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self-adjoint extensions of a symmetric operator


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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Formally Self-Adjoint and Self-Adjoint Differential OperatorsSelf-Adjoint Extensions
Self-Adjoint Operators
Self-adjoint extensions of a symmetric Operator
Spectral expansions and self-adjoint extensions
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 α < 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
§1.3(iv) Matrices as Linear Operators
Self-Adjoint Operators on 𝐄 n
Real symmetric ( 𝐀 = 𝐀 T ) and Hermitian ( 𝐀 = 𝐀 H ) matrices are self-adjoint operators on 𝐄 n . The spectrum of such self-adjoint operators consists of their eigenvalues, λ i , i = 1 , 2 , , n , and all λ i . … For self-adjoint 𝐀 and 𝐁 , if [ 𝐀 , 𝐁 ] = 𝟎 , see (1.2.66), simultaneous eigenvectors of 𝐀 and 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, , is a second order differential operator of the form … If Ψ ( x , t = 0 ) = χ ( x ) is an arbitrary unit normalized function in the domain of then, by self-adjointness, …
    §18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
    noting that the ψ 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 < b < a , then interchange a and b , and also μ and ν . 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 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 , , n . … …
    §19.16(iii) Various Cases of R a ( 𝐛 ; 𝐳 )
    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 a z ) ( 1 b z ) R n 1 ( z ; q a , q b , 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. …