Harris J. Silverstone

Harris J. Silverstone

Professor

Remsen 344
410-516-7431
hjsilverstone@jhu.edu
Curriculum Vitae

Biography
Publications

Professor Silverstone studies and publishes about expansions as tools in quantum chemistry. The question with some expansions is how fast do they converge, as with configuration interaction for atomic and molecular wave functions. (One answer: like l–4, where l is the highest angular momentum orbital used in the expansion.) With divergent expansions, questions are how fast do they diverge and how to decode the encrypted information, as with the power-series-in-electric-field-strength for of the energy of an atom in an electric field – Rayleigh-Schrödinger perturbation theory. (One set of answers: factorially fast; decode with Borel summation via Padé approximants. A second: decode via hyperasymptotic summation.) With divergent power series there are often additional divergent power series multiplied by an exponentially small factor. In the electric field case, this exponentially small series gives the tunneling field-ionization rate. The Jeffries-Wentzel-Kramers-Brillouin (JWKB) method expands the logarithm of the wave function in a divergent power series in ħ, and its associated exponentially small series are especially important for tunneling.

Recent open issues involve the 1/r12 Coulomb interaction in molecular wave functions: evaluation of multicenter integrals using numerical contour integration; the conditional convergence of the bipolar expansion when charge distributions overlap.

  1. A theoretical study of the electron spin resonance spectra of cycloheptatrienyl
    Harris Julian Silverstone
    Ph.D. Thesis, California Institute of Technology, 1964, (University Microfilms, Inc., Ann Arbor, Michigan, No. 65-6532). pdf 3.7 MB
  2. Study of molecular orbital degeneracy in C7H7
    H. J. Silverstone, D. E. Wood and H. M. McConnell
    J. Chem. Phys. 41, 2311-2323 (1964). pdf 1.1 MB
  3. Electron Correlation in nonclosed shell systems
    Harris J. Silverstone and Oktay Sinanoglu
    in Modern Quantum Chemistry - Istanbul Lectures, Part II, O. Sinano?lu, editor, (Academic Press, Inc., New York, 1965), pp. 99-121. pdf 1.6 MB
  4. Energy differences and Parr's Integral Hellmann-Feynman Theorem
    Harris J. Silverstone
    J. Chem. Phys. 43, 4537-4539 (1965). pdf 260 KB
  5. Toward a better 2p?-atomic orbital for ?-electron theory
    Harris J. Silverstone, Hubert W. Joy and Malcolm K. Orloff
    J. Am. Chem. Soc. 88, 1325-1327 (1966). pdf 336 KB
  6. Many-electron theory of nonclosed-shell atoms and molecules. I. Orbital wavefunction and perturbation theory
    Harris J. Silverstone and Oktay Sinano?lu
    J. Chem. Phys. 44, 1899-1907 (1966). pdf 700 KB
  7. Many-electron theory of nonclosed-shell atoms and molecules. II. Variational theory
    Harris J. Silverstone and Oktay Sinano?lu
    J. Chem. Phys. 44, 3608-3617 (1966). pdf 740 KB
  8. On the evaluation of two-center overlap and coulomb integrals with non-integer-n Slater type orbitals
    Harris J. Silverstone
    J. Chem. Phys. 45, 4337-4341 (1966). pdf 244 KB
  9. Symmetry properties of one- and two-electron correlation functions in the many-electron theory of atoms and molecules
    Harris J. Silverstone and Oktay Sinano?lu
    J. Chem. Phys. 46, 854-859 (1967). pdf 412 KB
  10. Nonempirical evaluation of ?-electron charge density dependence of proton isotropic hyperfine coupling constants. An application of the valence state model
    Kenneth G. Kay and Harris J. Silverstone
    J. Chem. Phys. 46, 2854-2855 (1967). pdf 172 KB
