# spectrum of a self-adjoint extension of a linear differential operator

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## 1—10 of 917 matching pages

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

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►There is also a notion of self-adjointness for unbounded operators, see §1.18(ix).
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►If $T$ is self-adjoint (bounded or unbounded) then $\sigma (T)$ is a closed subset of $\mathbb{R}$ and the residual spectrum is empty.
Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal.
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###### Self-adjoint extensions of a symmetric Operator

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

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►Square matices can be seen as

*linear operators*because $\mathbf{A}(\alpha \mathbf{a}+\beta \mathbf{b})=\alpha \mathbf{A}\mathbf{a}+\beta \mathbf{A}\mathbf{b}$ for all $\alpha ,\beta \in \u2102$ and $\mathbf{a},\mathbf{b}\in {\mathbf{E}}_{n}$, the space of all $n$-dimensional vectors. ►###### 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. …##### 4: 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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►A broad overview appears in Milson (2017).
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►The restriction to $n\ge 1$ is now apparent: (18.36.7) does not posses a solution if $y(x)$ is a constant.
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).

##### 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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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being ${L}^{2}$ and forming a complete set.
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►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space.
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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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##### 6: Bibliography R

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Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric.
IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.
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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials.
Canad. J. Math. 33 (4), pp. 915–928.
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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness.
Academic Press, New York.
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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters.
Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
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The effective resistance and inductance of a concentric main, and methods of computing the $\mathrm{ber}$ and $\mathrm{bei}$ and allied functions.
Philos. Mag. (6) 17, pp. 524–552.
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##### 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: 15.10 Hypergeometric Differential Equation

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►It has regular singularities at $z=0,1,\mathrm{\infty}$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively.
When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs ${f}_{1}(z)$, ${f}_{2}(z)$ of fundamental solutions.
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►Moreover, in (15.10.9) and (15.10.10) the symbols $a$ and $b$ are interchangeable.
►(c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\mathrm{\dots}$, then fundamental solutions in the neighborhood of $z=0$ are given by ${z}^{n-1}$ times those in (a) and (b), with $a$ and $b$ replaced throughout by $a+n-1$ and $b+n-1$, respectively.
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►(e) Finally, if $a-b+1$ equals $n=1,2,3,\mathrm{\dots}$, or $2-n=0,-1,-2,\mathrm{\dots}$, then fundamental solutions in the neighborhood of $z=\mathrm{\infty}$ are given by ${z}^{-a}$ times those in (a), (b), and (c) with $b$ and $z$ replaced by $a-c+1$ and $1/z$, respectively.
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##### 9: 30.2 Differential Equations

###### §30.2 Differential Equations

►###### §30.2(i) Spheroidal Differential Equation

… ► … ►The*Liouville normal form*of equation (30.2.1) is … ►

###### §30.2(iii) Special Cases

…##### 10: Bibliography

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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Spectrum line profiles: The Voigt function.
J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.
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An integral of products of ultraspherical functions and a
$q$-extension.
J. London Math. Soc. (2) 33 (1), pp. 133–148.
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Some basic hypergeometric extensions of integrals of Selberg and Andrews.
SIAM J. Math. Anal. 11 (6), pp. 938–951.
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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices.
Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
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