Expansions Of Green Functions Research Articles

Harmonic extension through cylindrical and conical surfaces

This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.

Toward an efficient second‐order method for computing the surface gravitational potential on spherical‐polar meshes

AbstractAstrophysical accretion discs that carry a significant mass compared with their central object are subject to the effect of self‐gravity. In the context of circumstellar discs, this can, for instance, cause fragmentation of the disc gas, and—under suitable conditions—lead to the direct formation of gas‐giant planets. If one wants to study these phenomena, the disc's gravitational potential needs to be obtained by solving the Poisson equation. This requires to specify suitable boundary conditions. In the case of a spherical‐polar computational mesh, a standard multipole expansion for obtaining boundary values is not practicable. We hence compare two alternative methods for overcoming this limitation. The first method is based on a known Green's function expansion (termed “CCGF”) of the potential, while the second (termed “James' method”) uses a surface screening mass approach with a suitable discrete Green's function. We demonstrate second‐order convergence for both methods and test the weak scaling behavior when using thousands of computational cores. Overall, James' method is found superior owing to its favorable algorithmic complexity of compared with the scaling of the CCGF method.

Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Increasing number of carrier electrons and nuclear charge in Hydrogenic systems associated with bound and ionised (scattered) states invites long-standing divergence issue of Schrödinger's equations due to Coulomb interactions. In this regard, He-isoelectronic series represents the benchmark for atoms and molecules within non-relativistic quantum limit. Schrödinger equations for such complex systems being transformed into either Whittaker-M or Associated Laguerre Polynomial, furnish multipole operators of Green's function expansion of Coulomb interaction a robust framework of lower and upper incomplete Gamma functions. The resulting monopole integral gives terminating, finitely summed, simple and analytical forms of Lauricella functions. This novel form succinctly remedies the paradox of Energy Contribution or Correction occurring due to divergent length scales among Hydrogenic bound states. Both singly and doubly excited spherically symmetric states of two-electron systems ( ) are examined to register full-scale third-order perturbation for monopole factor within 0.2–8.2% deviation of ground-state energies from experimental values. The deviation urges for further analytical treatment to dipole factor. Moreover, exchange of electronic coordinates by virtue of exchanging quantum numbers exhibits invariant integrals of multipoles which confirms reciprocity condition of Green's function. This methodology has achieved a new paradigm of employing generic H-like orbitals for electronic structure calculations.

Multi-electron anisotropic quantum dots/TMDCs/CNT families under magnetic field: analytical treatment to first Brillouin zone by Fermi liquid model

Acute Coulomb interaction of the two-dimensional systems has drawn special attention due to its unusual logarithmic Green function expansion. As the number of electrons (N) increases, Pauli Exclusion principle emerges inevitably with rapidly growing electronic correlations. Quantum dot, Transition metal dichalcogenides (TMDC) and Carbon nanotube (CNT) families of 2-D anisotropic mesoscopic systems are rich habitats of electrons. Schrödinger equations of such electrons in electrical confinement and transverse magnetic field can be recast in self-adjoint Whittaker-M functions facilitating each Coulomb interaction to terminable, exact and single summed Lauricella function via Chu-Vandermonde identity. For , multipoles of Green function expansion also succumb to terminable, single-summed, analytical integrals by inserting discretised closure relations. Thus, multi-configuration Slater determinantal states are employed for strong correlation of Fermi liquid model of first Brillouin zone (FBZ) within giga-units of reciprocal lattices (mesoscopic scale). Chemical potential, addition energies of WS, GaAs and model systems of dielectric constant have set benchmark at low and high confinement potentials, as a function of magnetic field and density of electrons. Because of sharp falls in surface integrals of both Newman and Dirichlet forms of Green function, Coulomb interaction takes to (or leads to) multipole expansion of generic coordinates. Formation of composite-fermions may be anticipated. At the most, octupole is sufficient for the convergence.

Analytical treatment to helium isoelectronic ions via Green function expansion of Coulomb interactions: perturbation calculations of ground-state energies with Hydrogenic orbitals

Analytical treatment to helium isoelectronic ions via Green function expansion of Coulomb interactions: perturbation calculations of ground-state energies with Hydrogenic orbitals

Contribution of quantum chromodynamics condensates to the operator product expansion of Green functions of chiral currents

In this paper, the basic properties of quantum chromodynamics (QCD) condensates are presented, together with theirs relation to the operator product expansion (OPE) and the two‐point and three‐point Green functions constructed of chiral currents. Next, our newest results for the contribution of the QCD condensates with dimension D < 6 to the Green functions calculated within the framework of χPT/RχT, i.e. chiral perturbation theory or resonance chiral theory, are discussed. This matching of the OPE and such effective theories can lead to some constraints on the coupling constants, thus allowing us to obtain some unknown parameters of the chiral/resonance Lagrangian.

