potential theory
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1: 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)). …2: 19.18 Derivatives and Differential Equations
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►The next four differential equations apply to the complete case of and in the form (see (19.16.20) and (19.16.23)).
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►Similarly, the function satisfies an equation of axially symmetric potential theory:
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3: 19.33 Triaxial Ellipsoids
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§19.33(ii) Potential of a Charged Conducting Ellipsoid
…4: Bibliography S
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Resonances in -body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory.
Ann. of Math. (2) 97, pp. 247–274.
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Mixed Boundary Value Problems in Potential Theory.
North-Holland Publishing Co., Amsterdam.
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Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory.
National Bureau of Standards Applied Mathematics Series, No.
19, U. S. Government Printing Office, Washington, D.C..
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5: 15.18 Physical Applications
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►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)).
6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►which appear in the quantum theory of binding or scattering of a particle in a spherically symmetric potential
in three dimensions, and where .
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7: Bibliography B
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The attractive Coulomb potential polynomials.
Constr. Approx. 1 (2), pp. 103–119.
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Random and Restricted Walks: Theory and Applications.
Gordon and Breach, New York.
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Uniform approximation for potential scattering involving a rainbow.
Proc. Phys. Soc. 89 (3), pp. 479–490.
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A class of solvable potentials.
Nuovo Cimento (10) 25, pp. 864–879.
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Stability of repulsive Bose-Einstein condensates in a periodic potential.
Phys. Rev. E (3) 63 (036612), pp. 1–11.
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8: 23.21 Physical Applications
§23.21 Physical Applications
… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form . The Weierstrass function plays a similar role for cubic potentials in canonical form . … ►Quantum field theory. See Witten (1987).
9: Bibliography D
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Multiplicative Number Theory.
3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
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Pólya’s Theory of Counting.
In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.),
pp. 144–184.
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Wave function for smooth potential and mass step.
Phys. Rev. A 59 (1), pp. 107–112.
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Elements of the Theory of Numbers.
Harcourt/Academic Press, San Diego, CA.
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Supersymmetry, shape invariance, and exactly solvable potentials.
Amer. J. Phys. 56, pp. 163–168.
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