Abstract
It is proved by functional analytic methods that for S-state solutions of Schrödinger's equation for the helium atom, Fock's expansion in powers of R 1/2 and R ln R, where R is the hyperspherical radius r 21 +r 22 , converges pointwise for all R, thereby generalising a result of Macek that the expansion converges in the mean for all R<1/2. It is shown that for any value (even complex) of the energy E, Schrödinger's equation, considered as a partial differential equation with no boundary condition at R=∞, has infinitely many solutions representable by an expansion of the type proposed by Fock. Some of the open problems are discussed in determining whether for E in the point spectrum of the atomic Hamiltonian the physical eigenfunction ΨE, which has exponential decay as R →∞, is representable by Fock's expansion.
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Morgan, J.D. Convergence properties of Fock's expansion for S-state eigenfunctions of the helium atom. Theoret. Chim. Acta 69, 181–223 (1986). https://doi.org/10.1007/BF00526420
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DOI: https://doi.org/10.1007/BF00526420