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On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives

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Summary.

The electronic Schrödinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry. The present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the Pauli principle. It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials. The result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces. It indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic Schrödinger equation and that it may even be possible to reduce the computational complexity of an N-electron problem to that of a one-electron problem.

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References

  1. Babenko, K.I.: Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Soviet Math. Dokl. 1, 672–675 (1960)

    MATH  Google Scholar 

  2. Bungartz, H.J.: Finite Elements of Higher Order on Sparse Grids, Habilitationsschrift, Fakultät für Informatik, TU München. Aachen: Shaker Verlag 1998

  3. DeVore, R.A., Konyagin, S.V., Temlyakov, V.N.: Hyperbolic wavelet approximation. Constr. Approx. 14, 1–26 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Flad, H.J., Hackbusch, W., Kolb, D., Schneider, R.: Wavelet approximation of correlated wave functions. I. Basics. J. Chem. Phys. 116, 9461–9657 (2002)

    Google Scholar 

  5. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergard Sørensen, T.: The electron density is smooth away from the nuclei. Commun. Math. Phys. 228, 401–415 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MATH  Google Scholar 

  7. Griebel, M., Knapek, S.: Optimized tensor-product approximation spaces. Constr. Approx. 16, 525–540 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hackbusch, W.: The efficient computation of certain determinants arising in the treatment of Schrödinger’s equation. Computing 67, 35–56 (2000)

    Article  MATH  Google Scholar 

  9. Helgaker, T., Jørgensen, P., Olsen, J.: Molecular electronic structure theory. New York: John Wiley and Sons, 2001

  10. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergard Sørensen, T.: Electron wavefunctions and densities for atoms. Ann. Henri Poincaré 2, 77–100 (2001)

    MathSciNet  Google Scholar 

  11. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Stremnitzer, H.: Local properties of coulombic wave functions. Commun. Math. Phys. 163, 185–215 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Hunziker, W., Sigal, I. M.: The quantum N-body problem. J. Math. Phys. 41, 3448–3510 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc. 70, 195–221 (1951)

    MathSciNet  MATH  Google Scholar 

  15. Kato, T.: On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure and Appl. Math. 10, 151–177 (1957)

    MathSciNet  MATH  Google Scholar 

  16. Parr, R.G., Yang, W.: Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press, 1994

  17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. San Diego: Academic Press, 1978

  18. Simon, B.: Schrödinger operators in the twentieth century. J. Math. Phys. 41, 3523–3555 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)

    Google Scholar 

  20. Szabo, A., Ostlund, N.S.: Modern Quantum Chemistry, Introduction to Advanced Electronic Structure Theory. New York: Dover Publications, 1996

  21. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Berlin: Springer-Verlag, 1932

  22. Yserentant, H.: On the electronic Schrödinger equation. Lecture Notes, Universität Tübingen 2003; accessible via http://na.uni-tuebingen.de

  23. 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

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  1. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Harry Yserentant

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  1. Harry Yserentant
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Correspondence to Harry Yserentant.

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Mathematics Subject Classification (1991): 35J10, 35B65, 41A63, 65D99

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Yserentant, H. On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98, 731–759 (2004). https://doi.org/10.1007/s00211-003-0498-1

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  • DOI: https://doi.org/10.1007/s00211-003-0498-1

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