External Ellipsoidal Harmonics for the Dunkl-Laplacian

Volkmer, Hans

doi:10.3842/sigma.2008.091

Cited by 4 publications

(3 citation statements)

References 19 publications

(24 reference statements)

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“…Recently, equation (1.3) has also found other applications in studies as diverse as the construction of ellipsoidal and sphero-conal h-harmonics of the DunklLaplacian [98], [99], the quantum asymmetric top [1,17,37], or certain quantum completely integrable system called the generalized Gaudin spin chains [40], and their thermodynamic limits.…”

Section: Generalized Lamé Equationmentioning

confidence: 99%

Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

Martínez–Finkelshtein

Рахманов

2011

Commun. Math. Phys.

104

186

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We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials -polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest.The problem has a rich variety of connections with other fields of analysis, some of them are briefly mentioned in the paper.

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Section: Generalized Lamé Equationmentioning

confidence: 99%

Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

Martínez–Finkelshtein

Рахманов

2011

Commun. Math. Phys.

104

186

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“…To our knowledge, up to now, D-harmonic functions have been studied only for [12]) or for Ω an ellipsoidal domain centered at the origin (see [20] and [21]). In [10], Trimèche and Mejjaoli proved that a function u ∈ C ∞ (R d ) is D-harmonic on R d if and only if u satisfies the following generalized spherical mean value property:…”

Section: Introductionmentioning

confidence: 99%

A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications

Gallardo

Rejeb

2015

Trans. Amer. Math. Soc.

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For a root system in R d furnished with its Coxeter-Weyl group W and a multiplicity nonnegative function k, we consider the associated commuting system of Dunkl operators D 1 , . . . , D d and the Dunkl-Laplacian Δ k = D 2 1 +. . .+D 2 d . This paper studies the properties of the functions u defined on an open W -invariant set Ω ⊂ R d and satisfying Δ k u = 0 on Ω (D-harmonicity).In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack's type theorem and a Bôcher's theorem for D-harmonic functions.

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“…Perhaps the biggest problem with the use of ellipsoidal harmonics [23][24][25][26][27][28][29][30][31][32][33][34] is their inability to reduce them, uniquely, to the corresponding harmonics for special, more symmetric, geometries. Of course, since the dimension of the subspace of harmonic polynomials of degree n is equal to 2n + 1, it follows that any such harmonic is representable as a linear combination of eigensolutions in any other of the degenerate coordinate systems.…”

mentioning

confidence: 99%

On the reducibility of the ellipsoidal system

Fragoyiannis

Vafeas

Dassios

2022

Math Methods in App Sciences

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Lamé introduced a triaxial ellipsoidal system, and via some ingenious arguments, he managed to spectrally decompose the Laplace operator and define ellipsoidal harmonic functions. Since then, many authors modified the Lamé system and proposed some related ellipsoidal coordinate variables. Perhaps the most important of them is the system introduced by Jacobi, in connection with the geodesic curves on an ellipsoid. All other attempts to introduce ellipsoidal coordinates are basically modifications of the Jacobi or the Lamé system. However, an important question in the theory of ellipsoidal harmonics is the way in which this theory reduces to the spheroidal and spherical systems. This is by no means a straightforward procedure, since the relative limiting cases lead to generic underdetermined forms. The basic difficulty is due to the different dimensionality of the singularity regions corresponding to each system. Note that this region has zero dimensions in the case of the sphere, one dimension in the case of the prolate spheroid, and two dimensions in the case of the oblate spheroid and the ellipsoid. The present work provides a systematic way to obtain these reductions and to establish a correspondence between ellipsoidal, spheroidal, and spherical harmonics. We have also demonstrated how our approach applies to the proposed modified Jacobi ellipsoidal system when we connect the Lamé with the Jacobi system through certain non-degenerable relations. The importance of these geometrical degeneracies lies in the fact that the solution of any boundary value problem in ellipsoidal geometry provides immediately the solutions in all special geometries of prolate and oblate spheroids, discs, needles, or spheres.

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External Ellipsoidal Harmonics for the Dunkl-Laplacian

Cited by 4 publications

References 19 publications

Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications

On the reducibility of the ellipsoidal system

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