100)] by proposing suitable sets of sparse grid trial functions so that the dimension of the trial spaces is reduced to a tractable level. An extensive bibliography that is relevant for the problem under consideration is also included.

MSC:
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J10 Schrödinger operator, Schrödinger equation
35B65 Smoothness and regularity of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
81P05 General and philosophical questions in quantum theory
81V25 Other elementary particle theory in quantum theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, S.: Lectures on the Exponential Decay of Solutions of Second-Order Elliptic Operators. Princeton: Princeton University Press, 1981
[2] Atkins, P.W., Friedman, R.S.: Molecular Quantum Mechanics. Oxford: Oxford University Press, 1997
[3] Babenko, K.I.: Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Soviet Math. Dokl. 1, 672–675 (1960) · Zbl 0102.05301
[4] Bungartz, H.J., Griebel, M.: Sparse Grids. Acta Numerica 2004, pp. 1–123 · Zbl 1118.65388
[5] Flad, H.J., Hackbusch, W., Kolb, D., Schneider, R.: Wavelet approximation of correlated wave functions. I. Basics. Journ. Chem. Phys. 116, 9641–9657 (2002)
[6] Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergard Sørensen, T.: Sharp regularity estimates for Coulombic many-electron wave functions. Commun. Math. Phys. 255, 183–227 (2005) · Zbl 1075.35063 · doi:10.1007/s00220-004-1257-6
[7] Gårding, L.: On the essential spectrum of Schrödinger operators. J. Funct. Anal. 52, 1–10 (1983) · Zbl 0522.35084 · doi:10.1016/0022-1236(83)90087-3
[8] Garcke, J., Griebel, M.: On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comput. Phys. 165, 694–716 (2000) · Zbl 0979.65101 · doi:10.1006/jcph.2000.6627
[9] Hackbusch, W.: The efficient computation of certain determinants arising in the treatment of Schrödinger’s equation. Computing 67, 35–56 (2000) · Zbl 0997.65075 · doi:10.1007/s006070170015
[10] Helgaker, T., Jørgensen, P., Olsen, J.: Molecular electronic structure theory. New York: John Wiley and Sons, 2001
[11] Hunziker, W., Sigal, I.M.: The quantum N-body problem. J. Math. Phys. 41, 3448–3510 (2000) · Zbl 0981.81026 · doi:10.1063/1.533319
[12] Luo, H., Kolb, D., Flad, H.J, Hackbusch, W., Koprucki, T.: Wavelet approximation of correlated wave functions. II. Hyperbolic wavelets and adaptive approximation schemes. J. Chem. Phys. 117, 3625–3638 (2002)
[13] Kato, T.: On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure and Appl. Math. 10, 151–177 (1957) · Zbl 0077.20904 · doi:10.1002/cpa.3160100201
[14] Le Bris, C. (Ed.): Handbook of Numerical Analysis, Vol. X: Computational Chemistry. Amsterdam: North Holland, 2003 · Zbl 1052.81001
[15] Messiah, A.: Quantum Mechanics. New York: Dover Publications, 2000 · Zbl 0102.42602
[16] Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. San Diego: Academic Press, 1978 · Zbl 0401.47001
[17] Shannon, C.E.: A mathematical theory of communication. Bell Sys. Tech. J. 27, 379–423, 623–656 (1948) · Zbl 1154.94303
[18] Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37, 10–21 (1949) · doi:10.1109/JRPROC.1949.232969
[19] Simon, B.: Schrödinger operators in the twentieth century. J. Math. Phys. 41, 3523–3555 (2000) · Zbl 0981.81025 · doi:10.1063/1.533321
[20] Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963) · Zbl 0202.39901
[21] Whittaker, J.M.: On the functions which are represented by the expansions of the interpolation theory. Proc. Roy. Soc. Edinburgh 35, 181–194 (1915) · JFM 45.0553.02
[22] Whittaker, J.M.: Interpolation Function Theory. Cambridge: Cambridge University Press, 1935 · Zbl 0012.15503
[23] Yserentant, H.: On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98, 731–759 (2004) · Zbl 1062.35100 · doi:10.1007/s00211-003-0498-1
[24] Yserentant, H.: On the electronic Schrödinger equation. Lecture Notes, Universität Tübingen 2003; accessible via the author’s homepage · Zbl 1062.35100
[25] Zenger, C.: Sparse grids. In: W. Hackbusch (ed.) Parallel Algorithms for Partial Differential Equations. Proceedings, Kiel 1990, pp. 241–251. Notes on Numerical Fluid Mechanics vol. 31, Braunschweig Wiesbaden: Vieweg, 1991 · Zbl 0763.65091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.