# self-adjoint differential operators

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## 3 matching pages

##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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###### 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

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►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.
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##### 3: 18.36 Miscellaneous Polynomials

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►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.
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►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.
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18.36.6
$${\int}_{0}^{\mathrm{\infty}}{\widehat{L}}_{n}^{(k)}\left(x\right){\widehat{L}}_{m}^{(k)}\left(x\right){\widehat{W}}_{k}(x)dx=\frac{(n+k)\mathrm{\Gamma}\left(n+k-1\right)}{(n-1)!}{\delta}_{n,m}.$$

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