# formally self-adjoint operators

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## 1—10 of 103 matching pages

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

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###### 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

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###### Formal Calculation of Determinants

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

… ►Real symmetric ($\mathbf{A}={\mathbf{A}}^{\mathrm{T}}$) and Hermitian ($\mathbf{A}={\mathbf{A}}^{\mathrm{H}}$) 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. …##### 3: 18.36 Miscellaneous Polynomials

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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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►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: 12.15 Generalized Parabolic Cylinder Functions

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►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function.
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##### 5: 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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►If $\mathrm{\Psi}(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ then, by self-adjointness,
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►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).
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►The radial operator (18.39.28)
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►With $N\to \mathrm{\infty}$ the functions normalized as $\delta (\u03f5-{\u03f5}^{\prime})$ with measure $dr$ are, formally,
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##### 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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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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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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##### 7: 25.17 Physical Applications

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►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.
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►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

##### 9: 18.38 Mathematical Applications

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

►in which $\mathrm{\Delta}$ is the forward difference operator (§3.6(i)).
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►Formal solutions are
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${c}_{0}=1$, and higher coefficients are determined by formal substitution.
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►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 ${\alpha}_{2}-{\alpha}_{1}=1,2,3,\mathrm{\dots}$.
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