Adventures in applied mathematics
Author:
C. L. Pekeris
Journal:
Quart. Appl. Math. 30 (1972), 67-83
DOI:
https://doi.org/10.1090/qam/99739
MathSciNet review:
QAM99739
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Abstract: A review is given of the work carried out at the Weizmann Institute during the past 25 years in the fields of terrestrial spectroscopy, the dynamo theory of the earth’s magnetic field, the tides in the world oceans, theoretical seismograms, hydrodynamic stability, atomic spectroscopy, and the Boltzmann integral equation. Some open problems in the solution of the Schrödinger wave equation are formulated.
N. F. Ness, J. C. Harrison and L. B. Slichter, Observations of the free oscillations of the earth, J. Geophys. Res. 66, 621–629 (1961)
C. L. Pekeris and H. Jarosch, The free oscillations of the earth, in Contributions in geophysics, Gutenberg Volume, Pergamon Press, London, 1958, pp. 171–192
C. L. Pekeris, Z. Alterman and H. Jarosch, Terrestrial spectroscopy, Nature 190, 498–500 (1961)
H. Benioff, B. Gutenberg and C. F. Richter, Progress report, Seismological Laboratory, California Institute of Technology, 1953, Trans. Amer. Geophys. Union 35, 979–987 (1954)
A. E. H. Love, Some problems of geodynamics, Cambridge University Press, Cambridge, 1911, p. 143
J. Larmor, Possible rotational origin of magnetic fields of sun and earth, British Assoc. Report, Bournmouth, 1919, p. 159; Electr. Review 85, 412 (1919)
T. G. Cowling, The magnetic field of sunspots, Mon. Not. Roy. Astron. Soc. 94, 39–48 (1934)
E. C. Bullard and H. Gellman, Homogeneous dynamos and terrestrial magnetism, Phil. Trans. Roy. Soc. A 247, 213–278 (1954)
R. D. Gibson and P. H. Roberts, The application of modern physics to the earth and planetary interiors, Wiley, London, 1969, pp. 577–602
F. E. M. Lilley, On kinematic dynamos, Proc. Roy. Soc. A 316, 153–167 (1970)
C. L. Pekeris, The magnetic field induced by the bodily tide in the core of the earth, Proc. Nat. Acad. Sci. 68, 1111–1113 (1971)
P. S. Laplace, Recherches sur quelques points du systéme du monde, Mém. de l’Acad. roy. des Sciences (1775); Oeuvres complètes, ix. 88, 187
M. Dishon and B. C. Heezen, Digital deep-sea sounding library: description and index list, Int’l. Hydrographic Rev. 45, 23–39 (1968)
C. L. Pekeris and Y. Accad, Solution of Laplace’s equations for the M$^{2}$ tide in the world oceans, Phil. Trans. Roy. Soc. A 265, 413–436 (1969)
D. E. Cartwright, Tides and waves in the vicinity of Saint Helena, Phil. Trans. Roy. Soc. A270, 603–649 (1971)
C. L. Pekeris, Theory of propagation of explosive sound in shallow water, Geological Society of America Memoir 27, 1948, pp. 1–117
- C. L. Pekeris, Solution of an integral equation occurring in impulsive wave propagation problems, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 439–443. MR 78575, DOI https://doi.org/10.1073/pnas.42.7.439
- Chaim L. Pekeris and Ivor M. Longman, Ray-theory solution of the problem of propagation of explosive sound in a layered liquid, J. Acoust. Soc. Amer. 30 (1958), 321–328. MR 93272, DOI https://doi.org/10.1121/1.1909585
C. L. Pekeris and B. Shkoller, Stability of plane Poiseuille flow to periodic disturbances of finite amplitude, III, Proc. Nat. Acad. Sci. 68, 1434–1435 (1971)
