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11: 28 Mathieu Functions and Hill’s Equation

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12: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28) …

13: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

14: 23 Weierstrass Elliptic and Modular
Functions

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15: 18.42 Software

… ►Also included is a website (CAOP) operated by a university department. …

16: 26.19 Mathematical Applications

… ►These have applications in operations research, probability theory, and statistics. …

17: Brian R. Judd

… ►Judd’s books include Operator Techniques in Atomic Spectroscopy, published by McGraw-Hill in 1963 and reprinted by Princeton University Press in 1998, Second Quantization and Atomic Spectroscopy, published by Johns Hopkins in 1967, Topics in Atomic and Nuclear Theory (with J. …

18: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …

19: 18.38 Mathematical Applications

… ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. … ►In the one-variable case the Dunkl operator eigenvalue equation … …

20: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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