# 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

… ►###### Self-Adjoint Operators

… ► … ►###### Self-adjoint extensions of a symmetric Operator

… ►###### Spectral expansions and self-adjoint extensions

…##### 2: 12.15 Generalized Parabolic Cylinder Functions

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►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

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

##### 4: 1.3 Determinants, Linear Operators, and Spectral Expansions

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###### §1.3(iv) Matrices as Linear Operators

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

… ►Real symmetric ($\mathbf{A}={\mathbf{A}}^{\mathrm{T}}$) and Hermitian ($\mathbf{A}={\mathbf{A}}^{\mathrm{H}}$) matrices are*self-adjoint*operators on ${\mathbf{E}}_{n}$. The*spectrum*of such self-adjoint operators consists of their eigenvalues, ${\lambda}_{i},i=1,2,\mathrm{\dots},n$, and all ${\lambda}_{i}\in \mathbb{R}$. … ►For self-adjoint $\mathbf{A}$ and $\mathbf{B}$, if $[\mathbf{A},\mathbf{B}]=\U0001d7ce$, see (1.2.66), simultaneous eigenvectors of $\mathbf{A}$ and $\mathbf{B}$ 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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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness.
Academic Press, New York.
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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators.
Academic Press, New York.
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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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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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##### 6: 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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►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

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

…##### 9: 18.38 Mathematical Applications

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►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). …