E. A. Hylleraas, Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium, Z. Physik 54, 347–366 (1929)
P. K. Kabir and E. E. Salpeter, Radiative corrections to the ground-state energy of the helium atom, Phys. Rev. 108, 1256–1263 (1957); E. E. Salpeter and M. H. Zaidi, Lamb shift excitation energy in the ground state of the helium atom, Phys. Rev. 125, 248–255 (1962); C. Schwartz, Lamb shift in the helium atom, Phys. Rev. 123, 1700–1705 (1961)
G. Herzberg, Ionization potentials and Lamb shifts of the ground states of $^{4}$He and $^{3}$He, Proc. Roy. Soc. A 248, 309–332 (1958)
A. S. Coolidge and H. M. James, On the convergence of the Hylleraas variational method, Phys. Rev. 51, 855–859 (1937)
- C. L. Pekeris, Ground state of two-electron atoms, Phys. Rev. (2) 112 (1958), 1649–1958. MR 101783
- C. L. Pekeris, $1$ $^{1}S$ and $2$ $^{3}S$ states of helium, Phys. Rev. (2) 115 (1959), 1216–1221. MR 108293
C. L. Pekeris, 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S states of Li$^{+}$, Phys. Rev. 126, 143–145 (1962)
C. L. Pekeris, 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S states of H$^{-}$ and of He, Phys. Rev. 126, 1470–1476 (1962)
G. Herzberg and H. Moore, The spectrum of Li$^{+}$, Can. J. Phys. 37, 1293–1313 (1959)
G. W. Series and K. Willis, Note on the Li II spectrum, Proc. Phys. Soc. 71, 274 (1958)
S. Werner, The spark spectrum of lithium, Nature 118, 154–155 (1926)
Y. G. Toresson and B. Edlén, The 1s2s$^{1}$S level of Li$^{+}$, Arkiv f. Fysik 23, 117–118 (1962)
S. Werner, Studier over spektroskopiske Lyskilder, H. Aschehoug & Co., Dansk Forlag, Copenhagen, 1927, p. 63
B. Schiff and C. L. Pekeris, f values for transitions between the 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S, and the 2$^{1}$P, 2$^{3}$P, 3$^{1}$P and 3$^{3}$P states in helium, Phys. Rev. 134, A638–A640 (1964)
H. G. Kuhn and J. M. Vaughan, Radiation width and resonance broadening in helium, Proc. Roy. Soc. A 277, 297–311 (1964)
J. M. Vaughan, Self-broadening and radiation width in the singlet spectrum of helium, Proc. Roy. Soc. A 295, 164–181 (1966)
H. G. Kuhn, E. L. Lewis and J. M. Vaughan, Enhancement of radiation damping by resonance coupling, Phys. Rev. Lett. 15, 687–688 (1965)
J. M. Burger and A. Lurio, Lifetime of the 2$^{1}$P$_{1}$ and 3$^{1}$P$_{1}$ states of atomic helium, Phys. Rev. A 3, 64–75 (1971)
Y. Accad, C. L. Pekeris and B. Schiff, The S and P states of the helium isoelectronic sequence up to $Z = 10$, Phys. Rev. A 4, 516–536 (1971)
B. Schiff, C. L. Pekeris and Y. Accad, f-values for transitions between low lying S and P states of the helium isoelectronic sequence up to $Z = 10$, Phys. Rev. A 4, 885-893 (1971)
A. Dalgarno and A. E. Kingston, Properties of metastable helium atoms, Proc. Phys. Soc. 72, 1053–1060 (1958)
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- C. L. Pekeris, Solution of the Boltzmann-Hilbert integral equation, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 661–669. MR 72367, DOI https://doi.org/10.1073/pnas.41.9.661
- C. L. Pekeris, Z. Alterman, L. Finkelstein, and K. Frankowski, Propagation of sound in a gas of rigid spheres, Phys. Fluids 5 (1962), 1608–1616. MR 152338, DOI https://doi.org/10.1063/1.1706572
