self-adjoint differential operators

SearchAdvancedHelp

(0.003 seconds)

3 matching pages

1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions

… ►

§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

… ►Consider on $X$ the linear formally self-adjoint second order differential operator … … ► …

2: 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. …

3: 18.36 Miscellaneous Polynomials

… ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►

18.36.6 $${\int }_0^{\mathrm{\infty }}{\widehat{L}}_n^{(k)}\left(x\right){\widehat{L}}_m^{(k)}\left(x\right){\widehat{W}}_k(x)dx=\frac{(n+k)\mathrm{\Gamma }\left(n+k-1\right)}{(n-1)!}{\delta }_{n,m}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , ${\widehat{L}}_n^{(k)}\left(x\right)$: exceptional Laguerre polynomial, $!$: factorial (as in $n!$), $\int$: integral, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

… ►Completeness follows from the self-adjointness of $T_k$, Everitt (2008). … ►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).