Nanoscale plasmonic slot waveguides for enhanced Raman spectroscopy

We theoretically investigate several types of plasmonic slot waveguides for enhancing the measured signal in Raman spectroscopy, which is a consequence of electric field and Purcell factor enhancements, as well as an increase in light-matter interaction volume and the Raman signal collection efficiency. An intuitive methodology is presented for calculating the accumulated Raman enhancement factor of an ensemble of molecules in waveguide sensing, which exploits an analytical photon Green function expansion in terms of the waveguide normal modes, and we combine this with a quantum optics formalism of the molecule-waveguide interaction to model Raman scattering. We subsequently show how integrated plasmonic slot waveguides can attain significantly higher Raman enhancement factors: $\sim$5.3$\times$ compared to optofluidic fibers and $\sim$3.7$\times$ compared to planar integrated dielectric waveguides, with a device size and thus analyte volume of at least three-orders of magnitude less. We also provide a comprehensive comparison between the different types of plasmonic slot waveguides based on the important figures-of-merit, and determine the optimal approaches to maximize Raman enhancement.

Quantum Interference, Graphs, Walks, and Polynomials

In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

Self-force and Green function in Schwarzschild spacetime via quasinormal modes and branch cut

The motion of a small compact object in a curved background spacetime deviates from a geodesic due to the action of its own field, giving rise to a self-force. This self-force may be calculated by integrating the Green function for the wave equation over the past worldline of the small object. We compute the self-force in this way for the case of a scalar charge in Schwarzschild spacetime, making use of the semianalytic method of matched Green function expansions. Inside a local neighborhood of the compact object, this method uses the Hadamard form for the Green function in order to render regularization trivial. Outside this local neighborhood, we calculate the Green function using a spectral decomposition into poles (quasinormal modes) and a branch cut integral in the complex frequency plane. We show that both expansions overlap in a sufficiently large matching region for an accurate calculation of the self-force to be possible. The scalar case studied here is a useful and illustrative toy model for the gravitational case, which serves to model astrophysical binary systems in the extreme mass-ratio limit.

Rapidly Convergent Eigenfunction Expansions of Green Functions for a Perfectly Conducting Wedge

Existing Green functions for a conducting wedge in the form of infinite series of orthogonal eigenfunctions are known to suffer from slow and conditional convergence 1) near the source point, due to the singularity of the Green functions at source points, and 2) when the source point is near the surface of the wedge, due to the presence of a nearby image source. As a result, such expansions are unsuited for the exact solution of singular integral equations wherein values of the Green functions at the source point do appear inside the integral. In this paper, alternative expansions are proposed which converge rapidly for any combination of the source and observation points. The main idea is to extract in closed form both the singular term and the nearby image term together with certain simple asymptotic terms. This helps isolate the inherent singularities and, thus, transform the remaining part of the Green function into a rapidly converging series of elementary terms. Numerical examples and case studies illustrate the stability and high accuracy of the new expansions. Application to the solution of an integral equation associated with wave diffraction by perfectly conducting curved strips in the vicinity of a wedge is exemplified.

Nyström Solution to Oblique Scattering of Arbitrarily Polarized Waves by Dielectric-Filled Slotted Cylinders

We investigate oblique diffraction of arbitrarily polarized waves by axially slotted, dielectric-filled, perfectly conducting circular cylinders with infinitesimally thin walls. The relevant 2 × 2 system of coupled singular integral-integrodifferential equations is solved by a highly accurate Nystrom method. For enhanced efficiency, several asymptotic terms are extracted from the kernels. This helps isolate the inherent singularities and, thus, transform the initial slowly converging Green-function expansions into rapidly converging series. As a consequence, all elements of the resulting Nystrom matrix take exact expressions in the form of very rapidly converging series of elementary terms. Numerical examples and case studies are presented that validate the algorithm, illustrate its rapid convergence, and bring to light the effect of changing several physical and geometrical parameters on the reflection, transmission, and absorption characteristics of the structure.

