potential theory

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11: Bibliography Q

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Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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External Links:

ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)

Referenced by:

§18.24.

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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

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External Links:

Document, Link

Referenced by:

§18.38(iii).

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12: Bibliography W

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

Ch.9.

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Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

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ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§29.7(ii).

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A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7 (2), pp. 273–286.

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Document, MathReview (T. B. Benjamin), ZentralBlatt

Referenced by:

§21.9.

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Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.

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ISSN 0010-3616, Document, MathReview (J. Stasheff), ZentralBlatt

Referenced by:

1st item.

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13: 22.19 Physical Applications

… ►where $V(x)$ is the potential energy, and $x(t)$ is the coordinate as a function of time $t$. The potential ►

22.19.5 $$V(x)=\pm \frac{1}{2}{x}^2\pm \frac{1}{4}\beta x^4$$

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Symbols:

$V(x)$: potential energy, $x(t)$: coordinate and $\beta$: real positive

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http://dlmf.nist.gov/22.19.E5

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Annotations for §22.19(ii), §22.19 and Ch.22

►plays a prototypal role in classical mechanics (Lawden (1989, §5.2)), quantum mechanics (Schulman (1981, Chapter 29)), and quantum field theory (Pokorski (1987, p. 203), Parisi (1988, §14.6)). Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)). …

14: 9.16 Physical Applications

… ►Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop. … ►Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)). … ►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. … ►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

15: Bibliography H

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Quantum theory for mathematicians. Graduate Texts in Mathematics, Vol. 267, Springer, New York.

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External Links:

ISBN 978-1-4614-7115-8; 978-1-4614-7116-5, Document, Link, MathReview Entry

Referenced by:

§1.18(x).

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Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.

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ISSN 0022-2488, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii), §31.17(ii).

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An Introduction to the Theory of Numbers. 5th edition, The Clarendon Press Oxford University Press, New York-Oxford.

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ISBN 0-19-853170-2; 0-19-853171-0, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§20.12(i), Ch.27.

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Molecular Electronic-Structure Theory. John Wiley & Sons, New York.

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Notes:

Paperback Reprint of 2004 Edition

External Links:

ISBN 9871118531471

Referenced by:

§18.39(iii).

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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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External Links:

Document

Referenced by:

§15.18.

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16: 18.39 Applications in the Physical Sciences

… ►defines the potential for a symmetric restoring force $-kx$ for displacements from $x=0$. … ►argument b) The Morse Oscillator … ►c) A Rational SUSY Potential argument ►The Schrödinger equation with potential … ►Now use spherical coordinates (1.5.16) with $r$ instead of $\rho$, and assume the potential $V$ to be radial. …

17: 5.20 Physical Applications

… ►Suppose the potential energy of a gas of $n$ point charges with positions $x_1,x_2,\mathrm{\dots },x_n$ and free to move on the infinite line , is given by ►

5.20.1

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Symbols:

$\ln z$: principal branch of logarithm function, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $W$: potential energy

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http://dlmf.nist.gov/5.20.E1

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Annotations for §5.20, §5.20 and Ch.5

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5.20.2 $$P(x_1,\mathrm{\dots },x_n)=C\exp \left(-W/(kT)\right),$$

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Symbols:

$\exp z$: exponential function, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, $T$: temperature and $C$: constant

Permalink:

http://dlmf.nist.gov/5.20.E2

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Annotations for §5.20, §5.20 and Ch.5

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Elementary Particles

►Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …

18: Bibliography G

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Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-0101-0, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

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WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.

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External Links:

FullText, MathReview (Ioannis K. Purnaras), ZentralBlatt

Referenced by:

§18.15(v), §18.15(v).

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Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

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External Links:

Document, MathReview (D. B. Fairlie)

Referenced by:

(9.12.22), (9.12.23), §9.12(vii), §9.17(iii).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Kinetic theory of linear shear flow. Phys. Fluids 1 (3), pp. 215–224.

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Referenced by:

§18.39(iii).

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19: Bibliography R

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Topics in Analytic Number Theory. Springer-Verlag, New York.

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Notes:

Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169

External Links:

MathReview (H. Halberstam), ZentralBlatt

Referenced by:

§1.8(iv), §20.7(viii), §20.7(viii), Ch.24.

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Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen.

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Referenced by:

§21.7(i).

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

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Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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ISBN 0-201-87073-8, MathReview (Norman J. Richert)

Referenced by:

§27.17.

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On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.

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External Links:

Document, MathReview (M. A. Harrison), ZentralBlatt

Referenced by:

§27.5.

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20: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►At positive energies $E>0$, $\rho \ge 0$, and: … ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►Both variable sets may be used for attractive and repulsive potentials: the $(ϵ,r)$ set cannot be used for a zero potential because this would imply $r=0$ for all $s$, and the $(\eta ,\rho )$ set cannot be used for zero energy $E$ because this would imply $\rho =0$ always. … ►

§33.22(vi) Solutions Inside the Turning Point

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