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General relativity and quantum mechanics: towards a generalization of the Lambert W function A Generalization of the Lambert W Function

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Abstract

We present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. This generalization expresses the exact solutions for general-relativistic self-gravitating N-body systems in one spatial and one time dimension, and a previously unknown mathematical link between the (1+1) gravity problem and the Schrödinger wave equation.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. (9th printing), Dover, New York, 1970

  2. Arfken, G.B.: Mathematical methods for physicists, 2nd ed., Academic Press, New York, 1970

  3. Luke, Y.L.: The special functions and their approximations. Vol. 1, Academic Press, New York, 1969

  4. Scott, T.C., Babb, J.F., Dalgarno, A., Morgan III, J.D.: J. Chem. Phys. 99, 2841–2854 (1993)

    Article  Google Scholar 

  5. Corless, R., Gonnet, D., Hare, E.G., Jeffrey, D.: Lambert's W function in maple. In: Scott, T.C. (ed) MapleTech 9, (Spring 1993); Corless, R., Gonnet, G., Hare, D.E.G., Jeffrey, D., Knuth, D.: Adv. Comp. Math. 5, 329–359 (1996)

    MATH  MathSciNet  Google Scholar 

  6. Valluri, S.R., Jeffrey, D.J., Corless, R.M.: Some applications of the Lambert W function to Physics. Can. J. Phys. 78, 823–831 (2000); see also Eric Weisstein's World of Physics, (22.02.2005)

    Article  Google Scholar 

  7. Cranmer, S.R.: Am. J. Phys. 72, 1397–1403 (2004). http://cfa-www.harvard.edu/scranmer/News2004/

    Google Scholar 

  8. Mann, R.B., Ohta, T.: Phys. Rev. D. 55, 4723–4747 (1997)

    Article  Google Scholar 

  9. Certain, P.R., Byers Brown, W.: Branch Point Singularities in the Energy of the Delta-Function Model of One-Electron Diatoms. Intern. J. Quantum Chem. 6, 131–142 (1972); Whitton, W.N., Byers Brown, W., Int. J. Quantum Chem. 10, 71–86 (1976)

    Article  Google Scholar 

  10. Martinez II, R.E.: Irrationality from tetration. in preparation, (2005)

  11. http://functions.wolfram.com/ElementaryFunctions/ProductLog/16/02/01/

  12. Malecki, J.J., Mann, R.B.: Phys. Rev. E. 69, 1–26 (2004)

    Google Scholar 

  13. Campbell, S.A.: Dynamics of continuous discrete and impulsive systems. 5, 225–235 (1999)

  14. MacDonald, N.: Biological delay systems: linear stability theory. Cambridge University Press, 1969; an der Heiden, U.: J. Math. Biol. 8, 345–364 (1979)

    MathSciNet  Google Scholar 

  15. Imambekov, A., Demler, E.: Exactly solvable one-dimensional Bose-Fermi mixture. private communication, 2005

  16. Mann, R.B., Scott, T.C., Martinez, R.E.: Lineal gravity and the Schroedinger wave equation. in preparation, 2005

  17. Scott, T.C., Aubert-Frécon, M., Grotendorst, J.: New approach for the electronic energies of the hydrogen molecular ion, Chem. Phys., in press (2005)

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Authors and Affiliations

  1. Institut für Physikalishe Chemie, RWTH-Aachen, Fachbereich Chemie, Universität Duisburg-Essen, and Zentralinstitut für Angewandte Mathematik (ZAM) Forschungszentrum Jülich GmbH, Germany

    Tony C. Scott

  2. Department of Physics, Professor of Physics and Applied Mathematics, University of Waterloo, Ontario, Canada, N2L 3G1

    Robert Mann

  3. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5

    Robert Mann

  4. Harvard University, 9 Oxford Street, Cambridge, MA, 02138, USA

    Roberto E. Martinez II

Authors
  1. Tony C. Scott
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  2. Robert Mann
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  3. Roberto E. Martinez II
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Correspondence to Tony C. Scott.

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Scott, T., Mann, R. & Martinez II, R. General relativity and quantum mechanics: towards a generalization of the Lambert W function A Generalization of the Lambert W Function. AAECC 17, 41–47 (2006). https://doi.org/10.1007/s00200-006-0196-1

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  • DOI: https://doi.org/10.1007/s00200-006-0196-1

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