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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
§1.18(iv) FormallySelf-adjoint Linear Second Order DifferentialOperators
Consider formally self-adjoint operators of the form …
2: 1.3 Determinants, Linear Operators, and Spectral Expansions
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. …
3: 30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
4: 15.10 Hypergeometric Differential Equation
§15.10(i) Fundamental Solutions
15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
This is the hypergeometric differential equation. …
5: 18.36 Miscellaneous Polynomials
Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … 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. …
18.36.6 $\int_{0}^{\infty}\hat{L}^{(k)}_{n}\left(x\right)\hat{L}^{(k)}_{m}\left(x\right% )\hat{W}_{k}(x)\,\mathrm{d}x=\frac{(n+k)\Gamma\left(n+k-1\right)}{(n-1)!}% \delta_{n,m}.$
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).
6: 12.15 Generalized Parabolic Cylinder Functions
This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
7: 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. … The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathcal{H}$, is a second order differential operator of the form … 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). … With $N\to\infty$ the functions normalized as $\delta(\epsilon-\epsilon^{\prime})$ with measure $\,\mathrm{d}r$ are, formally, …
8: Bibliography R
• M. Reed and B. Simon (1975) Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.
• M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.
• S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.
• G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.
• J. Rushchitsky and S. Rushchitska (2000) 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.
• 9: 10.22 Integrals
10.22.9 $\int_{0}^{x}J_{2n}\left(t\right)\,\mathrm{d}t=\int_{0}^{x}J_{0}\left(t\right)% \,\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1% }\left(t\right)\,\mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(% x\right),$ $n=0,1,\dots$.
Sufficient conditions for the validity of (10.22.77) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $\nu\geq-\tfrac{1}{2}$, or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\,\mathrm{d}x<\infty$ when $-1<\nu<-\tfrac{1}{2}$; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). … These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …A sufficient condition for the validity is $\int_{a}^{\infty}|f(y)|\,\mathrm{d}y<\infty$. …Sufficient conditions for the validity of (10.22.79) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $0<\nu\leq\tfrac{1}{2}$, or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\frac{1}{2}-\nu}|f(x)|\,\mathrm{d}x<\infty$ when $\tfrac{1}{2}<\nu<1$; see Titchmarsh (1962a, pp. 88–90). …
10: 25.17 Physical Applications
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. … 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). …