Classical Green Function Research Articles

Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional.

Uniform boundary Harnack principle and generalized triangle property

Let D be a bounded open subset in R d , d ⩾ 2 , and let G denote the Green function for D with respect to ( - Δ ) α / 2 , 0 < α ⩽ 2 , α < d . If α < 2 , assume that D satisfies the interior corkscrew condition; if α = 2 , i.e., if G is the classical Green function on D, assume—more restrictively—that D is a uniform domain. Let g = G ( · , y 0 ) ∧ 1 for some y 0 ∈ D . Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that G ( z , y ) / g ( z ) ⩽ C G ( x , y ) / g ( x ) when d ( z , x ) ⩽ d ( z , y ) . An intermediate step is the approximation G ( x , y ) ≈ | x - y | α - d g ( x ) g ( y ) / g ( A ) 2 , where A is an arbitrary point in a certain set B ( x , y ) . This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D × D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.

Coherent fluorescence resonance energy transfer between two dipoles: full quantum electrodynamics approach

The quantum electrodynamical theory of coherent fluorescence resonance energy transfer (FRET) between two dipoles (free atoms, molecules or dopant centres in crystalline matrices at low temperature conditions) in an arbitrary non-dissipative environment is elaborated using a canonical quantization scheme. It is shown that population dynamics of both atoms can be expressed in terms of a classical Green function. This general theory corresponds well to the problem of description of controllable quantum manipulations under the donor-acceptor system (e.g. its application for FRET scanning near-field optical microscopy-based quantum computing). In addition, its usefulness is illustrated for the case of coherent FRET in free space and for the case when atoms are placed near an ideally conducting plane. The possibility of creating an entangled state of three atoms is demonstrated.

Dyadic Green's functions of a laminar plate.

We introduce the concept of dyadic Green's functions of a laminar plate. These functions generalize classical Green's functions. In addition to relating displacements and stresses at the surface of a medium, they relate these quantities at both the top and the bottom surfaces of a medium of finite thickness and infinite extent in the transverse directions. We describe here the calculation of these functions in the spectral domain and provide some academic examples demonstrating their interest.

Coherent fluorescence resonance energy transfer between two dipoles: Full quantum electrodynamics approach

The quantum electrodynamical theory of coherent fluorescence resonance energy transfer (FRET) between two dipoles (free atoms, molecules or dopant centres in crystalline matrices at low temperature conditions) in an arbitrary non-dissipative environment is elaborated using a canonical quantization scheme. It is shown that population dynamics of both atoms can be expressed in terms of a classical Green function. This general theory corresponds well to the problem of description of controllable quantum manipulations under the donor-acceptor system (e.g. its application for FRET scanning near-field optical microscopy-based quantum computing). In addition, its usefulness is illustrated for the case of coherent FRET in free space and for the case when atoms are placed near an ideally conducting plane. The possibility of creating an entangled state of three atoms is demonstrated.

Comparison of the pluricomplex and the classical Green functions on convex domains of finite type

Comparison of the pluricomplex and the classical Green functions on convex domains of finite type

Planck's over 2 pi corrections in semiclassical formulas for smooth chaotic dynamics.

The validity of semiclassical expansions in the power of Planck's over 2 pi for the quantum Green's function have been extensively tested for billiards systems, but in the case of chaotic dynamics with smooth potential, even if formulas are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al. [Adv. Chem. Phys. 90, 105 (1995)], based on the classical Green's functions, we present an efficient method allowing the calculation of Planck's over 2 pi corrections for the propagator, the quantum Green's function, and their traces. In particular, we show that the previously published expressions for Planck's over 2 pi corrections to the traces are incomplete.

Mode expansion and photon operators in dispersive and absorbing dielectrics

We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.

