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1: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

►Let ${\widehat{\lambda }}_m$, $m=0,1,2,\mathrm{\dots }$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_m(x)$, $m=0,1,2,\mathrm{\dots }$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus …

2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► … ► The analogous orthonormality is … ► … ► … ►

§1.18(viii) Mixed Spectra and Eigenfunction Expansions

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3: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

4: 31.17 Physical Applications

… ►for the common eigenfunction $\mathrm{\Psi }(𝐱)=\mathrm{\Psi }(x_s,x_t,x_u)$, where $a$ is the coupling parameter of interacting spins. …The operators ${𝐉}^2$ and ${𝐻}_s$ admit separation of variables in $z_1,z_2$, leading to the following factorization of the eigenfunction $\mathrm{\Psi }(𝐱)$: ►

31.17.4 $$\mathrm{\Psi }(𝐱)={(z_1{z}_2)}^{-s-\frac{1}{4}}{((z_1-1)(z_2-1))}^{-t-\frac{1}{4}}{((z_1-a)(z_2-a))}^{-u-\frac{1}{4}}w(z_1)w(z_2),$$

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Symbols:

$z$: complex variable, $a$: complex parameter and $\mathrm{\Psi }(𝐱)$: common eigenfunction

Permalink:

http://dlmf.nist.gov/31.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.17(i), §31.17 and Ch.31

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5: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation … ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►with an infinite set of orthonormal $L^2$ eigenfunctions …

6: Bibliography T

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Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.

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External Links:

MathReview Entry

Referenced by:

(1.18.59), §1.18(ix), §1.18(v), §1.18(vi), §1.18(x), (10.22.78), §10.22(v), §18.39(ii).

Encodings:

BibTeX

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7: 1.13 Differential Equations

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

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Eigenvalues and Eigenfunctions

… ►The functions $u(x)$ which correspond to these being eigenfunctions. … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

8: 28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

►Two eigenfunctions correspond to each eigenvalue $a={\lambda }_{\nu }\left(q\right)$. …The other eigenfunction is ${\mathrm{me}}_{\nu }(-z,q)$, a Floquet solution with respect to $-\nu$ with $a={\lambda }_{\nu }\left(q\right)$. …

9: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. Period $\pi$ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas antiperiod $\pi$ means that $w(z+\pi )=-w(z)$. … ►

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

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Eigenvalues	Eigenfunctions	Periodicity	Parity
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10: 29.3 Definitions and Basic Properties

… ►The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. … ►

Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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