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1: 18.39 Applications in the Physical Sciences

… ►c) A Rational SUSY Potential argument … ►

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Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for $\mathrm{\hslash }=k=m=1$, of (18.39.13), in ${\psi }_1(x)$, ${\psi }_4(x)$ and those of the rational potential of (18.39.19), in ${\widehat{\psi }}_3(x)$, ${\widehat{\psi }}_6(x)$. … Magnify

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2: 18.36 Miscellaneous Polynomials

… ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

3: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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Document

Referenced by:

§1.18(viii).

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4: 18.38 Mathematical Applications

… ►Analogues of the original Dunkl operator (the rational case) were introduced by Heckman and Cherednik for the trigonometric case, and by Cherednik for the $q$-case. … ►The solved Schrödinger equations of §18.39(i) involve shape invariant potentials, and thus are in the family of supersymmetric or SUSY potentials. SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. … ►EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics. …

5: Bibliography S

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Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Comm. Math. Phys. 220 (1), pp. 165–229.

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ISSN 0010-3616, Document, MathReview (I. Dolgachev), ZentralBlatt

Referenced by:

§32.7(viii).

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Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.

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ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt

Referenced by:

§5.21, §5.23(iii).

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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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MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

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6: 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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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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Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.

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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.

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Notes:

International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.9(vi), §5.23(i).

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Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.

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ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.11(iii).

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7: 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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A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.

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ISSN 0003-9527, Document, MathReview (Richard Brown), ZentralBlatt

Referenced by:

§32.11(ii).

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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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Über die Integration logarithmisch-rationaler Differentiale. J. Reine Angew. Math. 3, pp. 101–159.

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External Links:

ZentralBlatt

Referenced by:

§25.12(i).

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8: Bibliography G

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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..

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ISSN 1751-8113, Document, Link, MathReview (Brian Simanek)

Referenced by:

§18.36(vi), §18.36(vi).

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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

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.

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External Links:

ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§32.10(ii), §32.8(i), §32.8(v).

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A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

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ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt

Referenced by:

§18.33(v).

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Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

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Notes:

Erratum: same journal 17 (1975), no. 7, p. 1361

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Document, MathReview (K. Veselic), ZentralBlatt

Referenced by:

4th item.

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Rational function approximations for Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 84, pp. 101–108.

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Notes:

Double-precision Fortran, maximum precision 8D.

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ISSN 0067-0049, Document, Code

Referenced by:

5th item, 1st item.

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10: Bibliography B

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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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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

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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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Referenced by:

§29.19(i).

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