as eigenvalues of q-difference operator
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1: 25.17 Physical Applications
§25.17 Physical Applications
►Analogies exist between the distribution of the zeros of on the critical line and of semiclassical quantum eigenvalues. 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. …2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►An essential feature of such symmetric operators is that their eigenvalues
are real, and eigenfunctions
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►These eigenvalues will be assumed distinct, i.
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►By Bessel’s differential equation in the form (10.13.1) we have the functions (, for see §10.2(ii)) as eigenfunctions with eigenvalue
of the self-adjoint extension of the differential operator
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►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal.
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3: 28.7 Analytic Continuation of Eigenvalues
§28.7 Analytic Continuation of Eigenvalues
►As functions of , and can be continued analytically in the complex -plane. The only singularities are algebraic branch points, with and finite at these points. …The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … ► …4: 29.21 Tables
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Ince (1940a) tabulates the eigenvalues , (with and interchanged) for , , and . Precision is 4D.
Arscott and Khabaza (1962) tabulates the coefficients of the polynomials in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues for , . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.
5: 29.20 Methods of Computation
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§29.20(i) Lamé Functions
… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►§29.20(ii) Lamé Polynomials
►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …6: 29.9 Stability
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►If is not an integer, then (29.2.1) is unstable iff or lies in one of the closed intervals with endpoints and , .
If is a nonnegative integer, then (29.2.1) is unstable iff or for some .
7: 28.34 Methods of Computation
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►Also, once the eigenvalues
, , and have been computed the following methods are applicable:
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§28.34(ii) Eigenvalues
►Methods for computing the eigenvalues , , and , defined in §§28.2(v) and 28.12(i), include: … ►8: 28.13 Graphics
9: 29.4 Graphics
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§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs
► ► … ►§29.4(ii) Eigenvalues of Lamé’s Equation: Surfaces
… ► …10: 28.6 Expansions for Small
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