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 $\zeta \left(s\right)$ 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: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. …The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … ► …

3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions … ► … ►These eigenvalues will be assumed distinct, i. … ►By Bessel’s differential equation in the form (10.13.1) we have the functions $\sqrt{x}{J}_{\nu }\left(x\sqrt{\lambda }\right)$ ($\lambda \ge 0$, for $J_{\nu }$ see §10.2(ii)) as eigenfunctions with eigenvalue $\lambda$ of the self-adjoint extension of the differential operator … ►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. …

4: 29.21 Tables

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Ince (1940a) tabulates the eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^{m+1}\left(k^2\right)$ (with $a_{\nu }^{2m+1}$ and $b_{\nu }^{2m+1}$ interchanged) for $k^2=0.1,0.5,0.9$, $\nu =-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. 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 $𝐌$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …

6: 29.9 Stability

… ►If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.

7: 28.34 Methods of Computation

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§28.34(ii) Eigenvalues

►Methods for computing the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$, defined in §§28.2(v) and 28.12(i), include: … ►

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

… ►Also, once the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$ have been computed the following methods are applicable: …

8: 28.13 Graphics

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§28.13(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$ for General $\nu$

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Figure 28.13.1: ${\lambda }_{\nu }\left(q\right)$ as a function of $q$ for $\nu =0.5(1)3.5$ and $a_n\left(q\right),b_n\left(q\right)$ for $n=0,1,2,3,4$ ($a$’s), $n=1,2,3,4$ ($b$’s). … Magnify

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Figure 28.13.2: ${\lambda }_{\nu }\left(q\right)$ for , $0\le q\le 10$. Magnify 3D Help

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9: 28.6 Expansions for Small $q$

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§28.6(i) Eigenvalues

►Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are: … ►The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … ►Higher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … ►Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. …

10: 18.38 Mathematical Applications

… ►A further operator, the so-called Casimir operator … ►

Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ►In the one-variable case the Dunkl operator eigenvalue equation … …