formally self-adjoint differential operators

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

Self-Adjoint and Symmetric Operators

… ►

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

… ►

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

… ► … ►Consider formally self-adjoint operators of the form …

2: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

Formal Calculation of Determinants

… ►

Self-Adjoint Operators on $\mathbf{E}_{n}$

… ►Real symmetric (

\mathbf{A}=\mathbf{A}^{\mathrm{T}}

) and Hermitian (

\mathbf{A}={\mathbf{A}}^{{\rm H}}

) matrices are self-adjoint operators on

\mathbf{E}_{n}

. The spectrum of such self-adjoint operators consists of their eigenvalues,

\lambda_{i},i=1,2,\dots,n

, and all

\lambda_{i}\in\mathbb{R}

. … ►For self-adjoint

\mathbf{A}

and

\mathbf{B}

, if

[{\mathbf{A}},{\mathbf{B}}]=\boldsymbol{{0}}

, see (1.2.66), simultaneous eigenvectors of

\mathbf{A}

and

\mathbf{B}

always exist. …

3: 30.2 Differential Equations

§30.2 Differential Equations

►

§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

…

4: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

►

§15.10(i) Fundamental Solutions

►

15.10.1

z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)
Permalink:: http://dlmf.nist.gov/15.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10 and Ch.15

►This is the hypergeometric differential equation. … ► …

5: 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}^{\infty}\hat{L}^{(k)}_{n}\left(x\right)\hat{L}^{(k)}_{m}\left(x\right% )\hat{W}_{k}(x)\,\mathrm{d}x=\frac{(n+k)\Gamma\left(n+k-1\right)}{(n-1)!}% \delta_{n,m}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\hat{L}^{(\NVar{k})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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).

6: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

7: 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. … ►The fundamental quantum Schrödinger operator, also called the Hamiltonian,

\mathcal{H}

, is a second order differential operator of the form … ►If

\Psi(x,t=0)=\chi(x)

is an arbitrary unit normalized function in the domain of

\mathcal{H}

then, by self-adjointness, … ►noting that the

\psi_{p,l}(r)

are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). … ►With

N\to\infty

the functions normalized as

\delta(\epsilon-\epsilon^{\prime})

with measure

\,\mathrm{d}r

are, formally, …

8: Bibliography R

… ►

M. Reed and B. Simon (1975) Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

ⓘ

External Links:: ISBN 978-0-12-585002-5, MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

►

M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

ⓘ

External Links:: ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

… ►

G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

ⓘ

External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

… ►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

ⓘ

External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

9: 10.22 Integrals

… ►

10.22.9

\int_{0}^{x}J_{2n}\left(t\right)\,\mathrm{d}t=\int_{0}^{x}J_{0}\left(t\right)% \,\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1% }\left(t\right)\,\mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(% x\right),

n=0,1,\dots

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer, $k$ : nonnegative integer and $x$ : real variable
A&S Ref:: 11.1.3, 11.1.4
Permalink:: http://dlmf.nist.gov/10.22.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

… ►Sufficient conditions for the validity of (10.22.77) are that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

when

\nu\geq-\tfrac{1}{2}

, or that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

and

\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\,\mathrm{d}x<\infty

when

-1<\nu<-\tfrac{1}{2}

; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). … ►These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …A sufficient condition for the validity is

\int_{a}^{\infty}|f(y)|\,\mathrm{d}y<\infty

. …Sufficient conditions for the validity of (10.22.79) are that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

when

0<\nu\leq\tfrac{1}{2}

, or that

\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty

and

\int_{0}^{1}x^{\frac{1}{2}-\nu}|f(x)|\,\mathrm{d}x<\infty

when

\tfrac{1}{2}<\nu<1

; see Titchmarsh (1962a, pp. 88–90). …

10: 25.17 Physical Applications

… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. … ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …

formally self-adjoint differential operators

1—10 of 400 matching pages

1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

Self-Adjoint and Symmetric Operators

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

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

2: 1.3 Determinants, Linear Operators, and Spectral Expansions

Formal Calculation of Determinants

Self-Adjoint Operators on 𝐄 n

3: 30.2 Differential Equations

§30.2 Differential Equations

§30.2(i) Spheroidal Differential Equation

§30.2(iii) Special Cases

4: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

§15.10(i) Fundamental Solutions

5: 18.36 Miscellaneous Polynomials

6: 12.15 Generalized Parabolic Cylinder Functions

7: 18.39 Applications in the Physical Sciences

8: Bibliography R

9: 10.22 Integrals

10: 25.17 Physical Applications

Self-Adjoint Operators on $\mathbf{E}_{n}$