Some properties of hyperspherical harmonics

Wen, Zhenhai; Avery, John

doi:10.1063/1.526621

Cited by 64 publications

(70 citation statements)

References 26 publications

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“…This equation can be treated perturbatively for small following the procedure in Sec. IV and by introducing the hyperspherical harmonics Y m l ( θ), which obey the eigenvalue equation [28] …”

Section: Intrinsically Spherical Growth and Rapid Rougheningmentioning

confidence: 99%

Statistics of interfacial fluctuations of radially growing clusters

Escudero

2011

Phys. Rev. E

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The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

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Section: Intrinsically Spherical Growth and Rapid Rougheningmentioning

confidence: 99%

Statistics of interfacial fluctuations of radially growing clusters

Escudero

2011

Phys. Rev. E

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“…However, the projection method allows one to obtain analytic results only for a limited set of functions having simple functional forms so that necessary integrals can be calculated analytically. We present here an alternative, differential approach, which generalizes earlier results by Sack [13] (who dealt with multipole expansions for the special case of two-center functions) as well as by Avery [14] and Wen and Avery [15] (who dealt with bicenter expansions using the vector differentiation technique in three dimensions [14] and m dimensions [15] for the special case of scalar functions). Our general results for the multipole expansion of T jm (a 1 , a 2 , .…”

Section: Introductionmentioning

confidence: 93%

“…Taking into account (14), the general equation (8) for the function (12) may be written (after some rearrangements of summation indices) as (15) where the differential operator  is defined by (16) Taking into account the known identity we obtain the multipole expansion for the function (12) in the following final form: (17) where the coefficients B are defined by (18) Note that the  -operator (16) commutes with the Ñ 2 -operator because  originates from Ñ-operators (see (8) and (A.2)). Thus, we can write the operator  ( j ) ll′ (r 1 ) in (18) before the sum over the index k.…”

Section: Multipole Expansion Of Two-center Functions Of the Form F (Rmentioning

confidence: 99%

Multipole expansions of irreducible tensor sets and some applications

Manakov

Meremianin

Starace

2001

J. Phys. B: At. Mol. Opt. Phys.

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A formal multipole expansion of tensor functions dependent on several vector parameters is obtained in an invariant differential form. We apply this result to derive multipole expansions of finite rotation matrices in terms of finite sums of bipolar harmonics. The multipole expansion in terms of bipolar harmonics of unit vectors r 1 and r 2 are analyzed for functions of the type f (r)Y lm (r) (with r = r 1 − r 2 ), which are important in two-center problems. As another example of the application of the multipole expansion technique, reduction formulae are obtained for tensor constructions which appear in analyses of angular distributions in photoprocesses which take exact account of non-dipole (or retardation) effects. As a result, the "photon factors" in the angular distribution for an arbitrary photoprocess are expressed in an invariant form involving only the photon polarization vector and spherical harmonics of the photon wavevector.

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“…Note that those were not conventional harmonics as defined in [8,12] but the set of harmonics labeled by the individual angular momenta quantum numbers of particles [14]. The expression for CMC was then extracted from the series by using some specific gauge convention (eqs.…”

Section: Discussionmentioning

confidence: 99%

“…In order to calculate this integral we note that the Gegenbauer polynomial in (47) is proportional to the scalar product of six-dimensional hyperspherical harmonics [8,12],…”

Section: The Hyperspherical Expansion Of the Multipole Coeffi-cientsmentioning

confidence: 99%

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

Meremianin

2008

Few-Body Syst

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The perturbation theory with respect to the potential energy of three particles is considered.The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner D-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the S-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed. * Electronic address: meremianin@phys.vsu.ru

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Some properties of hyperspherical harmonics

Cited by 64 publications

References 26 publications

Statistics of interfacial fluctuations of radially growing clusters

Statistics of interfacial fluctuations of radially growing clusters

Multipole expansions of irreducible tensor sets and some applications

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

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