formally self-adjoint operators

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-Adjoint and Symmetric Operators

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

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§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

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Formal Calculation of Determinants

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Self-Adjoint Operators on ${𝐄}_n$

… ►Real symmetric ($𝐀={𝐀}^{\mathrm{T}}$) and Hermitian ($𝐀={𝐀}^{\mathrm{H}}$) matrices are self-adjoint operators on ${𝐄}_n$. The spectrum of such self-adjoint operators consists of their eigenvalues, ${\lambda }_i,i=1,2,\mathrm{\dots },n$, and all ${\lambda }_i\in ℝ$. … ►For self-adjoint $𝐀$ and $𝐁$, if $[𝐀,𝐁]=𝟎$, see (1.2.66), simultaneous eigenvectors of $𝐀$ and $𝐁$ always exist. …

3: 18.36 Miscellaneous Polynomials

… ►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. … ►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).

4: 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. …

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. … ►If $\mathrm{\Psi }(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ 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). … ►The radial operator (18.39.28) … ►With $N\to \mathrm{\infty }$ the functions normalized as $\delta (ϵ-{ϵ}^{\prime })$ with measure $dr$ are, formally, …

6: Bibliography R

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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

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External Links:

ISBN 978-0-12-585002-5, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:

ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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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

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On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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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.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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7: 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). …

8: 10.22 Integrals

… ►These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …

9: 18.38 Mathematical Applications

… ►However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►In the one-variable case the Dunkl operator eigenvalue equation … …

10: 2.9 Difference Equations

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2.9.2 $${\mathrm{\Delta }}^{2}w(n)+(2+f(n))\mathrm{\Delta }w(n)+(1+f(n)+g(n))w(n)=0,$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Delta }$: forward difference operator, $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution

Permalink:

http://dlmf.nist.gov/2.9.E2

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Annotations for §2.9(i), §2.9 and Ch.2

►in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). … ►Formal solutions are … ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►The coefficients $b_s$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha }_2-{\alpha }_1=0$, or with $b_0=1$ when ${\alpha }_2-{\alpha }_1=1,2,3,\mathrm{\dots }$. …