J-Matrix Theory of Scattering
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1: 15.18 Physical Applications
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►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)).
►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).
2: Ian J. Thompson
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► 1953 in New Zealand) has been since 2006 a Theoretical Nuclear Physicist in the Nuclear Theory and Modeling Group of the Lawrence Livermore National Laboratory, Livermore, California.
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►Thompson has published papers on special functions, and numerous papers in theoretical nuclear physics, especially in scattering theory.
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3: 27.17 Other Applications
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►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art.
Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.
4: 14.31 Other Applications
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§14.31(ii) Conical Functions
… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►§14.31(iii) Miscellaneous
►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). … ►Legendre functions of complex degree appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …5: 29.19 Physical Applications
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►Bronski et al. (2001) uses Lamé functions in the theory of Bose–Einstein condensates.
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§29.19(ii) Lamé Polynomials
… ►Macfadyen and Winternitz (1971) finds expansions for the two-body relativistic scattering amplitudes. …6: Brian D. Sleeman
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►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine.
He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D.
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7: T. Mark Dunster
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►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory.
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8: William P. Reinhardt
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►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions.
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9: Bibliography N
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Scattering theory of waves and particles.
Dover Publications, Inc., Mineola, NY.
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Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten.
Chelsea Publishing Co., New York (German).
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High-frequency scattering by an impenetrable sphere.
Ann. Physics 34 (1), pp. 23–95.
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Diffraction Effects in Semiclassical Scattering.
Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.
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10: 9.16 Physical Applications
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►Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop.
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►Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)).
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►Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics.
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►A study of the semiclassical description of quantum-mechanical scattering is given in Ford and Wheeler (1959a, b).
In the case of the rainbow, the scattering amplitude is expressed in terms of , the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics.
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