Coulomb Problem in 3D

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1: Bibliography H

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Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:

A reprinting with corrections appeared in 2000.

External Links:

ISBN 3-540-56670-8, MathReview, ZentralBlatt

Referenced by:

§3.7(v), §3.7(i).

Encodings:

BibTeX

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Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-60452-9, MathReview, ZentralBlatt

Referenced by:

§3.7(v).

Encodings:

BibTeX

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Some problems of “Partitio Numerorum” (VI): Further researches in Waring’s Problem. Math. Z. 23, pp. 1–37.

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Notes:

Reprinted in Collected Papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others), Vol. I, pp. 469–505, Clarendon Press, Oxford, 1966.

External Links:

ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.13(iii).

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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Mathematical Problems of Radiative Equilibrium. Cambridge Tracts in Mathematics and Mathematical Physics No. 31, Cambridge University Press, Cambridge.

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Notes:

Reprinted by Stechert-Hafner, Inc., New York, 1964

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.19(ix), §8.24(iii).

Encodings:

BibTeX

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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Assume that $𝒟(T)$ is dense in $V$, i. … ► $u_{\lambda }\in 𝒟(T)$, corresponding to distinct eigenvalues, are orthogonal: i. … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … … ►, $𝒟(T)\subset 𝒟(T^{\ast })$ and $T^{\ast }v=Tv$ for $v\in 𝒟(T)$. …

3: 18.39 Applications in the Physical Sciences

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§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

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The Quantum Coulomb Problem

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The Relativistic Quantum Coulomb Problem

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The Quantum Coulomb Problem: Scattering States

… ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. …

4: Bibliography S

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

Referenced by:

8th item.

Encodings:

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

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The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

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Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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External Links:

ISSN 0020-7225, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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

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External Links:

MathReview (R. Shail), ZentralBlatt

Referenced by:

§10.22(iv).

Encodings:

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5: Errata

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The following additions were made in Chapter 18:

• In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),

• A new introduction, minor changes in the presentation, and three new paragraphs.

• Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

• Line numbers and two extra rows were added to Table 18.8.1.

• Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

• Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

• Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

• Equation (18.15.4_5).

• The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

• Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

• Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

• A new introduction.

• Equation (18.21.13).

• Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

• In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

• Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

• A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).

• Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).

• This short section has been expanded, with Equation (18.32.2).

• Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

• This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

• This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

• This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

• The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

• This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

  The original text of 18.39(i) Quantum Mechanics was:

  “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:

  errata.1 $$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=i\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$

  where $\mathrm{\hslash }$ is the reduced Planck’s constant. On substituting $\psi (x,t)=\eta (x)\zeta (t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta (x)$ is

  errata.2 $$\frac{d^2\eta }{{dx}^2}+\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(x)\right)\eta =0.$$

  For a harmonic oscillator, the potential energy is given by

  errata.3 $$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$

  where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant $E$ (the total energy of the particle) must satisfy

  errata.4 $$E=E_n=\left(n+\frac{1}{2}\right)\mathrm{\hslash }\omega ,$$ $n=0,1,2,\mathrm{\dots }$.

  The corresponding eigenfunctions are

  errata.5 $${\eta }_n(x)={\pi }^{-\frac{1}{4}}{2}^{-\frac{1}{2}n}{(n!b)}^{-\frac{1}{2}}{H}_n\left(x/b\right){\mathrm{e}}^{-x^2/2b^2},$$

  where $b={(\mathrm{\hslash }/m\omega )}^{1/2}$, and $H_n$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

  A second example is provided by the three-dimensional time-independent Schrödinger equation

  errata.6 $${\nabla }^2\psi +\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(𝐱)\right)\psi =0,$$

  when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

  For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

• The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

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22.9.8 $$s_{1,3}^{(4)}{s}_{2,3}^{(4)}+s_{2,3}^{(4)}{s}_{3,3}^{(4)}+s_{3,3}^{(4)}{s}_{1,3}^{(4)}=\frac{{\kappa }^2-1}{k^2}$$

22.9.9 $$c_{1,3}^{(4)}{c}_{2,3}^{(4)}+c_{2,3}^{(4)}{c}_{3,3}^{(4)}+c_{3,3}^{(4)}{c}_{1,3}^{(4)}=-\frac{\kappa (\kappa +2)}{{(1+\kappa )}^2}$$

22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2)$$

Originally all the functions $s_{m,p}^{(4)}$, $c_{m,p}^{(4)}$ , $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ in Equations (22.9.8), (22.9.9) and (22.9.10) were written incorrectly with $p=2$. These functions have been corrected so that they are written with $p=3$. In the sentence just below (22.9.10), the expression $s_{m,2}^{(4)}{s}_{n,2}^{(4)}$ has been corrected to read $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$.

Reported by Juan Miguel Nieto on 2019-11-07

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28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots }$$

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Reported 2017-02-02 by Daniel Karlsson.

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables $a$ and $x=\lambda a$. It has been extended to allow for complex variables $a$ and $z=\lambda a$ (and we have replaced $x$ with $z$ in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from to the sector , and the interval of validity of (8.11.7) was increased from $\lambda >1$ to the sector $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$. A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$. Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

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