Erratum: Use of Green Functions in Atomic and Molecular Calculations. II. The Green Function of the Helium Atom

The question of non-uniqueness in boundary integral equation formulations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of coefficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

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Simplifying the one-loop five graviton amplitude in type IIB string theory

International Journal of Modern Physics A ◽

10.1142/s0217751x17500749 ◽

2017 ◽

Vol 32 (14) ◽

pp. 1750074 ◽

Cited By ~ 10

Author(s):

Anirban Basu

Keyword(s):

Green Function ◽

String Theory ◽

Fundamental Domain ◽

Green Functions ◽

Considerable Simplification ◽

The Green Function ◽

Type Iib String Theory ◽

The One

We consider the [Formula: see text] and [Formula: see text] terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of [Formula: see text]. Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the [Formula: see text] and [Formula: see text] terms where each of these two links are given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.

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Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0255 ◽

2016 ◽

Vol 472 (2191) ◽

pp. 20160255 ◽

Cited By ~ 9

Author(s):

Oscar P. Bruno ◽

Stephen P. Shipman ◽

Catalin Turc ◽

Stephanos Venakides

Keyword(s):

Green Function ◽

Dimensional Space ◽

Three Dimensional ◽

Integral Equation Method ◽

Boundary Integral ◽

Green Functions ◽

Lattice Sums ◽

Periodic Arrays ◽

The Green Function ◽

Three Dimensional Space

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain ‘Wood frequencies’ at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function—that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

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Population Field Theory with Applications to Tag Analysis and Fishery Modeling: the Empirical Green Function

Canadian Journal of Fisheries and Aquatic Sciences ◽

10.1139/f93-274 ◽

1993 ◽

Vol 50 (11) ◽

pp. 2491-2512 ◽

Cited By ~ 1

Author(s):

Carlos A. M. Salvadó

Keyword(s):

Green Function ◽

Transition Probability ◽

Space Time ◽

Point Sources ◽

Closed Form Expression ◽

Model Parameters ◽

Green Functions ◽

Field Equations ◽

Empirical Green Function ◽

The Green Function

A theoretical framework is proposed for analyzing fish movement and modeling the associated dynamics using tagging data. When tagged fish are released in an area small compared with the domain of the fish population and over a period short compared with the time they take to disperse throughout their domain, the pattern of movement approximates a point-source solution of the underlying population dynamics. A method of point sources (Green functions) is invoked for representing the solution of the tagged and untagged fish field equations (partial differential equations) in terms of integral equations. As an approximate representation of a tagging experiment, the Green function is interpreted as the probability density of survival and movement from point to point in space–time. The Green functions are constructed empirically using one parameter, catchability, as the ratio of population density of tagged fish divided by the number of tagged fish released. The number of tagging experiments necessary to characterize the population is dictated by the dependence of catchability on space–time. The moments of the Green function are used to calculate model parameters and lead to the identification of a closed form expression for the transition probability densities of the model assumed.

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Spectra of Collective Excitations and Low-Frequency Asymptotics of Green’s Functions in Uniaxial and Biaxial Ferrimagnetics

East European Journal of Physics ◽

10.26565/2312-4334-2019-1-04 ◽

2019 ◽

Keyword(s):

Asymptotic Behavior ◽

Green Function ◽

Wave Vector ◽

Easy Axis ◽

Green’S Functions ◽

Low Frequency ◽

Easy Plane ◽

Collective Excitations ◽

Green Functions ◽

The Green Function

The paper studies the dynamic description of uniaxial and biaxial ferrimagnetics with spin s=1/2 in alternative external field. The nonlinear dynamic equations with sources are obtained, on basis on which low-frequency asymptotics of two-time Green functions in the uniaxial and biaxial cases of the ferrimagnet are obtained. Energy models are constructed that are specific functions of Casimir invariants of the algebra of Poisson brackets for magnetic degrees of freedom. On their basis, the question of the stable magnetic states has been solved for the considered systems. These equations were linearized, an explicit form of the collective excitations spectra was found, and their character was analyzed. The article studies the uniaxial case of a ferrimagnet, as well as biaxial cases of an antiferromagnet, easy-axis and easy-plane ferrimagnets. It is shown that for a uniaxial antiferromagnet the spectrum of magnetic excitations has a Goldstone character. For biaxial ferrimagnetic materials, it was found that the spectrum has either a quadratic character or a more complex dependence on the wave vector. It is shown that in the uniaxial case of an antiferromagnet the Green function of the type Gsα,sβ(k,0), Gsα,nβ(k,0) and Gsα,sβ(0,ω) have regular asymptotic behavior, and the Green function of type Gnα,nβ(k,0)≈1/k2 and Gsα,nβ(0,ω)≈1/ω, Gnα,nβ(0,ω)≈1/ω2 have a pole feature in the wave vector and frequency. Biaxial ferrimagnetic states have another type of the features of low-frequency asymptotics of the Green's functions. In the case of a ferrimagnet, the “easy-axis” of the asymptotic behavior of the Green functions Gsα,sβ(0,ω), Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,sβ(k,0), Gsα,nβ(k,0), Gnα,nβ(k,0) have a pole character. For the case of the “easy-plane” type ferrimagnet, the asymptotics of the Green functions Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,nβ(k,0), Gnα,nβ(k,0), have a pole character, and the Green function Gsα,sβ(k,ω) contains both the pole component and the regular part. A comparative analysis of the low-frequency asymptotics of Green functions shows that the nature of magnetic anisotropy significantly effects the structure of low-frequency asymptotics for uniaxial and biaxial cases of ferrimagnet. Separately, we note the non-Bogolyubov character of the Green function asymptotics for ferrimagnet with biaxial anisotropy Gnα,nβ(k,0)≈1/k4.

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