- Chaim L. Pekeris and Zipora Alterman, Solution of the Boltzmann-Hilbert integral equation. II. The coefficients of viscosity and heat conduction, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 998–1007. MR 93106, DOI https://doi.org/10.1073/pnas.43.11.998
- David Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Publishing Company, New York, N.Y., 1953 (German). MR 0056184
N. F. Ness, J. C. Harrison and L. B. Slichter, Observations of the free oscillations of the earth, J. Geophys. Res. 66, 621–629 (1961)
C. L. Pekeris and H. Jarosch, The free oscillations of the earth, in Contributions in geophysics, Gutenberg Volume, Pergamon Press, London, 1958, pp. 171–192
C. L. Pekeris, Z. Alterman and H. Jarosch, Terrestrial spectroscopy, Nature 190, 498–500 (1961)
H. Benioff, B. Gutenberg and C. F. Richter, Progress report, Seismological Laboratory, California Institute of Technology, 1953, Trans. Amer. Geophys. Union 35, 979–987 (1954)
A. E. H. Love, Some problems of geodynamics, Cambridge University Press, Cambridge, 1911, p. 143
J. Larmor, Possible rotational origin of magnetic fields of sun and earth, British Assoc. Report, Bournmouth, 1919, p. 159; Electr. Review 85, 412 (1919)
T. G. Cowling, The magnetic field of sunspots, Mon. Not. Roy. Astron. Soc. 94, 39–48 (1934)
E. C. Bullard and H. Gellman, Homogeneous dynamos and terrestrial magnetism, Phil. Trans. Roy. Soc. A 247, 213–278 (1954)
R. D. Gibson and P. H. Roberts, The application of modern physics to the earth and planetary interiors, Wiley, London, 1969, pp. 577–602
F. E. M. Lilley, On kinematic dynamos, Proc. Roy. Soc. A 316, 153–167 (1970)
C. L. Pekeris, The magnetic field induced by the bodily tide in the core of the earth, Proc. Nat. Acad. Sci. 68, 1111–1113 (1971)
P. S. Laplace, Recherches sur quelques points du systéme du monde, Mém. de l’Acad. roy. des Sciences (1775); Oeuvres complètes, ix. 88, 187
M. Dishon and B. C. Heezen, Digital deep-sea sounding library: description and index list, Int’l. Hydrographic Rev. 45, 23–39 (1968)
C. L. Pekeris and Y. Accad, Solution of Laplace’s equations for the M$^{2}$ tide in the world oceans, Phil. Trans. Roy. Soc. A 265, 413–436 (1969)
D. E. Cartwright, Tides and waves in the vicinity of Saint Helena, Phil. Trans. Roy. Soc. A270, 603–649 (1971)
C. L. Pekeris, Theory of propagation of explosive sound in shallow water, Geological Society of America Memoir 27, 1948, pp. 1–117
C. L. Pekeris, Solution of an integral equation occurring in impulsive wave propagation problems, Proc. Nat. Acad. Sci. 42, 439–443 (1956)
C. L. Pekeris, I. M. Longman and H. Lipson, Application of ray theory to the problem of long-range propagation of explosive sound in a layered liquid, Bull. Seism. Soc. Amer. 49, 247–250 (1959)
C. L. Pekeris and B. Shkoller, Stability of plane Poiseuille flow to periodic disturbances of finite amplitude, III, Proc. Nat. Acad. Sci. 68, 1434–1435 (1971)
E. A. Hylleraas, Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium, Z. Physik 54, 347–366 (1929)