  11. Series expansion for two-center noninteger-n overlap integrals
    Harris J. Silverstone
    J. Chem. Phys. 46, 4368-4376 (1967). pdf 468 KB
  12. Series expansion for two-center noninteger-n coulomb integrals
    Harris J. Silverstone
    J. Chem. Phys. 46, 4377-4380 (1967). pdf 204 KB
  13. Analytical evaluation of three- and four-center integrals of r_{12}^{-1} with Slater-type orbitals
    Harris J. Silverstone
    Proc. Natl. Acad. Sci. U.S. 58, 34-36 (1967). pdf 292 KB
  14. 1em>Expansion about an arbitrary point of three-dimensional functions involving spherical harmonics by the Fourier transform convolution theorem
    Harris J. Silverstone
    J. Chem. Phys. 47, 537-540 (1967). pdf 224 KB
  15. 1Valence states of carbon in ?-electron systems. I. Alternant hydrocarbon ground states
    Harris J. Silverstone
    J. Chem. Phys. 47, 1384-1392 (1967). pdf 548 KB
  16. Valence states of carbon in ?-electron system. II. Excited states and ions
    Hubert W. Joy and Harris J. Silverstone
    Molec. Phys. 13, 149-156 (1967). pdf 692 KB
  17. Energy calculation by the method of local moments
    Harris J. Silverstone, Moon-Lung Yin and R. L. Somorjai
    J. Chem. Phys. 47, 4824-4827 (1967). pdf 244 KB
  18. Analytical evaluation of multicenter integrals of r_{12}^{-1} with Slater-type atomic orbitals. I. (1-2)-type three-center integrals
    Harris J. Silverstone
    J. Chem. Phys. 48, 4098-4106 (1968). pdf 540 KB
  19. Analytical evaluation of multicenter integrals of r_{12}^{-1}with Slater-type atomic orbitals. II. Three-center nuclear attraction integrals
    Harris J. Silverstone
    J. Chem. Phys. 48, 4106-4108 (1968). pdf 208 KB
  20. Analytical evaluation of multicenter integrals of r_{12}^{-1} with Slater-type atomic orbitals. III. (2-2)-type three-center integrals
    Harris J. Silverstone and Kenneth G. Kay
    J. Chem. Phys. 48, 4108-4115 (1968). pdf 428 KB
  21. Modified perturbation theory for atoms and molecules based on a Hartree-Fock \phi_0
    Harris J. Silverstone and Moon-Lung Yin
    J. Chem. Phys. 49, 2026-2030 (1968). pdf 404 KB
  22. 2Some aspects of electron correlation in open-shell states. 2^{1}P and 2^{3}P helium
    Moon-Lung Yin and Harris J. Silverstone
    J. Chem. Phys. 49, 3160-3164 (1968). pdf 400 KB
  23. Long-range behavior of Hartree-Fock orbitals
    Nicholas C. Handy, Michael T. Marron, and Harris J. Silverstone
    Phys. Rev. 180, 45-48 (1969). pdf 556 KB
  24. Convergence properties of certain formulas for multicenter electron repulsion integrals obtained from the bipolar expansion of r_{12}^{-1}
    Harris J. Silverstone and Kenneth G. Kay
    J. Chem. Phys. 50, 5045-5047 (1969). pdf 228 KB
  25. Analytical evaluation of multicenter integrals of r_{12}^{-1} with Slater-type atomic orbitals. IV. Four-center integrals by Taylor series method
    Kenneth G. Kay and Harris J. Silverstone
    J. Chem. Phys. 51, 956-961 (1969). pdf 344 KB
  26. Dirac delta functions in the Laplace-type expansion of r^{n}Y^{l}_{m}(\theta,\phi)
    Kenneth G. Kay, H. David Todd, and Harris J. Silverstone
    J. Chem. Phys. 51, 2359-2362 (1969). pdf 220 KB
  27. Bipolar expansion for r^{n}Y^{l}_{m}(\theta,\phi)
    Kenneth G. Kay, H. David Todd, and Harris J. Silverstone
    J. Chem. Phys. 51, 2363-2367 (1969). pdf 328 KB
  28. Method of local configuration interaction applied to electronic systems. Hydrogen atom and hydrogen molecule ion
    Stuart M. Rothstein, James E. Welsch, and Harris J. Silverstone
    J. Chem. Phys. 51, 2932-2936 (1969). pdf 320 KB
  29. Analytical evaluation of multicenter integrals of r_{12}^{-1} with Slater-type atomic orbitals. V. Four-center integrals by Fourier-transform method
