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The hyperbolic cross space approximation of electronic wavefunctions. (English) Zbl 1116.78007
Author’s abstract: 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 continues the author’s recent work [Numer. Math. 98, 731–759 (2004; Zbl 1062.35100)] on the regularity of its solutions, the electronic wavefunctions. It was shown in the mentioned article that these wavefunctions possess square integrable high-order mixed weak derivatives as they are needed to justify and underpin sparse grid and hyperbolic cross space approximation techniques theoretically. How fast the norms of these derivatives can increase with the number of electrons is studied in the present article. Provided that all quantities are properly related to a characteristic lengthscale of the considered atomar or molecular system, the norms of the derivatives remain bounded independent of the number of electrons and the number, the positions, and the charges of the nuclei.

MSC:
78A35 Motion of charged particles
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
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