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 $ℝ$ and the residual spectrum is empty. Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. … ►

Self-adjoint extensions of a symmetric Operator

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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 $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors. ►

Self-Adjoint Operators on ${𝐄}_n$

… ►Real symmetric ($𝐀={𝐀}^{\mathrm{T}}$) and Hermitian ($𝐀={𝐀}^{\mathrm{H}}$) matrices are self-adjoint operators on ${𝐄}_n$. The spectrum of such self-adjoint operators consists of their eigenvalues, ${\lambda }_i,i=1,2,\mathrm{\dots },n$, and all ${\lambda }_i\in ℝ$. … ►For self-adjoint $𝐀$ and $𝐁$, if $[𝐀,𝐁]=𝟎$, see (1.2.66), simultaneous eigenvectors of $𝐀$ and $𝐁$ 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 $n\ge 1$ 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). … ►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 $\mathscr{H}$ form a discrete, normed, and complete basis for a Hilbert space. … ►If $\mathrm{\Psi }(x,t=0)=\chi (x)$ is an arbitrary unit normalized function in the domain of $\mathscr{H}$ 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

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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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Document

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6th item.

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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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ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

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ISBN 978-0-12-585002-5, MathReview Entry

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§1.18(x).

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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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English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

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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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Document, ZentralBlatt

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§10.73(iii).

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7: 10.22 Integrals

… ►If , 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 $z=0,1,\mathrm{\infty }$, 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. … ►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. … ►(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. …

9: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

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

§30.2(iii) Special Cases

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10: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.

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Document

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§7.19(iv), §7.21.

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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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External Links:

ISSN 0024-6107, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§10.22(iv), §14.30(ii).

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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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ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§17.13, §17.13.

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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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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

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