    Kenneth G. Kay and Harris J. Silverstone
    J. Chem. Phys. 51, 4287-4304 (1969). pdf 840 KB
  30. Rational function approximation for atomic and molecular wavefunctions
    Gail Wilson and Harris J. Silverstone
    Int. J. Quantum Chem 3, 1067-1068 (1969). pdf 160 KB
  31. Explicit formulas for the nth order wavefunction and energy in nondegenerate Rayleigh-Schrödinger perturbation theory
    Harris J. Silverstone and Thomas T. Holloway
    J. Chem. Phys. 52, 1472-1475 (1970). pdf 212 KB
  32. Solution of the Hartree-Fock problem by expansion onto nested bases
    Michael T. Marron, Nicholas C. Handy, Robert G. Parr, and Harris J. Silverstone
    Int. J. Quantum Chem. 4, 245-255 (1970). pdf 856 KB
  33. Analytical evaluation of three-center one-electron integrals of r^{n}Y^{l}_{m}(\theta,\phi) with Slater-type atomic orbitals
    Harris J. Silverstone and H. David Todd
    Int. J. Quantum Chem Symposium 4, 371-383 (1971). pdf 968 KB
  34. Unified treatment of two-center overlap, coulomb, and kinetic-energy integrals
    H. David Todd, Kenneth G. Kay, and Harris J. Silverstone
    J. Chem. Phys. 53, 3951-3956 (1970). pdf 452 KB
  35. Comment on "The evaluation of three-center nuclear attraction integrals"
    Kenneth G. Kay, Harris J. Silverstone, and B. Kleinman
    J. Chem. Phys. 53, 2545-2546 (1970). pdf 116 KB
  36. Analytical evaluation of multicenter integrals of r_{12}^{-1} with Slater-type atomic orbitals. VI. Asymptotic expansions for large internuclear distances
    Kenneth G. Kay and Harris J. Silverstone
    J. Chem. Phys. 53, 4269-4285(1970). pdf 980 KB
  37. Explicit solution for the wavefunction and energy in degenerate Rayleigh-Schrödinger perturbation theory
    Harris J. Silverstone
    J. Chem. Phys. 54, 2325-2335 (1971). pdf 900 KB
  38. Kinetic energy expectation values with discontinuous approximate wave functions
    Harris J. Silverstone and Edward W. Stuebing
    Phys. Rev. A 5, 1092-1093 (1972). pdf 284 KB
  39. Explicit formulas in degenerate Rayleigh-Schrödinger perturbation theory for the energy and wavefunction, based on a formula of Lagrange
    Harris J. Silverstone and Thomas T. Holloway
    Phys. Rev A 4, 2191-2199 (1971). pdf 1.2 MB
  40. Complex variable theory
    Harris J. Silverstone
    Chapter 3 in Physical Chemistry, An Advanced Treatise, Vol. XI, Mathematical Methods, Edited by Douglas Henderson (Academic Press, Inc., New York, 1975), pp. 151-26pdf 6.8 MB
  41. 1. Piecewise polynomial electronic wavefunctions
    Jose Luis Gazquez and Harris J. Silverstone
    J. Chem. Phys. 67, 1887-1898 (1977). pdf 856 KB
  42. Hartree-Fock equations for open shell states
    Harris J. Silverstone
    J. Chem. Phys. 67, 4172-4180 (1977). pdf 632 KB
  43. Perturbation theory of resonant states induced by an electrostatic field: one-dimensional model
    Tokio Yamabe, Akitomo Tachibana and Harris J. Silverstone
    J. Phys. B 10, 2083-2100 (1977). pdf 904 KB
  44. Theory of the ionization of the hydrogen atom by an external electrostatic field
    Tokio Yamabe, Akitomo Tachibana and Harris J. Silverstone
    Phys. Rev. A 16, 877-890 (1977). pdf 1.7 MB
  45. Expansion of a function about a displaced center
    Harris J. Silverstone and Richard K. Moats
    Phys. Rev. A 16, 1731-1732 (1977). pdf 116 KB
  46. Piecewise polynomial basis functions for configuration interaction and many-body perturbation theory calculations. The radial limit of helium
    Harris J. Silverstone, Dennis P. Carroll, and David M. Silver
    J. Chem. Phys. 68, 616-618 (1978). pdf 208 KB
