An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem
Authors:
C. D. Luning and W. L. Perry
Journal:
Quart. Appl. Math. 35 (1977), 257-268
MSC:
Primary 34B25; Secondary 81.34
DOI:
https://doi.org/10.1090/qam/445056
MathSciNet review:
445056
Full-text PDF Free Access
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Abstract: Development of an iterative solution technique for a certain nonlinear eigenvalue problem supplies an iterative solution technique for the ion case and isolated neutral atom case boundary-value problems for the Thomas-Fermi equation.
L. H. Thomas, The calculation of atomic fields, Proc. Camb. Phil. Soc. 23, 542–548 (1927)
E. Fermi, Un metodo statistico par la determinazione di alcune proprietá dell’ atome, Rend. Accad. Naz. del Lincei. CI. sci. fis., mat. e nat. (6) 6, 602–607 (1927)
P. Csavinszky, Calculation of diamagnetic susceptibilities of ions using a universal approximate analytical solution of the Thomas-Fermi equation, Bull. Amer. Phys. Soc. (2) 18, 726–727 (1973)
P. Csavinszky, Universal approximate analytical solution of the Thomas-Fermi equation for ions, Phys. Rev. A(3) 8, 1688–1701 (1973)
Y. Bae Suh, Perturbation calculation using Thomas-Fermi model, Phys. Lett. 49A, 99–100 (1974)
E. H. Lieb and B. Simon, Thomas-Fermi theory revisited, Phys. Rev. Lett. 31, 681–683 (1973)
- Elliott H. Lieb and Barry Simon, The Thomas-Fermi theory of atoms, molecules and solids, Advances in Math. 23 (1977), no. 1, 22–116. MR 428944, DOI https://doi.org/10.1016/0001-8708%2877%2990108-6
P. Gombás, Die statistische Theorie des Atoms, Springer, Berlin, 1949
N. H. March, The Thomas-Fermi approximation in quantum mechanics, Adv. Phys. 6, 1–101 (1957)
- James S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975), 339–360. MR 367368, DOI https://doi.org/10.1137/1017036
E. B. Baker, The application of the Thomas-Fermi statistical model to the calculation of potential distribution of positive ions, Phys. Rev. 36, 630–647 (1930)
A. Sommerfeld, Asymptotische Integration der Differential-Gleichung des Thomas-Fermischen Atoms, Z. Phys. 78, 283–308 (1932)
- Harold T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York, 1962. MR 0181773
V. Bush and S. H. Caldwell, Thomas-Fermi equation solution by the differential analyzer, Phys. Rev. (2) 38, 1898–1901 (1931)
R. V. Ramnath, A new analytical approximation for the Thomas-Fermi model in atomic physics, J. Math. Anal. Appl. 31, 285–296 (1970)
R. P. Feynman, N. Metropolis, and E. Teller. Equations of state of elements based on the generalized Fermi-Thomas theory, Phys. Rev. 75, 1561–1573 (1949)
- Einar Hille, Some aspects of the Thomas-Fermi equation, J. Analyse Math. 23 (1970), 147–170. MR 279376, DOI https://doi.org/10.1007/BF02795497
A. Mambriani, Su un teorema relativo alle equazioni differenziali ordinarie del 2d ordine. Rend. Accad. Naz. del Lincei, CI. sci. fis., mat. e nat. (6) 9, 620–622 (1929)
G. Scorza-Dragoni, Su un’ equazione differenziale particolare, Rend. Accad. Naz. del Lincei, CI. sci. fis., mat. e nat. (6) 9, 623–625 (1929)
- James L. Reid, Solution to a nonlinear differential equation with application to Thomas-Fermi equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 53 (1972), 376–379 (1973) (English, with Italian summary). MR 340691
- James L. Reid and Richard J. De Puy, Derivation of modified Thomas-Fermi and Emden equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 54 (1973), 529–532 (1974) (English, with Italian summary). MR 355150
