formally self adjoint linear operator
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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions
… ►§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators
… ► … ► … ► …2: 1.3 Determinants, Linear Operators, and Spectral Expansions
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Formal Calculation of Determinants
… ►Self-Adjoint Operators on
… ►Real symmetric () and Hermitian () matrices are self-adjoint operators on . The spectrum of such self-adjoint operators consists of their eigenvalues, , and all . … ►For self-adjoint and , if , see (1.2.66), simultaneous eigenvectors of and always exist. …3: 25.17 Physical Applications
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►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.
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►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).
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4: 12.15 Generalized Parabolic Cylinder Functions
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►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function.
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5: 18.39 Applications in the Physical Sciences
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►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.
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►If is an arbitrary unit normalized function in the domain of then, by self-adjointness,
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►noting that the 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).
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►The radial operator (18.39.28)
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►With the functions normalized as with measure are, formally,
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6: Bibliography R
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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials.
Canad. J. Math. 33 (4), pp. 915–928.
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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness.
Academic Press, New York.
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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators.
Academic Press, New York.
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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters.
Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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7: 18.36 Miscellaneous Polynomials
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►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the polynomials, self-adjointness implying both orthogonality and completeness.
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►Completeness follows from the self-adjointness of , Everitt (2008).
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►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).
8: 1.1 Special Notation
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real variables. | |
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adjoint of the square matrix | |
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linear operator defined on a manifold | |
adjoint of defined on the dual manifold |
9: 2.9 Difference Equations
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►or equivalently the second-order homogeneous linear difference equation
…in which is the forward difference operator (§3.6(i)).
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►Formal solutions are
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, and higher coefficients are determined by formal substitution.
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►The coefficients and constant are again determined by formal substitution, beginning with when , or with when .
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10: 18.38 Mathematical Applications
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►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.
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►A further operator, the so-called Casimir operator
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