P. K. Kabir and E. E. Salpeter, Radiative corrections to the ground-state energy of the helium atom, Phys. Rev. 108, 1256–1263 (1957); E. E. Salpeter and M. H. Zaidi, Lamb shift excitation energy in the ground state of the helium atom, Phys. Rev. 125, 248–255 (1962); C. Schwartz, Lamb shift in the helium atom, Phys. Rev. 123, 1700–1705 (1961)
G. Herzberg, Ionization potentials and Lamb shifts of the ground states of $^{4}$He and $^{3}$He, Proc. Roy. Soc. A 248, 309–332 (1958)
A. S. Coolidge and H. M. James, On the convergence of the Hylleraas variational method, Phys. Rev. 51, 855–859 (1937)
C. L. Pekeris, Ground state of two-electron atoms, Phys. Rev. 112, 1649–1658 (1958)
C. L. Pekeris, 1$^{1}$S and 2$^{3}$S states of helium, Phys. Rev. 115, 1216–1221 (1959)
C. L. Pekeris, 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S states of Li$^{+}$, Phys. Rev. 126, 143–145 (1962)
C. L. Pekeris, 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S states of H$^{-}$ and of He, Phys. Rev. 126, 1470–1476 (1962)
G. Herzberg and H. Moore, The spectrum of Li$^{+}$, Can. J. Phys. 37, 1293–1313 (1959)
G. W. Series and K. Willis, Note on the Li II spectrum, Proc. Phys. Soc. 71, 274 (1958)
S. Werner, The spark spectrum of lithium, Nature 118, 154–155 (1926)
Y. G. Toresson and B. Edlén, The 1s2s$^{1}$S level of Li$^{+}$, Arkiv f. Fysik 23, 117–118 (1962)
S. Werner, Studier over spektroskopiske Lyskilder, H. Aschehoug & Co., Dansk Forlag, Copenhagen, 1927, p. 63
B. Schiff and C. L. Pekeris, f values for transitions between the 1$^{1}$S, 2$^{1}$S, and 2$^{3}$S, and the 2$^{1}$P, 2$^{3}$P, 3$^{1}$P and 3$^{3}$P states in helium, Phys. Rev. 134, A638–A640 (1964)
H. G. Kuhn and J. M. Vaughan, Radiation width and resonance broadening in helium, Proc. Roy. Soc. A 277, 297–311 (1964)
J. M. Vaughan, Self-broadening and radiation width in the singlet spectrum of helium, Proc. Roy. Soc. A 295, 164–181 (1966)
H. G. Kuhn, E. L. Lewis and J. M. Vaughan, Enhancement of radiation damping by resonance coupling, Phys. Rev. Lett. 15, 687–688 (1965)
J. M. Burger and A. Lurio, Lifetime of the 2$^{1}$P$_{1}$ and 3$^{1}$P$_{1}$ states of atomic helium, Phys. Rev. A 3, 64–75 (1971)
Y. Accad, C. L. Pekeris and B. Schiff, The S and P states of the helium isoelectronic sequence up to $Z = 10$, Phys. Rev. A 4, 516–536 (1971)
B. Schiff, C. L. Pekeris and Y. Accad, f-values for transitions between low lying S and P states of the helium isoelectronic sequence up to $Z = 10$, Phys. Rev. A 4, 885-893 (1971)
A. Dalgarno and A. E. Kingston, Properties of metastable helium atoms, Proc. Phys. Soc. 72, 1053–1060 (1958)
G. N. Watson, A treatise on Bessel functions, Macmillan, New York, 1944, p. 145
C. L. Pekeris, Solution of the Boltzmann-Hilbert integral equation, Proc. Nat. Acad. Sci. 41, 661–669 (1955)
C. L. Pekeris, Z. Alterman, L. Finkelstein and K. Frankowski, Propagation of sound in a gas of rigid spheres, Phys. Fluids 5, 1608–1616 (1962)
C. L. Pekeris and Z. Alterman, Solution of the Boltzmann-Hilbert integral equation II. The coefficients of viscosity and heat conduction, Proc. Nat. Acad. Sci. 43, 998–1007 (1957)
D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea, New York, 1953, p. 267
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Article copyright:
© Copyright 1972
American Mathematical Society
