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essentially self-adjoint operator


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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Self-Adjoint and Symmetric Operators
Formally Self-Adjoint and Self-Adjoint Differential OperatorsSelf-Adjoint Extensions
Self-Adjoint Operators
Self-adjoint extensions of a symmetric Operator
2: 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).
3: 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. …
4: 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 Ψ ( x , t = 0 ) = χ ( x ) is an arbitrary unit normalized function in the domain of then, by self-adjointness, … 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). … The radial operator (18.39.28) … The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as x 1 . …
5: 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. …
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: 10.22 Integrals
    These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix). …
    8: Guide to Searching the DLMF
    Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: …
  • Boolean operator:

    and and or

  • proximity operator:

    adj, prec/n, and near/n, where n is any positive natural number.

  • $ stands for any number of alphanumeric characters
    (the more conventional * is reserved for the multiplication operator)
    DLMF search recognizes just the essential font differences, that is, the font style differences deemed important for the DLMF contents: …
    9: Preface
     Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST. …
    10: 14.32 Methods of Computation
    Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. …