attractive
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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►At positive energies , , and:
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►Both variable sets may be used for attractive and repulsive potentials: the set cannot be used for a zero potential because this would imply for all , and the set cannot be used for zero energy because this would imply always.
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►The functions defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977).
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Attractive potentials: | , . |
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2: 33.1 Special Notation
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►The main functions treated in this chapter are first the Coulomb radial functions , , (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions , , , (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.
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3: 18.39 Applications in the Physical Sciences
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►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with protons, involve Laguerre polynomials of fractional index.
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►As in this subsection both positive (repulsive) and negative (attractive) Coulomb interactions are discussed, the prefactor of in (18.39.43) has been set to , rather than the of (18.39.28) implying that is an attractive interaction, being repulsive.
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►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3).
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►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified:
…The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009).
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4: 18.40 Methods of Computation
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►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
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5: 3.6 Linear Difference Equations
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►Difference equations are simple and attractive for computation.
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6: Bibliography B
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The attractive Coulomb potential polynomials.
Constr. Approx. 1 (2), pp. 103–119.
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7: Bibliography S
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Coulomb functions for attractive and repulsive potentials and for positive and negative energies.
Comput. Phys. Comm. 146 (2), pp. 225–249.
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