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

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##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
There is also a notion of self-adjointness for unbounded operators, see §1.18(ix). … 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. …
##### 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: 1.3 Determinants, Linear Operators, and Spectral Expansions
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 $\alpha,\beta\in\mathbb{C}$ and $\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}$, the space of all $n$-dimensional vectors.
###### Self-AdjointOperators on $\mathbf{E}_{n}$
Real symmetric ($\mathbf{A}=\mathbf{A}^{\mathrm{T}}$) and Hermitian ($\mathbf{A}={\mathbf{A}}^{{\rm H}}$) matrices are self-adjoint operators on $\mathbf{E}_{n}$. The spectrum of such self-adjoint operators consists of their eigenvalues, $\lambda_{i},i=1,2,\dots,n$, and all $\lambda_{i}\in\mathbb{R}$. … For self-adjoint $\mathbf{A}$ and $\mathbf{B}$, if $[{\mathbf{A}},{\mathbf{B}}]=\boldsymbol{{0}}$, see (1.2.66), simultaneous eigenvectors of $\mathbf{A}$ and $\mathbf{B}$ always exist. …
##### 4: 18.36 Miscellaneous Polynomials
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). … The restriction to $n\geq 1$ 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). … 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
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. … 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. … However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of $\mathcal{H}$ form a discrete, normed, and complete basis for a Hilbert space. … If $\Psi(x,t=0)=\chi(x)$ is an arbitrary unit normalized function in the domain of $\mathcal{H}$ then, by self-adjointness, … 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). …
##### 6: Bibliography R
• H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.
• M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.
• M. Reed and B. Simon (1975) Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.
• E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
• A. Russell (1909) 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.
• ##### 7: 10.22 Integrals
If $0, 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: 15.10 Hypergeometric Differential Equation
It has regular singularities at $z=0,1,\infty$, 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. … 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 $2-n=0,-1,-2,\dots$, 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. … (e) Finally, if $a-b+1$ equals $n=1,2,3,\dots$, or $2-n=0,-1,-2,\dots$, then fundamental solutions in the neighborhood of $z=\infty$ are given by $z^{-a}$ times those in (a), (b), and (c) with $b$ and $z$ replaced by $a-c+1$ and $\ifrac{1}{z}$, respectively. …
##### 9: 30.2 Differential Equations
###### §30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
##### 10: Bibliography
• M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
• B. H. Armstrong (1967) Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.
• R. Askey, T. H. Koornwinder, and M. Rahman (1986) An integral of products of ultraspherical functions and a $q$-extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.
• R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.
• J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.