Narrow Results

This paper primarily lies in presenting existence and uniqueness analysis of boundary fractional difference equations of a special Riemann–Liouville operators classes. To this end, we first develop Green’s function to the corresponding fractional boundary value problems and provide boundary conditions to find the constants. Then we study the existence of solutions and we examine the bounded of their solutions. Eventually, two numerical examples are given to demonstrate the efficiency and uniqueness behavior of the boundary value problem. Furthermore, such fractional problems can typically be converted to a direct fractional problem with certain boundary or initial conditions in transport in porous media.

This article is concerned with the GW method for finite electronic systems. In the first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then give a rigorous mathematical formulation of the ${\mathrm{\text{GW}}}^0$ equations, and study the well-posedness of these equations, proving the existence of a unique solution in a perturbative regime.

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equation on a comb, the time-fractional equation for the Green’s functions appears, due to these specific geometric constraints.

We review the beginning of the effective Lagrangian in QED that was first introduced in the literature by W. Heisenberg and H. Euler in 1936. Deviating from their way of calculating the one-loop effective correction to the classical Maxwell Lagrangian, we use Green's functions and adopt the Fock–Schwinger proper-time method. The important role of the Heisenberg–Euler effective Lagrangian is explicitly demonstrated for low-energy photon–photon processes.

The theory of Green's functions (GF) for thin films was formulated, where site indices were different also along the direction of broken symmetry. This approach results in the layer dependence of correlation functions in films, which did not appear in the previous investigations of thin films. The magnetizations of three layer thin ferromagnetic (FM) and four layer thin ferroelectric (FE) films were investigated using the GFs found in this way. It was shown that the transition temperature varies from layer to layer, which is the obvious consequence of the generalization of the formalism of the GF calculation for films.

The idea of spontaneous symmetry breaking in many-body physics from personal perspective (Bose-gas, nuclear structure and a new approach of Generalized Density Matrix).

This paper presents a method for thermal simulation of a plate fin heat sink using an analytical solution of the three-dimensional heat equation resulting from an appropriate plate fin heat sink thermal model. The steady-state temperature map on the surface of the plate fin heat sink is computed analytically employing the Green's function method. The entire solution methodology is illustrated for the particular examples of the plate fin heat sink subjected to non-uniform heat density on the surfaces. The simulation results are validated with experimental measurements.

In this paper, a complete investigation of the magnetic properties of the Hubbard model on the AB₂ chain is presented. We evaluate the magnetic phase diagram as a function of the electron–electron interaction and band filling by using the Green's function method within the Hartree–Fock approximation. Our results are in agreement with Lieb's theorem and the system shows ferromagnetic, ferrimagnetic, and paramagnetic phases.

Electron subsystem of ultrathin films was analyzed using Green's function method including quantum size effect and effect of boundaries on Hamiltonian parameters. We have calculated basic physical properties of electrons in crystalline films: energy spectra, possible states, space distribution of electrons and the position of Fermi level, which enabled the complete insight into the thermodynamic or conducting characteristics of observed film-structure. The comparison with crystal bulk have shown that electronic properties of the materials are strongly influenced by both the sample dimensions and boundary conditions. The numerical calculations performed for very thin crystalline metallic-like films show that localized states and spatial distribution of the (quasi)free electrons might be manipulated by varying the surface parameters which is significant for operation of devices based on thin films.

This paper models three-dimensional wave propagation around two-dimensional rigid acoustic screens, with minimal thickness (approaching zero), and placed in a fluid layer. Rigid or free boundaries are prescribed for the flat fluid surfaces.

The problem is computed using the Traction Boundary Element Method (TBEM), which is appropriate for modeling thin-body inclusions, overcoming the difficulty posed by the conventional direct Boundary Element Method (BEM).

The problem is solved as a summation of two-dimensional problems for different wave numbers along the direction for which the geometry does not vary. The source in each problem is a spatially sinusoidal harmonic line load. The influence of the horizontal boundaries of the fluid medium on the final wave field is computed analytically using appropriate 2.5D Green's functions for each model developed. Thus, only the boundary of the rigid acoustic screen needs to be discretized by boundary elements.

The computations are performed in the frequency domain and are subsequently inverse Fourier transformed to obtain time domain results. Complex frequencies are used to avoid aliasing phenomena in the time domain results.

We consider a special family of occupation-time derivatives, namely proportional step options introduced by [18]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

A new analytical method is developed for transient solutions for bars in longitudinal vibration, shafts in torsional vibration and strings in transverse vibration. These vibrating continua have nonuniform distributions of geometric and material properties, and are governed by second-order partial differential equations (wave equations). Based on a transfer function formulation and a residue formula, the proposed method delivers exact solutions for several types of nonuniform systems subject to external, boundary and initial disturbances. The proposed method does not need system eigenfunctions, and is numerically efficient as it only involves simple operations of two-by-two matrices.