  47. Perturbation theory of the Stark effect in hydrogen to arbitrarily high order
    Harris J. Silverstone
    Phys. Rev. A 18, 1853-1864 (1978). pdf 1.4 MB
  48. Piecewise polynomial configuration interaction natural orbital study of 1s^2 helium
    Dennis P. Carroll, Harris J. Silverstone, and Robert Melville Metzger
    J. Chem. Phys. 71, 4142-4163 (1979). pdf 1.5 MB
  49. Pseudo-eigenvalue equation for natural orbitals of two-electron systems and long-range behavior
    Harris J. Silverstone, Dennis P. Carroll, and Robert Melville Metzger
    J. Chem. Phys. 70, 5919 (1979). pdf 244 KB
  50. Calculation of Stark effect energy shifts by Padé approximants of Rayleigh-Schrödinger perturbation theory
    Harris J. Silverstone and Peter M. Koch
    J. Phys. B 12, L537-L541 (1979). pdf 284 KB
  51. 1. Stark effect in hydrogen: dispersion relation, asymptotic formulas, and calculation of the ionization rate via high-order perturbation theory
    Harris J. Silverstone, Barry G. Adams, Ji?í ?ížek, and Peter Otto
    Phys. Rev. Lett. 43, 1498-1501 (1979). pdf 536 KB
  52. Bender-Wu formulas for degenerate eigenvalues
    B. G. Adams, J. E. Avron, J. ?ížek, P. Otto J. Paldus, R. K. Moats, and Harris J. Silverstone
    Phys. Rev. A 21, 1914-1916 (1980). pdf 360 KB
  53. Asymptotic behavior of atomic Hartree-Fock orbitals
    George S. Handler, Darwin W. Smith, and Harris J. Silverstone
    J. Chem. Phys. 73, 3936-3939 (1980). pdf 176 KB
  54. Practical recursive solution of degenerate Rayleigh-Schrödinger perturbation theory and application to high-order calculations of the Zeeman effect in hydrogen
    Harris J. Silverstone and Richard K. Moats
    Phys. Rev. A 23, 1645-1654 (1981). pdf 1.4 MB
  55. Long-range behavior of electronic wavefunctions; generalized Carlson-Keller expansion
    Harris J. Silverstone
    Phys. Rev. A 23, 1030-1037 (1981). pdf 1.2 MB
  56. High-order perturbation theory of the imaginary part of the resonance eigenvalues of the Stark effect in hydrogen and of the anharmonic oscillator with negative anharmonicity
    Harris J. Silverstone, Evans Harrell, and Christina Grot
    Phys. Rev. A 24, 1925-1934 (1981). pdf 1.3 MB
  57. Unified derivation of the perturbation series for the real and imaginary parts of the energy of hydrogen in the Stark effect and of the negatively anharmonic oscillator
    Harris J. Silverstone
    Int. J. Quantum Chem. 21, 125-131 (1982). pdf 576 KB
  58. Recent progress on three-center integrals using Fourier-transform-based analytical formulas
    H. David Todd, Kenneth C. Daiker, Robert D. Cloney, S. J., Richard K. Moats, and Harris J. Silverstone
    in ETO Multicenter Molecular Integrals, Charles A. Weatherford and Herbert W. Jones, editors, (D. Reidel Publishing Company, Boston, 1982), pp. 73-8pdf 2.2 MB
  59. 1/R Expansion for H_{2}^{+}: analyticity, summability, asymptotics, and calculation of exponentially small terms
    Robert J. Damburg, Rafail Kh. Propin, Sandro Graffi, Vincenzo Grecchi, Evans M. Harrell II, Ji?í ?ížek, Josef Paldus, and Harris J. Silverstone
    Phys. Rev. Lett. 52, 1112-1115 (1984). pdf 592 KB
  60. On definitions of L convergence of atomic correlation energies
    Karol Jankowski, Daniel W. Zaharevitz, and Harris J. Silverstone
    J. Chem. Phys. 82, 1969-1972 (1985). pdf 412 KB
  61. 1. Resonances and convergence of perturbation theory for N-body atomic systems in external AC-electric field
    S. Graffi, V. Grecchi, and H. J. Silverstone
    Ann. Inst. Henri. Poincaré (A) 42, 215-234 (1985). pdf 1.4 MB
  62. Complex energies from real perturbation series for the LoSurdo-Stark effect in hydrogen by Borel-Padé approximants
