DVR (discrete variable representations)

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1: 18.38 Mathematical Applications

… ►See also the paragraph on DVRs, below. … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►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. The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate $x$, is replaced by its values at a finite set of zeros of appropriate OP’s resulting in expansions using functions localized at these points. …Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. …

2: 18.39 Applications in the Physical Sciences

… ►The spectrum is entirely discrete as in §1.18(v). … ►As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role. … ►

§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. … ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate, $x$. …

3: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

4: 18.27 $q$-Hahn Class

… ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. … ►

§18.27(vii) Discrete $q$-Hermite I and II Polynomials

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Discrete $q$-Hermite I

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Discrete $q$-Hermite II

… ►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

5: 20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). … ►However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). …

6: Alexander I. Bobenko

… ► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R.  Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P. …

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►These sets may be discrete, continuous, or a combination of both, as discussed in the following three subsections. … ►

§1.18(v) Point Spectra and Eigenfunction Expansions

… ► … ► …

8: Bibliography Y

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

Encodings:

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9: 18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

… ►The Askey scheme, depicted in Figure 18.21.1, gives a graphical representation of these limits. ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). … ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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10: Bibliography F

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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External Links:

ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

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Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

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External Links:

ISSN 1058-6458, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.16.13), §25.16(ii), §25.16(ii), §25.16.

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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External Links:

ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

Encodings:

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

Encodings:

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