We consider the second-order linear differential equation

A boundary element method (BEM) is presented to study 3D wave scattering by cracks in a cylinder. Green's functions needed in the kernel of boundary integral equations in BEM are derived with the help of guided wave functions. Guided wave modes in the cylinder are obtained by a semi-analytical finite element (SAFE) method. Green's functions are constructed numerically by superposition of guided wave modes. In this method, the cylinder is discretized in the radial direction into several coaxial circular cylinders (sub-cylinders) and the radial dependence of the displacement in each sub-cylinder is approximated by quadratic interpolation polynomials. A numerical procedure is used here to accurately calculate the Cauchy's principal value (CPV) and weakly singular integrals. The multi-domain technique is employed here to model the crack surface. Numerical results are presented to show the effectiveness of the proposed solution.

In this paper, a method for construction of the time-dependent approximate Green's function for the initial boundary value problem in a radially multilayered cylinder is suggested. This method is based on determination of the eigenvalues and the orthogonal set of the eigenfunctions; regularization of the Dirac delta function in the form of the Fourier series with a finite number of terms; expansion of the unknown Green's function in the form of Fourier series with unknown coefficients and computation of a finite number of unknown Fourier coefficients. Computational experiment confirms the robustness of the method for the approximate computation of the Dirac delta function and Green's function.

In this paper, DeWitt’s formalism for field theories is presented; it provides a framework in which the quantization of fields possessing infinite-dimensional invariance groups may be carried out in a manifestly covariant (non-Hamiltonian) fashion, even in curved space-time. Another important virtue of DeWitt’s approach is that it emphasizes the common features of apparently very different theories such as Yang–Mills theories and General Relativity; moreover, it makes it possible to classify all gauge theories in three categories characterized in a purely geometrical way, i.e. by the algebra which the generators of the gauge group obey; the geometry of such theories is the fundamental reason underlying the emergence of ghost fields in the corresponding quantum theories, too. These “tricky extra particles”, as Feynman called them in 1964, contribute to a physical observable such as the stress-energy tensor, which can be expressed in terms of Feynman’s Green function itself. Therefore, an entire section is devoted to the study of the Green functions of the neutron scalar meson: in flat space-time, the choice of a particular Green’s function is the choice of an integration contour in the “momentum” space; in curved space-time the momentum space is no longer available, and the definition of the different Green functions requires a careful discussion itself. After the necessary introduction of bitensors, world function and parallel displacement tensor, an expansion for the Feynman propagator in curved space-time is obtained. Most calculations are explicitly shown.

Explicit results for a piezoelectric half-space x₂ ≥ 0 subject to linearly-varying surface loadings along x₃ axis are derived. The extended Stroh formalism is employed to provide three-dimensional solutions with the generalized displacement vector u expressed as a function of (z, x₂, x₃). A general polynomial solution for u with order of m in x₃ is suggested and it provides a particularly efficient solution for half-space problem with loadings on the surface. A simple uniform surface loading is considered first to clarify the derivations. Then explicit solution in case of a linearly-varying surface loading along x₃-direction is obtained. In addition, the Green's function for a piezoelectric half-space with a linearly-varying surface line loading along x₃-axis is constructed.

Generally, one thinks of a “quantum jump” as the process in which an electron “jumps” between a pair of quantum states, even as the process is treated within perturbation theory. This jump of an electron has remained a key point of conservative (i.e., traditional) quantum mechanics. But, the question of the time dependence of such a transition, e.g. the time for an atom to be ionized by radiation, is somewhat different than this view. A detailed approach in which an incoming wave first polarizes the quantum states and then completes the transition has allowed for a detailed discussion of the smooth transition of the electron from one state to the next. Here, we will discuss the history of the process, and illustrate the approach with the question of “how long does it take for an electron to emit a phonon?” The entire process arises from the proper application of wave mechanics and obviates the need to even consider a discussion of quantum jumps.

In this Chapter, we review a state-of-the-art theoretical formalism for modeling quantum transport properties of molecular scale conductors. The formalism is based on carrying out density functional theory (DFT) analysis within the Keldysh non-equilibrium Green's function (NEGF) framework. It allows predictions of nonlinear and nonequilibrium charge transport through nano-electronic devices without using phenomenological parameters. We present relevant details to the physical, mathematical, and numerical backgrounds of this formalism. An example will be given to illustrate the application of this technique.