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## 1—10 of 103 matching pages

##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(iv) FormallySelf-adjoint Linear Second Order Differential Operators
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: 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. … 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).
##### 4: 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. …
##### 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. … 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). … The radial operator (18.39.28) … With $N\to\infty$ the functions normalized as $\delta(\epsilon-\epsilon^{\prime})$ with measure $\,\mathrm{d}r$ are, formally, …
##### 6: 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.
• ##### 7: 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). …
##### 8: 10.22 Integrals
These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …
##### 9: 18.38 Mathematical Applications
However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … A further operator, the so-called Casimir operator
###### Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials
In the one-variable case the Dunkl operator eigenvalue equation … …
##### 10: 2.9 Difference Equations
2.9.2 $\Delta^{2}w(n)+(2+f(n))\Delta w(n)+(1+f(n)+g(n))w(n)=0,$ $n=0,1,2,\dots$,
in which $\Delta$ is the forward difference operator3.6(i)). … Formal solutions are … $c_{0}=1$, and higher coefficients are determined by formal substitution. … The coefficients $b_{s}$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when $\alpha_{2}-\alpha_{1}=0$, or with $b_{0}=1$ when $\alpha_{2}-\alpha_{1}=1,2,3,\dots$. …