Recombination yield of geminate radical pairs in high magnetic fields: general results and application to free diffusion

We have derived a general and exact expression for the quantum yield of geminate radical pairs (RPs) in high magnetic fields. We consider the dynamical S–T 0 mixing caused by differences in Zeeman and hyperfine interactions, intraradical ( T 1 and T 2) relaxation, and a first-order scavenging process. The recombination process and the exchange interaction are assumed to be isotropic and of the contact type. In order to solve the stochastic Liouville equation, we introduce effective methods to minimize the dimensionality of the problem and to calculate the matrix elements of the superoperators in an arbitrary coupled basis. Our general expression is expressed in terms of the classical Green's function for the relative motion of the radicals, and it is therefore valid for any kind of relative motion. Various limiting cases of the general expression, which are of particular interest, are derived. Explicit analytical formulas for freely diffusing RPs are given, and the high accuracy of the expressions is illustrated by comparison with accurate numerical results.

Quantum analysis and the classical analysis of spontaneous emission in a microcavity

Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.

Development of one-energy group, two-dimensional, frequency dependent detector adjoint function based on the nodal method

One-energy group, two-dimensional computer code was developed to calculate the response of a detector to a vibrating absorber in a reactor core. A concept of local/global components, based on the frequency dependent detector adjoint function, and a nodalization technique were utilized. The frequency dependent detector adjoint functions presented by complex equations were expanded into real and imaginary parts. In the nodalization technique, the flux is expanded into polynomials about the center point of each node. The phase angle and the magnitude of the one-energy group detector adjoint function were calculated for a detector located in the center of a 200×200 cm reactor using a two-dimensional nodalization technique, the computer code EXTERMINATOR, and the analytical solution. The purpose of this research was to investigate the applicability of a polynomial nodal model technique to the calculations of the real and the imaginary parts of the detector adjoint function for one-energy group two-dimensional polynomial nodal model technique. From the results as discussed earlier, it is concluded that the nodal model technique can be used to calculate the detector adjoint function and the phase angle. Using the computer code developed for nodal model technique, the magnitude of one energy group frequency dependent detector adjoint function and the phase angle were calculated for the detector located in the center of a 200×200 cm homogenous reactor. The real part of the detector adjoint function was compared with the results obtained from the EXTERMINATOR computer code as well as the analytical solution based on a double sine series expansion using the classical Green's Function solution. The values were found to be less than 1% greater at 20 cm away from the source region and about 3% greater closer to the source compared to the values obtained from the analytical solution and the EXTERMINATOR code. The currents at the node interface matched within 1% of the average value at the interface. The phase angle varied from 0.1° to 0.4° compared to 0.2° calculated using the point reactor zero power transfer function.

Exact Solutions of Covariant Wave Equations with a Multipole Source Term on Curved Spacetimes

A formalism is presented for calculating exactsolutions of covariant inhomogeneous scalar and tensorwave equations whose source terms are arbitrary ordermultipoles on a curved background spacetime. The developed formalism is based on the theory ofthe higher-order fundamental solutions for wave equationwhich are the distributions that satisfy theinhomogeneous wave equation with the corresponding order covariant derivatives of the Dirac deltafunction on the right-hand side. Like the classicalGreen's function for a scalar wave equation, thehigher-order fundamental solutions contain a direct termwhich has support on the light cone as well as a tailterm which has support inside the light cone. Knowinghow to compute the fundamental solutions of arbitraryorder, one can find exact multipole solutions of wave equations on curved spacetimes. Wepresent complete recurrent algorithms for calculatingthe arbitrary-order fundamental solutions and the exactmultipole solutions in a form convenient for practical computations. As an example we apply thealgorithm to a massless scalar wave field on aparticular Robertson-Walker spacetime.

Complete algorithms for calculating the higher-order fundamental solutions of wave equations

Complete recurrent algorithms for calculating the higher-order fundamental solutions of covariant linear wave equations for scalar and tensor wave fields on an arbitrary curved spacetime are derived. The higher-order fundamental solutions are the distributions that satisfy the wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. Like the classical Green's function for a scalar wave equation, the higher-order fundamental solutions contain the terms which have support on, and only on, the lightcone as well as tail terms which have support inside the lightcone. With the help of the higher-order fundamental solutions found it is possible to compute the exact multipole solutions of wave equations in a form convenient for practical computations. As applications we consider the exact field of a dipole source of variable strength travelling in an arbitrary curved spacetime and the tail term of scalar multipole waves in the Friedman dust-dominated universe for the case of minimal coupling.