    Valter Franceschini, Vincenzo Grecchi, and Harris J. Silverstone
    Phys. Rev. A 32, 1338-1340 (1985). pdf 556 KB
  63. Observations on the summability of confluent hypergeometric functions and on semiclassical quantum mechanics
    Harris J. Silverstone, Sachiko Nakai, and Jonathan G. Harris
    Phys. Rev. A 32, 1341-1345 (1985). pdf 964 KB
  64. Asymptotics of high-order perturbation theory for the one-dimensional anharmonic oscillator by quasisemiclassical methods
    Harris J. Silverstone, Jonathan G. Harris, Ji?í ?ížek, and Josef Paldus
    Phys. Rev. A 32, 1965-1980 (1985). pdf 2.6 MB
  65. 1/R Expansion for H_{2}^{+}: analyticity, summability, and asymptotics
    S. Graffi, V. Grecchi, E. M. Harrell II, and H. J. Silverstone
    Ann. Phys. (NY) 165, 441-483 (1985). pdf 3.3 MB
  66. Piecewise polynomial Hylleraas configuration interaction wavefunctions for helium
    D. W. Zaharevitz, H. J. Silverstone, and D. M. Silver
    in Supercomputer Applications, Proceedings of the Supercomputer Applications Symposium cosponsored by the Purdue University Computing Center, the Purdue Center of Parallel and Vector Computing, and Control Data, held October 31 - November 1, 1984, in West Lafayette, Indiana, edited by R. W. Numrich (Plenum Press, New York, 1985), pp. 263-271. pdf 812 KB
  67. 1/R Expansion for H_{2}^{+}: calculation of exponentially small terms and asymptotics
    Ji?í ?ížek, Robert J. Damburg, Sandro Graffi, Vincenzo Grecchi, Evans M. Harrell II, Jonathan G. Harris, Sachiko Nakai, Josef Paldus, Rafail Kh. Propin, and Harris J. Silverstone
    Phys. Rev. A 33, 12-54 (1986). pdf 8.8 MB
  68. Reality and complexity in asymptotic expansions for eigenvalues and eigenfunctions, with application to the JWKB connection-formula problem
    Harris J. Silverstone
    Int. J. Quantum Chem. 29, 261-272 (1986). pdf 1.2 MB
  69. JWKB connection-formula problem revisited via Borel summation
    Harris J. Silverstone
    Phys. Rev. Lett. 55, 2523-2526 (1985). pdf 228 KB
  70. LoSurdo-Stark effect for a hydrogenic impurity in a thin layer: Two-dimensional model
    Kazuyoshi Tanaka, Masahiro Kobashi, Tokushige Shichiri, Tokio Yamabe, David M. Silver, and Harris J. Silverstone
    Phys. Rev. B 35, 2513-2516 (1987). pdf 476 KB
  71. 1. Classical resonance overlapping and quantum avoided crossings
    Sandro Graffi, Thierry Paul, and Harris J. Silverstone
    Phys. Rev. Lett. 59, 255-258 (1987). pdf 548 KB
  72. On the use of asymptotic expansions
    Modris Gailitis and Harris J. Silverstone
    Theor. Chim. Acta 73, 105-114 (1988). pdf 464 KB
  73. Resonance overlapping in classical mechanics and avoided crossings in quantum mechanics
    S. Graffi, T. Paul, and H. J. Silverstone
    Phys. Rev. A 37, 2214-2219 (1988). pdf 1.3 MB
  74. Transition from classical to quantum mechanics: x^4 perturbed harmonic oscillator
    Gabriel Alvarez, Sandro Graffi, and Harris J. Silverstone
    Phys. Rev. A 38, 1687-1696 (1988). pdf 1.6 MB
  75. Resonance Contributions to the Photoionization Spectrum of Atomic Hydrogen in an Electric Field
    Gabriel Alvarez and Harris J. Silverstone
    Phys. Rev. Lett. 63, 1364-1367 (1989). pdf 596 KB
  76. Photoionization cross section: exact expansion over resonances and natural line shape
    Gabriel Alvarez and Harris J. Silverstone
    Phys. Rev. A 40, 3690-3697 (1989). pdf 1.1 MB
  77. High-order perturbation theory and its application to atoms in strong fields
    Harris J. Silverstone
    in Atoms in Strong Fields, edited by C. A. Nicolaides, C. W. Clark, and M. H. Nayfeh, (Plenum, New York, 1990), pp. 295–30pdf 1.4 MB