A Lippmann–Schwinger approach for the determination of photoionization and photodetachment cross sections based on a partial wave Green's function expansion and configuration interaction wave functions

A computational approach is presented for determining partial differential cross sections for molecular photodetachment and photoionization. The approach, which extends a recent treatment of negative ion photodetachment, uses multireference configuration interaction wave functions to describe both the target or residual molecule and its interaction with the scattered electron and can treat states of the residual molecule which can be described adiabatically or are non-adiabatically coupled by conical intersections. Antisymmetry is taken into account in an approximate manner and a Lippmann–Schwinger equation is used to treat the interaction of the ionized/detached electron with the residual molecule. A partial wave expansion of the requisite Green's functions enables a unified treatment of photoionization and photodetachment and achieves computational efficiencies. A matrix representation of the quantities appearing in the iterated solution to the Lippmann–Schwinger equation using the radial Green's function permits the iterated solution to be summed eliminating potential convergence issues. Two methods for determining the radial Green's functions are presented. In the standard approach the Green's function is determined using the regular and irregular solutions to the corresponding homogeneous equation. The numerical issues associated with this well-studied form of the Green's function are non-trivial. An alternative approach is introduced which uses the complex scaling technique to construct the relevant matrix elements of the Green's function. The degree of simplification achieved in this way is significant.

Long time asymptotic expansions of heat kernels for perturbed Hill operators

We obtain, by exploiting Jost functions, asymptotic expansions of Green functions of perturbed Hill operators with respect to the spectral parameter around the minima of the continuous spectra. From the Green function expansions, we derive asymptotic expansions for large time of heat kernels for perturbed Hill operators.

Quantum field theory in the presence of a medium: Green's function expansions

Based on a canonical approach and functional-integration techniques, a series expansion of Green's function of a scalar field, in the presence of a medium, is obtained. A series expansion for Lifshitz-energy, in finite-temperature, in terms of the susceptibility of the medium is derived and the whole formalism is generalized to the case of electromagnetic field in the presence of some dielectrics. A covariant formulation of the problem is presented.

The absence of QCD β-function factorization property of the generalized Crewther relation in the ʼt Hooft [formula omitted]-based scheme

We apply the ʼt Hooft MS¯-based scheme to study the scheme-dependence of the QCD generalization of Crewther relation for the product of the normalized non-singlet perturbative contributions to the e+e−-annihilation Adler function and to the Bjorken sum rule of the polarized lepton–nucleon deep-inelastic scattering process. We prove that after the transformations from the pure MS¯-scheme to the ʼt Hooft scheme the characteristic MS¯-scheme theoretical property of this relation, namely the factorization of the β-function in its conformal symmetry breaking part, disappears. Another “non-comfortable” theoretical consequence of the application of this prescription in N=1 SUSY QED model is mentioned. It is shown, that within the ʼt Hooft scheme the expansions of Green functions in terms of the Lambert function is simplified in high orders of perturbation theory. This may be considered as the attractive feature of the ʼt Hooft scheme, which manifest itself in high-order perturbative phenomenological applications.

Resonance theory for discrete models: Methodology and isolated resonances

We here consider open quantum systems defined on discretized space, motivated by experimental and theoretical interest in the electronic conduction through nanoscale devices such as molecular junctions and quantum dots. We particularly focus on effects of resonances on the conductance through the systems. We develop a method of calculating the conductance with the use of Green's function expansion with respect to the eigenstates of the effective Hamiltonian for the open quantum systems. Unlike previous methodologies where one can treat only narrow resonances far from the band edges in a satisfactory manner with a Lorentzian profile, our method provides a novel resonance profile which can be used to describe any isolated resonance in the spectrum even close to the band edges.

High-order time expansion path integral ground state

The feasibility of path integral Monte Carlo ground state calculations with very few beads using a high-order short-time Green's function expansion is discussed. An explicit expression of the evolution operator which provides dramatic enhancements in the quality of ground-state wave functions is examined. The efficiency of the method makes possible to remove the trial wave function and thus obtain completely model-independent results still with a very small number of beads. If a single iteration of the method is used to improve a given model wave function, the result is invariably a shadow-type wave function, whose precise content is provided by the high-order algorithm employed.

Green's function expansion for exponentially graded elasticity

AbstractNew computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

Table of Feynman diagrams of the interacting Fermion Green's function

AbstractThe Feynman diagrams of the Green's function expansion of fermions interacting with a nonrelativistic 2‐body interaction are displayed in first, second, and third perturbative order of the interaction as 2, 10, and 74 diagrams, respectively. A name convention for the diagrams is proposed and then used to tabulate the diagrams of fourth to seventh order. The Hartree–Fock approximation summons up 2, 8, 40, and 224 of them in first through fourth order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

AbstractPeaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007