M. Abramowitz and I. Stegun, Handbook of Mathematical functions, Dover, New York, 1965
R. Courant and D. Hilbert, Methods of mathematical physics, Wiley, New York, 1953
- Samuel Karlin, Positive operators, J. Math. Mech. 8 (1959), 907–937. MR 0114138, DOI https://doi.org/10.1512/iumj.1959.8.58058
M. Krasnoselskii, Topological methods in nonlinear integral equations, Pergamon, New York, 1964
L. H. Thomas, The calculation of atomic fields, Proc. Camb. Phil. Soc. 23, 542–548 (1927)
E. Fermi, Un metodo statistico par la determinazione di alcune proprietá dell’ atome, Rend. Accad. Naz. del Lincei. CI. sci. fis., mat. e nat. (6) 6, 602–607 (1927)
P. Csavinszky, Calculation of diamagnetic susceptibilities of ions using a universal approximate analytical solution of the Thomas-Fermi equation, Bull. Amer. Phys. Soc. (2) 18, 726–727 (1973)
P. Csavinszky, Universal approximate analytical solution of the Thomas-Fermi equation for ions, Phys. Rev. A(3) 8, 1688–1701 (1973)
Y. Bae Suh, Perturbation calculation using Thomas-Fermi model, Phys. Lett. 49A, 99–100 (1974)
E. H. Lieb and B. Simon, Thomas-Fermi theory revisited, Phys. Rev. Lett. 31, 681–683 (1973)
E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules, and solids, Advances in Mathematics, to appear 1976
P. Gombás, Die statistische Theorie des Atoms, Springer, Berlin, 1949
N. H. March, The Thomas-Fermi approximation in quantum mechanics, Adv. Phys. 6, 1–101 (1957)
J. S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. (2) 17, 339–360 (1975)
E. B. Baker, The application of the Thomas-Fermi statistical model to the calculation of potential distribution of positive ions, Phys. Rev. 36, 630–647 (1930)
A. Sommerfeld, Asymptotische Integration der Differential-Gleichung des Thomas-Fermischen Atoms, Z. Phys. 78, 283–308 (1932)
H. T. Davis, Introduction to nonlinear differential and integral equations, Dover, New York, 1962
V. Bush and S. H. Caldwell, Thomas-Fermi equation solution by the differential analyzer, Phys. Rev. (2) 38, 1898–1901 (1931)
R. V. Ramnath, A new analytical approximation for the Thomas-Fermi model in atomic physics, J. Math. Anal. Appl. 31, 285–296 (1970)
R. P. Feynman, N. Metropolis, and E. Teller. Equations of state of elements based on the generalized Fermi-Thomas theory, Phys. Rev. 75, 1561–1573 (1949)
E. Hille, Some aspects of the Thomas-Fermi equation, J. d’Analyse Math. 23, 147–170 (1970)
A. Mambriani, Su un teorema relativo alle equazioni differenziali ordinarie del 2d ordine. Rend. Accad. Naz. del Lincei, CI. sci. fis., mat. e nat. (6) 9, 620–622 (1929)
G. Scorza-Dragoni, Su un’ equazione differenziale particolare, Rend. Accad. Naz. del Lincei, CI. sci. fis., mat. e nat. (6) 9, 623–625 (1929)
J. L. Reid, Solution to a nonlinear differential equation with application to Thomas-Fermi equations, Rend. Accad. Naz. del Lincei, CI. Sci. fis., mat. e nat. 53, 376–379 (1972)
J. L. Reid and R. J. Depuy, Derivation of modified Thomas-Fermi and Emden equations, Atti. Accad. Naz. del Lincei 54, 529 (1973)
M. Abramowitz and I. Stegun, Handbook of Mathematical functions, Dover, New York, 1965
R. Courant and D. Hilbert, Methods of mathematical physics, Wiley, New York, 1953
S. Karlin, Positive operators, J. Math. Mech. 8, 907–937 (1959)
M. Krasnoselskii, Topological methods in nonlinear integral equations, Pergamon, New York, 1964
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Article copyright:
© Copyright 1977
American Mathematical Society
