# self-adjoint extensions of a symmetric operator

(0.006 seconds)

## 1—10 of 901 matching pages

##### 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: 18.36 Miscellaneous Polynomials
This inequality is violated for $n=1$ if $\alpha<-1$, seemingly precluding such an extension of the Laguerre OP’s into that regime. … 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). … 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: 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. …
##### 5: Bibliography R
• E. M. Rains (1998) Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.
• 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.
• ##### 6: 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, …
###### §18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
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). …
##### 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: 19.16 Definitions
###### §19.16(i) Symmetric Integrals
A fourth integral that is symmetric in only two variables is defined by … which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$. … …
##### 9: 18.38 Mathematical Applications
The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … A symmetric Laurent polynomial is a function of the form …Define a further operator $K_{2}$ by …A further operator, the so-called Casimir operatorThe Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_{n}(z;a,b,c,d\,|\,q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d\,|\,q)$, where $R_{n}(z)$ is given in (18.28.1_5). …
##### 10: 17.16 Mathematical Applications
Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. In Lie algebras Lepowsky and Milne (1978) and Lepowsky and Wilson (1982) laid foundations for extensive interaction with $q$-series. …