Circular means of fine green's functions and the longest arc relation

An inquality of the first author involving circular means of classical Green's functions is extended to fine Green's functions on fine domains. The inequality leads to a more natural proof of the longest arc relation recently proved by the authors.

Comparison of the pluricomplex and the classical Green functions.

Comparison of the pluricomplex and the classical Green functions.

A relation between the pluricomplex and the classical Green functions in the unit ball of 𝐂ⁿ

We give a sharp upper bound for the quotient of the pluricomplex and the classical Green functions in the unit ball ofCn\mathbf {C}^n.

Recipes for the classical kernel functions associated to a multiply connected domain in the plane

I have recently shown that the Bergman kernel associated to a finitely connected domain in the plane is given as an explicit rational combination of finitely many basic functions of one complex variable. In this paper, it is proved that all the basic functions and constants in the new formula for the Bergman kernel can be evaluated using onedimensional integrals and simple linear algebra. In fact, all integrals used in the computations are line integrals over boundary curves; at no point is an integral with respect to area measure required. From a theoretical perspective, these results lead to an understanding of the complexity of the Bergman kernel. From a practical point of view, they give an efficient method to numerically compute the Bergman kernel. Similar results are also proved for the Szego kernel function, the Poisson kernel, and the classical Green's function.

On the time dependence of the wave resistance of a body accelerating from rest

We consider the large-time behaviour of the disturbances associated with an initial acceleration of a body in or near a free surface. As a canonical problem, we study the case of a body starting impulsively from rest to a constant velocity U. For a constant-strength translating point source (or dipole) started impulsively, it is known that the unsteady part of the Green function oscillates at the critical frequency ωc=g/4U (g is the gravitational acceleration) with an amplitude that decays with time, t, as t−1/2 and t−1 for t [Gt ] 1 in two (Havelock 1949) and three (Wehausen 1964) dimensions respectively. These classical results turn out to be non-realistic in that for an actual body, the associated source strengths are in general time dependent and a priori unknown, and must together satisfy the kinematic condition on the body boundary. We consider such an initial-boundary-value problem using a transient wave-source distribution on the body surface. Through asymptotic analyses, the unsteady behaviour of the solution at large time is obtained explicitly. Specifically, we show that for a general class of bodies satisfying a simple geometric condition (Γ ≠ 0), the decay rate of the transient oscillations (at frequency ωc) in the wave resistance and velocity potential is an order of magnitude faster: as t−3/2 and t−2, as t → ∞, in two and three dimensions respectively. For body geometries satisfying T = 0, for which the single source is a special case, the classical Green function results are recovered. These results are confirmed by an analysis in the frequency domain and substantiated by direct time-domain numerical simulations.

Third-harmonic wave diffraction by a vertical cylinder

The diffraction of regular waves by a vertical circular cylinder in finite depth water is considered, within the frame of potential theory. The wave slope kA is assumed to be small so that successive boundary value problems at orders kA, k2A2, and k3A3 can be formulated. Here we focus on the third-order (k3A3) problem but restrict ourselves to the triple-frequency component of the diffraction potential. The method of resolution is based on eigenfunction expansions and on the integral equation technique with the classical Green function expressed in cylindrical coordinates. Third-order (triple-frequency) loads are calculated and compared with experimental measurements and approximate methods based on long-wave theories.

Complexity of the classical kernel functions of potential theory

We show that the Bergman, Szego, and Poisson kernels associated to a finitely connected domain in the plane are all composed of finitely many easily computed functions of one variable. The new formulas give rise to new methods for computing the Bergman and Szeg\H o kernels in which all integrals used in the computations are line integrals; at no point is an integral with respect to area measure required. The results mentioned so far can be interpreted as saying that the kernel functions are simpler than one might expect. However, we also prove that the kernels cannot be too simple by showing that the only finitely connected domains in the plane whose Bergman or Szeg\H o kernels are rational functions are the obvious ones. This leads to a proof that the classical Green's function associated to a finitely connected domain in the plane is the logarithm of a rational function if and only if the domain is simply connected and rationally equivalent to the unit disc.