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The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces

Abstract

We continue the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schrödinger operators and their exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order \({\vartheta}\) for \({\vartheta < 3/4}\) . The bound 3/4 is best possible and can neither be reached nor surpassed. Such results are important for the study of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.

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Correspondence to Harry Yserentant.

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This research was supported by the DFG-Priority Program 1324 and the DFG-Research Center Matheon.

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Kreusler, HC., Yserentant, H. The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces. Numer. Math. 121, 781–802 (2012). https://doi.org/10.1007/s00211-012-0447-y

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Mathematics Subject Classification (2000)

  • 35J10
  • 35B65
  • 41A25
  • 41A63