axially symmetric potential theory

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11: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

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

Referenced by:

§27.5.

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12: 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)). …

13: 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. …

14: Bibliography E

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The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.

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Document, ZentralBlatt

Referenced by:

§15.18.

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On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

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Notes:

With a loose Russian summary

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ISSN 0011-4642, MathReview (D. Hinton)

Referenced by:

§1.13(viii).

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A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.

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Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

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Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In Sturm-Liouville theory, pp. 45–74.

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Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

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15: 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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Notes:

Revised and with a preface by Hugh L. Montgomery

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ISBN 0-387-95097-4, MathReview, ZentralBlatt

Referenced by:

§27.12.

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Pólya’s Theory of Counting. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), pp. 144–184.

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ISBN 0898741726

Referenced by:

§26.20.

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Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.

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Document

Referenced by:

§15.18.

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

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Supersymmetry, shape invariance, and exactly solvable potentials. Amer. J. Phys. 56, pp. 163–168.

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Referenced by:

§18.38(iii).

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16: Bille C. Carlson

… ►If some of the parameters are equal, then the $R$-function is symmetric in the corresponding variables. This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-Adjoint and Symmetric Operators

… ►One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner. … ►which appear in the quantum theory of binding or scattering of a particle in a spherically symmetric potential $V(r)$ in three dimensions, and where $r\in [0,\mathrm{\infty })$. … ►A linear operator $T$ with dense domain is called symmetric if … ►

Self-adjoint extensions of a symmetric Operator

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18: Bibliography G

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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

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ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§18.28(xi).

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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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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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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 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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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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Document, Link

Referenced by:

§18.38(iii).

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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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