separation of variables in 3D

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1: 19.39 Software

… ►In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter. …References to research software that is available in other ways is listed separately. ►A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C14). … ►The variables are real and the functions are $R$-functions. …

2: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. …$p({𝒟}^{\prime }3,n)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). …

3: 31.10 Integral Equations and Representations

… ►where ${𝒟}_z$ is Heun’s operator in the variable $z$ : … ►

Kernel Functions

… ►where $\sigma$ is a separation constant. … ►where ${𝒟}_z$ is given by (31.10.4). … ►where ${\sigma }_1$ and ${\sigma }_2$ are separation constants. …

4: 1.9 Calculus of a Complex Variable

… ►A function $f(z)$ is analytic in a domain $D$ if it is analytic at each point of $D$. … ►If $f(z)$ is analytic in an open domain $D$, then each of its derivatives $f^{\prime }(z)$, $f^{\prime \prime }(z)$, $\mathrm{\dots }$ exists and is analytic in $D$. … ►Suppose $f(z)$ is analytic in a domain $D$ and $C_1,C_2$ are two arcs in $D$ passing through $z_0$. … ►Suppose the series ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(z)$, where $f_n(z)$ is continuous, converges uniformly on every compact set of a domain $D$, that is, every closed and bounded set in $D$. …for any finite contour $C$ in $D$. …

5: 18.39 Applications in the Physical Sciences

… ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►allows anharmonic, or amplitude dependent, frequencies of oscillation about $x_e$, and also escape of the particle to $x=+\mathrm{\infty }$ with dissociation energy $D$. … ►

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

… ►Since the operators $T_e+V(r)$ and ${\mathrm{L}}^2$ commute and have simultaneous eigenfunctions (see §1.3(iv)), the wave function $\mathrm{\Psi }(r,\theta ,\varphi )$ separates as … ►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A Hilbert space $V$ is separable if there is an (at most countably infinite) orthonormal set $\{v_n\}$ in $V$ such that for every $v\in V$ … ►Assume that $𝒟(T)$ is dense in $V$, i. … ► $u_{\lambda }\in 𝒟(T)$, corresponding to distinct eigenvalues, are orthogonal: i. … ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that $𝒟(T)=𝒟(T^{\ast })$, as $f(x)$ and $g(x)$ satisfy the same boundary conditions and thus define the same domains. … ►, $𝒟(T)\subset 𝒟(T^{\ast })$ and $T^{\ast }v=Tv$ for $v\in 𝒟(T)$. …

7: Errata

… ►

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.

… ►

The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

… ►

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$. The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma (a,z)$. In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$.

Reported 2017-07-10 by Kurt Fischer.

… ►

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.

… ►

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).

…