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1: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►At positive energies $E>0$, $\rho \ge 0$, and: ► ►►►

Attractive potentials:	, .
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $\eta =0$.
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… ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. ► ►►►

Attractive potentials:	, $r>0$.
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $r=0$.
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Attractive potentials:	, .
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $\kappa =0$.
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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Unlike in the example in the paragraph above, in 3-dimensions a “dip below zero, or a potential well” in $V(r)$ does not always correspond to the existence of a discrete part of the spectrum. …

3: 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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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

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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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4: Bibliography D

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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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Sur les zéros réels des polynômes de Bernoulli. Ann. Inst. Fourier (Grenoble) 41 (2), pp. 267–309 (French).

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ISSN 0373-0956, MathReview (F. Beukers), ZentralBlatt

Referenced by:

§24.12(i).

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On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.

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ISSN 0065-1036, Document, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.12(iv).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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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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5: 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. …

6: Bibliography H

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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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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

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

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Document

Referenced by:

§15.18.

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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

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7: Bibliography B

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

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The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.

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ISSN 0176-4276, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

§18.39(iv), §18.39(iv).

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Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.

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External Links:

Document, ZentralBlatt

Referenced by:

§36.14(iii), §9.16.

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A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.

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Document, MathReview (S. Rosendorff), ZentralBlatt

Referenced by:

§15.18.

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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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External Links:

Document

Referenced by:

§29.19(i).

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8: Bibliography G

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New inequalities for the zeros of Jacobi polynomials. SIAM J. Math. Anal. 18 (6), pp. 1549–1562.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(vi).

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New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.9(i), §13.9(ii).

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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.16(iv), §18.16(iv), §18.16(vi).

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On the zeros of the Scorer functions. J. Approx. Theory 120 (2), pp. 253–266.

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ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§9.12(ix), §9.17(v), 5th item.

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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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9: 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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External Links:

Document, ZentralBlatt

Referenced by:

§15.18.

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On the zeros of the Weierstrass $\mathrm{\wp }$-function. Math. Ann. 258 (4), pp. 399–407.

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ISSN 0025-5831, Document, MathReview (Toyokazu Hiramatsu), ZentralBlatt

Referenced by:

§23.13.

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Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

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ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

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Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

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ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(i).

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

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10: Bibliography S

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Coulomb functions for attractive and repulsive potentials and for positive and negative energies. Comput. Phys. Comm. 146 (2), pp. 225–249.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§1.17(ii), §33.1, §33.14(i), §33.14(ii), §33.14(iii), §33.14(iv), §33.14(v), §33.16(i), §33.16(ii), §33.16(iii), §33.16(v), §33.17, §33.19, §33.20(i), §33.20(ii), §33.20(iii), §33.21(i), Ch.33.

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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Resonances in $n$-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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ISSN 0003-486X, Document, Link, MathReview (K. Kumar)

Referenced by:

§1.18(viii).

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Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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MathReview (R. Shail), ZentralBlatt

Referenced by:

§10.22(iv).

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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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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

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