self-adjoint extensions of a symmetric operator

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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

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Self-Adjoint Operators

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Self-adjoint extensions of a symmetric Operator

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Spectral expansions and self-adjoint extensions

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2: 12.15 Generalized Parabolic Cylinder Functions

… ►can be viewed as a generalization of (12.2.4). This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).

3: 18.36 Miscellaneous Polynomials

… ►This inequality is violated for $n=1$ if , seemingly precluding such an extension of the Laguerre OP’s into that regime. … ►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. … ►A broad overview appears in Milson (2017). … ►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: 1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iv) Matrices as Linear Operators

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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. …

5: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

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BibTeX

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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).

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BibTeX

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6: 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, $\mathscr{H}$, is a second order differential operator of the form … ►If $\mathrm{\Psi }(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ then, by self-adjointness, … ►

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

… ►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). …

7: 10.22 Integrals

… ►If , then interchange $a$ and $b$, and also $\mu$ and $\nu$. If $b=a$, then … ►In (10.22.66)–(10.22.70) $a,b,c$ are positive constants. … ►(Thus if $a,b,c$ are the sides of a triangle, then $A^{\frac{1}{2}}$ is the area of the triangle.) … ►These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …

8: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►A fourth integral that is symmetric in only two variables is defined by … ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. … … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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9: 18.38 Mathematical Applications

… ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►A symmetric Laurent polynomial is a function of the form …Define a further operator $K_2$ by …A further operator, the so-called Casimir operator … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). …

10: 17.16 Mathematical Applications

… ►Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. In Lie algebras Lepowsky and Milne (1978) and Lepowsky and Wilson (1982) laid foundations for extensive interaction with $q$-series. …