Terms Of Green Function Research Articles

Quantum Mechanics and Quantum Field Theory: Algebraic and Geometric Approaches

This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.

An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx+2uxxyy+uyyyy=f(u) . To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE.

Scattering in Algebraic Approach to Quantum Theory—Associative Algebras

The definitions of scattering matrix and inclusive scattering matrix in the framework of formulation of quantum field theory in terms of associative algebras with involution are presented. The scattering matrix is expressed in terms of Green functions on shell (LSZ formula), and the inclusive scattering matrix is expressed in terms of generalized Green functions on shell. The expression for inclusive scattering matrix can be used also for quasi-particles (for elementary excitations of any translation-invariant stationary state, for example, for elementary excitations of equilibrium state). An interesting novelty is the consideration of associative algebras over real numbers.

Hadamard\u2019s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Analytical solution of a fractional differential equation in the theory of viscoelastic fluids

The aim of this paper is to present analytical solutions of fractional delay differential equations (FDDEs) of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. This results presented in two condition, in first step when 0 ≤ α, β ≤ 1/2 and in the second step we considered 1/2 ≤ α, β ≤ 1, for each step 1,2 for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples.

Accelerating Uncertainty Quantification for Nonlinear Inverse Scattering Problems With High Contrast Media by Direct Envelope Methods and Krylov Subspace Iterative Integral Equation Solvers

Uncertainty quantification related to nonlinear inverse scattering problems often involves the posterior covariance matrix that cannot be effectively estimated but still need to be considered as an important part of the nonlinear inverse scattering problem. The objective of this work is to show how to take advantage of a Bayesian framework to estimate the uncertainty of velocity reconstruction in the nonlinear inverse scattering imaging. The key ingredients are twofold. On the one hand, we rely on a Kalman Filter method, equipped with an optimization scheme, to solve the inverse problem. We use the distorted Born iterative method to formulate the sensitivity kernel. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method, in which the Green’s functions are based on integral equations. On the other hand, we use the direct envelope methods to provide the initial guess (large-scale smooth model) to overcome the nonlinearity in the inverse scattering problems. We first investigate the uncertainty of velocity imaging in high contrast media. Then we show by means of Lippmann-Schwinger-type equations that the uncertainty of multiparameter inverse scattering problems can be addressed. Numerical results dealing with the uncertainty of nonlinear inverse scattering problems in the case of both isotropic and anisotropic media highlight the important role played by uncertainty quantification.

Effective dispersion coefficients for the upscaling of pore-scale mixing and reaction

We upscale reactive mixing using effective dispersion coefficients to capture the combined effect of pore-scale heterogeneity and molecular diffusion on the evolution of the mixing interface between two initially segregated dissolved species. Effective dispersion coefficients are defined in terms of the average spatial variance of the solute distribution evolving from a pointlike injection, that is, the transport Green function. We numerically investigate the temporal behavior of the longitudinal effective dispersion coefficients for two porous media of different pore-scale heterogeneity as measured by the statistics of the flow speed, and different Péclet numbers. We find that the effective dispersion coefficients evolve with time, or equivalently travel distance. As the solute samples the pore-scale flow heterogeneity due to advection and transverse diffusion, the effective dispersion coefficients evolve from the value of molecular diffusion to the corresponding hydrodynamic dispersion coefficients. Thus, at times smaller than the diffusion time over a characteristic pore length, the effective dispersion coefficients can be significantly smaller than the hydrodynamic dispersion coefficients. This difference can explain frequently observed mismatches between pore-scale reactive mixing data, and predictions using Darcy scale transport descriptions based on hydrodynamic dispersion coefficients that are constant in time. This suggests that the notion of incomplete mixing on the support scale can be quantified in terms of effective pore-scale dispersion coefficients. We use effective dispersion in order to approximate the transport Green function in terms of a Gaussian-shaped distribution that is characterized by the effective variance. This is approximation is termed dispersive lamella. Based on this representation, we study reactive mixing between two initially segregated solutes. The dispersive lamella approach accurately predicts the evolution of the product mass of an instantaneous bimolecular reaction obtained from direct numerical simulations. This demonstrates that effective dispersion is an accurate measure for width of the mixing interface between the two reacting species. These results shed some new light on pore-scale mixing, the notion of incomplete mixing, and its prediction and upscaling in terms of an effective mixing model.

Integral Equation Methods With Multiple Scattering and Gaussian Beams in Inhomogeneous Background Media for Solving Nonlinear Inverse Scattering Problems

I develop inverse scattering methods for velocity reconstruction in the subsurface, based on multiple scattering theory, Gaussian beams and nonlinear Born approximation. This method is based on a new version of the distorted Born iterative inverse scattering method. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method. I propose three direct scattering methods for forward modeling, namely, Gaussian beam integral equation propagators, nonlinear Born approximation, and multiple scattering. First, I apply a sensitivity kernel incorporating the Gaussian beam summation-based integral equation method and compute the background medium Green's function in an inhomogeneous medium. Then I extend an approximate solution of the Lippmann-Schwinger equation, referred to as nonlinear Born approximation to inverse scattering problem. I apply the nonlinear Born approximation to forward modeling at each iteration. Furthermore, I combine the Gaussian beams with the multiple scattering theory for nonlinear scattering problems. I obtain the inversion results using the proposed approaches, which shows the capability for inverse scattering imaging. I have carried out several numerical experiments that involve reconstructing the resampled Marmousi and Society of Exploration Geophysicists (SEG)/European Association of Geoscientists and Engineers (EAGE) salt models from its smoothed version.

