essentially selfadjoint operator
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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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SelfAdjoint and Symmetric Operators
… ►Formally SelfAdjoint and SelfAdjoint Differential Operators: SelfAdjoint Extensions
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… ► … ►Selfadjoint extensions of a symmetric Operator
…2: 18.36 Miscellaneous Polynomials
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►These results are proven in Everitt et al. (2004), via construction of a selfadjoint Sturm–Liouville operator which generates the ${L}_{n}^{(k)}(x)$ polynomials, selfadjointness implying both orthogonality and completeness.
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►Completeness follows from the selfadjointness of ${T}_{k}$, Everitt (2008).
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►Completeness and orthogonality follow from the selfadjointness of the corresponding Schrödinger operator, GómezUllate and Milson (2014), Marquette and Quesne (2013).
3: 12.15 Generalized Parabolic Cylinder Functions
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►This equation arises in the study of nonselfadjoint elliptic boundaryvalue problems involving an indefinite weight function.
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4: 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 selfadjoint 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 selfadjointness,
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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 selfadjoint, or from the direct derivation of Dunkl (2003).
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►The radial operator (18.39.28)
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►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to 1$.
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5: 1.3 Determinants, Linear Operators, and Spectral Expansions
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§1.3(iv) Matrices as Linear Operators
… ►SelfAdjoint Operators on ${\mathbf{E}}_{n}$
… ►Real symmetric ($\mathbf{A}={\mathbf{A}}^{\mathrm{T}}$) and Hermitian ($\mathbf{A}={\mathbf{A}}^{\mathrm{H}}$) matrices are selfadjoint operators on ${\mathbf{E}}_{n}$. The spectrum of such selfadjoint operators consists of their eigenvalues, ${\lambda}_{i},i=1,2,\mathrm{\dots},n$, and all ${\lambda}_{i}\in \mathbb{R}$. … ►For selfadjoint $\mathbf{A}$ and $\mathbf{B}$, if $[\mathbf{A},\mathbf{B}]=\U0001d7ce$, see (1.2.66), simultaneous eigenvectors of $\mathbf{A}$ and $\mathbf{B}$ always exist. …6: Bibliography R
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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, SelfAdjointness.
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—ChebyshevHermite, Mathieu, Whittaker—in Twophase 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: 10.22 Integrals
8: Guide to Searching the DLMF
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Boolean operator:
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9: Preface
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► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST.
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