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  • 1
    Electronic Resource
    Electronic Resource
    JWKB connection-formula problem revisited again via Borel summation (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 4417-4429 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Silverstone [Phys. Rev. Lett. 55, 2523–2526 (1985)] has expressed the opinion that the traditional version of the JWKB connection formulas associated with a classical turning point is incorrect and that the correct version follows from the Borel-based summability of the asymptotic expansions for the Airy functions. In the present paper we show that this assertion is incorrect. After the Borel summation of the asymptotic expressions for the Airy functions has actually been performed explicitly, there are no connection problems, and thus no connection formulas, but exact relations expressing the functions Ai(z) and Bi(z) in terms of the Whittaker function, although with two defects: Due to the derivation there appear false restrictions on arg z, and there is an artificial cut that coincides with the Stokes line along the real z-axis. We also show that these exact relations can be obtained more appropriately in a direct way with the aid of standard formulas available in handbooks, whereby the defects mentioned do not appear. On the other hand, when one uses truncated asymptotic series, the Stokes phenomenon gives rise to connection problems, and the connection formulas, which one uses for handling these connection problems, are inherently one-directional. This property of the connection formulas is demonstrated and discussed anew, in general terms as well as with special regard to the Airy functions. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532517
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  • 2
    Electronic Resource
    Electronic Resource
    The energy levels and the corresponding normalized wave functions for a model of a compressed atom (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1813-1826 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In the model of a compressed atom (or ion) considered in the present paper the boundary condition associated with the corresponding uncompressed atom, i.e., the condition that the radial wave function must vanish at r=∞, is replaced by the boundary condition that the radial wave function must have a node at the finite distance r=a. The treatment of the problem of obtaining the energy shift due to the compression is based on the phase-integral method developed by Fröman and Fröman, an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage. The treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by Furry [Phys. Rev. 71, 360 (1947)] and Yngve [J. Math. Phys. 13, 324 (1972)], in which one also uses exact formulas in the calculations and makes all approximations in the final stage. Since compression of an atom gives rise to very subtle effects, rigorous methods are indispensible for obtaining accurate and reliable analytical final formulas. As an application, the resulting general formulas are particularized to the case of a hydrogenic atom, and a numerical illustration of the accuracy of the formulas is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527441
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  • 3
    Electronic Resource
    Electronic Resource
    Comments on the paper "On the WKBJ approximation'' [J. Math. Phys. 28, 556 (1987)] (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 912-912 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Attention is drawn to the fact that the "standard form for the generalized WKBJ approximation'' of El Sawi [J. Math. Phys. 28, 556 (1987)] already had been derived by N. Fröman [Ark. Fys. 32, 541 (1966)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527988
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  • 4
    Electronic Resource
    Electronic Resource
    Determination of the potential from experimental data on energies and widths of quasi-stationary levelsDedicated to Professor Ivar Waller. (1989)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 35 (1989), S. 751-760 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Formulas for the energies and widths of shape resonances, i.e., quasi-stationary levels for a quantal particle moving in a spherically symmetric potential, valid also close to the top of the potential barrier, have been published by Drukarev, Fröman, and Fröman. On the basis of these formulas, in the present paper the potential is explicitly expressed in terms of experimental data on the energies and widths of the quasi-stationary states. The treatment is related to previous treatments by Wheeler and by Cole and Good, Jr., which are in turn related to the Rydberg-Klein-Rees method.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560350611
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