Scattering and Absorption of Light by a Monolayer of Spherical Particles under Oblique Illumination

The problem of light scattering and absorption by a monolayer of homogeneous spherical particles at an arbitrary angle of incidence of a plane wave, which determines the phase shift between the averaged fields in the particles, has been solved. The solution is based on the quasi-crystalline approximation of the theory of multiple scattering of waves, the mean-field approximation, and the multipole expansion of the fields and the tensor Green function in terms of vector spherical wave functions. Formulas for determining the angular distribution of the incoherently scattered field intensity, the incoherent scattering, absorption, coherent transmission, and reflection coefficients have been derived. The dependences of these characteristics on the direction of illumination are illustrated using partially ordered monolayers of dielectric and semiconductor particles as an example. It is shown that: (i) under oblique illumination the angular distribution of the scattered radiation in azimuthal scattering angle for any polarization of the incident wave (except for p and s) is asymmetric relative to the plane of incidence, (ii) the maximum of the angular distribution of the radiation transmitted through the monolayer can approach its normal with increasing angle of incidence and/or concentration of particles, (iii) there exist conditions under which the bulk of the scattered radiation is concentrated in the direction opposite to the direction of the incident radiation, and (iv) the dependence of the monolayer absorption coefficient on the illumination angle has extrema whose positions and magnitudes are determined by the polarization of the incident wave.

Green function and resolvent operator of a Schrödinger equation with general point interaction

In this paper, we investigate the time independent Schrodinger equation which has complex valued potential function under the general point interaction. We construct Green function of this problem and we find the resolvent of the problem in terms of Green function.

Perturbations of an extremal Kerr spacetime: Analytic framework and late-time tails

We develop a complete and systematic analytical approach to field perturbations of extremal Kerr spacetime based on the formalism of Mano, Suzuki and Takasugi (MST) for the Teukolsky equation. Analytical expressions for the radial solutions and frequency-domain Green function in terms of infinite series of special functions are presented. As an application, we compute, for the first time, the leading late-time behavior due to the branch point at zero frequency of scalar, gravitational, and electromagnetic field perturbations on and off the event horizon. We also use the MST method to compute the leading behavior of the Green function modes near the branch point at the superradiant bound frequency and show that this behavior agrees with existing results in the literature using a different method.

Superconductivity of strongly correlated electrons on the honeycomb lattice

A microscopic theory of the electronic spectrum and of superconductivity within the t-J model on the honeycomb lattice is developed. We derive the equations for the normal and anomalous Green functions in terms of the Hubbard operators by applying the projection technique. Superconducting pairing of d + id′-type mediated by the antiferromagnetic exchange is found. The superconducting Tc as a function of hole doping exhibits a two-peak structure related to the van Hove singularities of the density of states for the two-band t-J model. At half-filling and for large enough values of the exchange coupling, gapless superconductivity may occur. For small doping the coexistence of antiferromagnetic order and superconductivity is suggested. It is shown that the s-wave pairing is prohibited, since it violates the constraint of no-double-occupancy.Graphical abstract

Analytical Solution of a Fractional Differential Equation in the Theory of Viscoelastic Fluids

The aim of this paper is to present analytical solutions of fractional delay differential equations of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. Different situations for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples.

Spatial Order and Absorption of light by Monolayer of Silicon Nano- and Submicrometer-Sized Particles

The effect of the spatial order of a monolayer of monodisperse spherical crystalline silicon nano- and submicrometer-sized particles upon its absorption coefficient is theoretically investigated. The calculation method is based on the quasicrystalline approximation of the theory of multiple scattering of waves and multipole expansion of electromagnetic fields and tensor Green function in terms of vector spherical wave functions. The results can be used for an enhancement of light harvesting in a solar cells design.

Iterated extended Kalman filter method for time‐lapse seismic full‐waveform inversion

ABSTRACTTime‐lapse seismic data is useful for identifying fluid movement and pressure and saturation changes in a petroleum reservoir and for monitoring of CO2 injection. The focus of this paper is estimation of time‐lapse changes with uncertainty quantification using full‐waveform inversion. The purpose of also estimating the uncertainty in the inverted parameters is to be able to use the inverted seismic data quantitatively for updating reservoir models with ensemble‐based methods. We perform Bayesian inversion of seismic waveform data in the frequency domain by combining an iterated extended Kalman filter with an explicit representation of the sensitivity matrix in terms of Green functions (acoustic approximation). Using this method, we test different strategies for inversion of the time‐lapse seismic data with uncertainty. We compare the results from a sequential strategy (making a prior from the monitor survey using the inverted baseline survey) with a double difference strategy (inverting the difference between the monitor and baseline data). We apply the methods to a subset of the Marmousi2 P‐velocity model. Both strategies performed well and relatively good estimates of the monitor velocities and the time‐lapse differences were obtained. For the estimated time‐lapse differences, the double difference strategy gave the lowest errors.

