# eigenfunctions

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##### 1: 28.30 Expansions in Series of Eigenfunctions
###### §28.30(i) Real Variable
Let $\widehat{\lambda}_{m}$, $m=0,1,2,\dots$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$, $m=0,1,2,\dots$, 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 …
##### 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 $\Psi(\mathbf{x})=\Psi(x_{s},x_{t},x_{u})$, where $a$ is the coupling parameter of interacting spins. …The operators $\mathbf{J}^{2}$ and $H_{s}$ admit separation of variables in $z_{1},z_{2}$, leading to the following factorization of the eigenfunction $\Psi(\mathbf{x})$:
31.17.4 $\Psi(\mathbf{x})=(z_{1}z_{2})^{-s-\frac{1}{4}}((z_{1}-1)(z_{2}-1))^{-t-\frac{1% }{4}}\*((z_{1}-a)(z_{2}-a))^{-u-\frac{1}{4}}w(z_{1})w(z_{2}),$
##### 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
• E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.
• ##### 7: 1.13 Differential Equations
###### 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
###### §28.12(ii) Eigenfunctions$\operatorname{me}_{\nu}\left(z,q\right)$
Two eigenfunctions correspond to each eigenvalue $a=\lambda_{\nu}\left(q\right)$. …The other eigenfunction is $\operatorname{me}_{\nu}\left(-z,q\right)$, a Floquet solution with respect to $-\nu$ with $a=\lambda_{\nu}\left(q\right)$. …
##### 9: 28.2 Definitions and Basic Properties
###### §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 $w(z+\pi)=-w(z)$. …
##### 10: 29.3 Definitions and Basic Properties
The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$. …