separation of variables

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1: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

2: 31.17 Physical Applications

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§31.17(i) Addition of Three Quantum Spins

►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … ►The operators ${𝐉}^2$ and ${𝐻}_s$ admit separation of variables in $z_1,z_2$, leading to the following factorization of the eigenfunction $\mathrm{\Psi }(𝐱)$: … ►For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). …

3: 12.17 Physical Applications

… ►Setting $w=U(\xi )V(\eta )W(\zeta )$ and separating variables, we obtain … ►Miller (1974) treats separation of variables by group theoretic methods. …

4: 13.28 Physical Applications

… ►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …

5: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …

6: 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 … ►

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

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7: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(iv) Separation of Variables

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8: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(iv) Separation of Variables

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9: Bibliography K

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The group $\mathrm{O}(4)$, separation of variables and the hydrogen atom. SIAM J. Appl. Math. 30 (4), pp. 630–664.

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External Links:

Document, MathReview (Anthony J. Bracken), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.

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External Links:

ISBN 0-582-98807-1, MathReview (H. Kilp), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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10: Bibliography O

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Triorthogonal systems in spaces of constant curvature in which the equation ${\mathrm{\Delta }}_{2}u+\lambda u=0$ allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

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Notes:

In Russian.

External Links:

MathReview (H. Busemann), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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