Scattering of Polarized and Natural Light by a Monolayer of Spherical Homogeneous Spatially Ordered Particles under Normal Illumination

Expressions for determining the optical characteristics of a monolayer of homogeneous monodisperse spherical particles under normal illumination by a plane electromagnetic wave with arbitrary polarization and nonpolarized radiation have been obtained. They are based on the quasi-crystalline approximation (QCA) of the theory of multiple scattering of waves (TMSW) and the multipole decomposition of fields and the tensor Green function in terms of the vector spherical wave functions. The influence of the polarization state of the incident wave on the angular structure of the radiation that is scattered by a partially ordered monolayer and a monolayer with a imperfect lattice has been considered.

General Green's function formalism for layered systems: Wave function approach

The single-particle Green's function (GF) of mesoscopic structures plays a central role in mesoscopic quantum transport. The recursive GF technique is a standard tool to compute this quantity numerically, but it lacks physical transparency and is limited to relatively small systems. Here we present a numerically efficient and physically transparent GF formalism for a general layered structure. In contrast to the recursive GF that directly calculates the GF through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint not only allows us to reproduce existing results in a concise and physically intuitive manner, but also provides analytical expressions of the GF in terms of a generalized scattering matrix. This identifies the contributions from each individual scattering channel to the GF and hence allows this information to be extracted quantitatively from dual-probe STM experiments. The simplicity and physical transparency of the formalism further allows us to treat the multiple reflection analytically and derive an analytical rule to construct the GF of a general layered system. This could significantly reduce the computational time and enable quantum transport calculations for large samples. We apply this formalism to perform both analytical analysis and numerical simulation for the two-dimensional conductance map of a realistic graphene p-n junction. The results demonstrate the possibility of observing the spatially-resolved interference pattern caused by negative refraction and further reveal a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene.

Self-consistent Green's function embedding for advanced electronic structure methods based on a dynamical mean-field concept

We present an embedding scheme for periodic systems that facilitates the treatment of the physically important part (here the unit cell) with advanced electronic-structure methods, that are computationally too expensive for periodic systems. The rest of the periodic system is treated with computationally less demanding approaches, e.g., Kohn-Sham density-functional theory, in a self- consistent manner. Our scheme is based on the concept of dynamical mean-field theory (DMFT) formulated in terms of Green functions. In contrast to the original DMFT formulation for correlated model Hamiltonians, we here consider the unit cell as local embedded cluster in a first-principles way, that includes all electronic degrees of freedom. Our real-space dynamical mean-field embedding (RDMFE) scheme features two nested Dyson equations, one for the embedded cluster and another for the periodic surrounding. The total energy is computed from the resulting Green functions. The performance of our scheme is demonstrated by treating the embedded region with hybrid functionals and many-body perturbation theory in the GW approach for simple bulk systems. The total energy and the density of states converge rapidly with respect to the computational parameters and approach their bulk limit with increasing cluster (i.e., unit cell) size.

Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions

We study certain spectral aspects of the Sturm-Liouville problem with a finite number of interior singularities. First, for self-adjoint realization of the considered problem, we introduce a new inner product in the direct sum of the $L_{2}$ spaces of functions defined on each of the separate intervals. Then we define some special solutions and construct the Green function in terms of them. Based on the Green function, we establish an eigenfunction expansion theorem. By applying the obtained results we extend and generalize such important spectral properties as the Parseval and Carleman equations, Rayleigh quotient, and Rayleigh-Ritz formula (minimization principle) for the considered problem.

Gravitational waves in fourth order gravity

In the post-Minkowskian limit approximation, we study gravitational wave solutions for general fourth-order theories of gravity. Specifically, we consider a Lagrangian with a generic function of curvature invariants \(f(R, R_{\alpha\beta}R^{\alpha\beta}, R_{\alpha\beta\gamma\delta }R^{\alpha\beta\gamma\delta})\). It is well known that when dealing with General Relativity such an approach provides massless spin-two waves as propagating degree of freedom of the gravitational field while this theory implies other additional propagating modes in the gravity spectra. We show that, in general, fourth order gravity, besides the standard massless graviton is characterized by two further massive modes with a finite-distance interaction. We find out the most general gravitational wave solutions in terms of Green functions in vacuum and in presence of matter sources. If an electromagnetic source is chosen, only the modes induced by \(R_{\alpha\beta}R^{\alpha\beta}\) are present, otherwise, for any \(f(R)\) gravity model, we have the complete analogy with tensor modes of General Relativity. Polarizations and helicity states are classified in the hypothesis of plane wave.