  78. Photoionization of Atomic Hydrogen in an Electric Field
    Gabriel Alvarez, Robert J. Damburg, and Harris J. Silverstone
    Phys. Rev. A 44, 3060-3082 (1991). pdf 3.3 MB
  79. EPR Spectroscopy of Interdoublet Transitions in High-Spin Iron: Applications to Transferrin Oxalate
    Kutbuddin S. Doctor, B. J. Gaffney, Gabriel Alvarez, and Harris J. Silverstone
    J. Phys. Chem. 97, 3028-3033 (1993). pdf 1.2 MB
  80. Simulation of the EMR Spectra of High-Spin Iron in Proteins
    Betty J. Gaffney and Harris J. Silverstone
    Chapter 1 of Biological Magnetic Resonance, Volume 13: EMR of Paramagnetic Molecules, edited by Lawrence J. Berliner and Jacques Reuben (Plenum, New York, 1993), pp. 1-5pdf 4.7 MB
  81. 1. Large-field behavior of the LoSurdo-Stark resonances in atomic hydrogen
    Gabriel Alvarez and Harris J. Silverstone
    Phys. Rev. A 50, 4679-4699 (1994). pdf 2.6 MB
  82. Exact expansion methods for atomic hydrogen in an external electrostatic field: divergent perturbation series, Borel summability, semiclassical approximation, and expansion of photoionization cross-section over resonance eigenvalues
    Harris J. Silverstone
    Chapter 10 of Modern Electronic Structure Theory, edited by David R. Yarkony (World Scientific Publishing Co., Singapore, 1995). pdf 2.3 MB
  83. Simulation methods for looping transitions
    Betty J. Gaffney and Harris J. Silverstone
    J. Magn. Reson. 134, 57-66 (1998). pdf 328 KB
  84. On the summations involving Wigner rotation matrix elements
    Shan-Tao Lai, Pancracio Palting, Ying-Nan Chiu, and Harris J. Silverstone
    J. Math. Chem. 24, 123-132 (1998). pdf 80 KB
  85. Connection formula, hyperasymptotics, and Schrödinger eigenvalues: dispersive hyperasymptotics and the anharmonic oscillator
    Gabriel Álvarez, Christopher J. Howls, and Harris J. Silverstone
    in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, edited by Christopher J. Howls, Takahiro Kawai, and Yoshitsugu Takei (Kyoto University Press, Kyoto, 2000), pp. 121-13pdf 1.2 MB
  86. Anharmonic oscillator discontinuity formulae up to second-exponentially-small order
    Gabriel Álvarez, Christopher J. Howls, and Harris J. Silverstone
    J. Phys. A 35, 4003-4016 (2002). pdf 168 KB
  87. Dispersive hyperasymptotics and the anharmonic oscillator
    Gabriel Álvarez, Christopher J. Howls, and Harris J. Silverstone
    J. Phys. A 35, 4017-4042 (2002). pdf 328 KB
  88. JWKB method as an exact technique
    Hujun Shen and Harris J. Silverstone
    Int. J. Quantum Chem. 99, 336-352 (2004). pdf 204 KB
  89. On the computation of (2-2)-three-center Slater-type-orbital integrals of 1/r_{12} using Fourier-transform-based analytical formulas
    Danko Antolovic and Harris J. Silverstone
    Int. J. Quantum Chem. 100, 146-154 (2004). pdf 184 KB
  90. On the bidirectionality of the JWKB connection-formula at a linear turning point
    Hujun Shen and Harris J. Silverstone
    Collect. Czech. Chem. Commun. (Collection of Czechoslovak Chemical Communications) 70, 740-754 (2005). pdf 128 KB
  91. Observations on the JWKB treatment of the quadratic barrier
    Hujun Shen and Harris J. Silverstone
    in Algebraic analysis of differential equations: from microlocal analysis to exponential asymptotics: Festschrift in honor of Takahiro Kawai, T. Aoki, H. Majima, Y. Takei, and N. Tose, eds., (Springer-Verlag, 2008), pp. 237-250. pdf 212 KB
  92. Rereading Langer's influential 1937 JWKB paper: the unnecessary Langer transformation; the two hbar's
    Tatsuya Koike and Harris J. Silverstone J. Phys. A: Math. Theor. 42 (2009) 495206 (11pp) pdf 156 KB