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We show that a Green function solution can be given for a class of nonhomogeneous nonlinear systems having relevance in quantum field theory. This in turn means that a quantum field theory in the strong coupling limit can be formulated and the spectrum obtained.

The Fradkin–Schwinger functional methods to represent a Green function in an external gravitational field are used to study the eikonal and the next-to-eikonal limit, including the nonlinear gravitational interactions, of the scattering amplitudes of an ultra-relativistic scalar particle on a static super-massive scalar target in the nearly forward limit. The functional approach confirms the exponentiation of the leading eikonal which also applies to the first non-leading power in the energy of the light particle, moreover includes the interaction at impact parameter much larger than the Schwarzschild radius associated with the center of mass energy in the ultra-relativistic limit.

Interacting Bose gas confined by an external potential is studied using Green functions in spectral representation. The calculation is presented transparently using the equation of motion method. With this, the interplay between the condensed and the non-condensed atoms is inevitably seen. An expression for the condensate number at finite temperature is obtained in the lowest and first orders, from which depletion of bosons from the ground state is qualitatively analyzed. Finally, we discuss the behaviour of the specific heat of a trapped interacting Bose gas in the quasi-continuum limit.

In this paper we investigated the causality problem present in the recent published work about the Doplicher–Fredenhagen–Roberts–Amorim (DFRA) noncommutative framework which analyzed the complex scalar field. We described the noncommutative Klein–Gordon equation with a source term. Its solution was accomplished through the retarded, advanced and causal Green functions constructed in this noncommutative ten-dimensional DFRA space–time. We believe that this solution constitutes the first step in the elaboration of a quantum field theory using the DFRA formalism where the well known noncommutative parameter from now on is an ordinary coordinate of the system and therefore has a canonical conjugate momentum. We discussed the dimensional reduction of this noncommutative space–time (D = 10) in order to recover commutativity in D = 4 space–time, which is not the usual one in noncommutativity literature.

Quantum steeplechase is the study of a Luttinger liquid (LL) in one dimension in the presence of a finite number of barriers and wells clustered around an origin. The powerful nonchiral bosonization technique (NCBT) is introduced to write down closed formulas for the two-point functions in the sense of the random phase approximation (RPA). Unlike g-ology-based methods that are tied to the translationally invariant, free particle basis, the NCBT explicitly makes use of the translationally noninvariant single particle wave functions. The present method, that provides the most singular part of the asymptotically exact Green function in a closed form, is in contrast to competing methods that require a combination of renormalization group and/or numerical methods in addition to the bosonization techniques.

We develop a theoretical approach to calculate the scattering and localization properties in the inhomogeneous hexagonal nanostructure system with adsorbed chain. Our numerical results are presented for three different positions of the defect, Top, Hollow and Bridge. The breakdown of translation symmetry induced by the inhomogeneity, gives rise to localized elastic wave modes at its neighborhood. The Landauer and Buttiker method is used to analyze the elastic wave scattering phenomena for atomic sites that constitute a minimum representation set in the neighborhood of the defect. The transmission and reflection probabilities as well as the phonons average transmittance across and the vibration density of states are determined for a large band of incident scattering energies. The coherent coupling between the localized elastic wave and the propagating modes of the perfect hexagonal structure leads to Fano resonances in the phononic conductance spectra of the studied structure. This is directly related to the measurable physical thermal conductivity of the perturbed domain.

For the scalar wave equation in periodic structures, we establish rigorously the manifestation of a spectral bandgap as a frequency interval of exponentially small transmission of energy through a slab of the crystal when the thickness of the slab tends to infinity. We extend the result to slabs that contain a planar defect for frequencies that do not coincide with guided mode frequencies for the defect. For one-dimensional crystals, we prove that the transmission approaches a nonzero value at guided mode frequencies and give an explicit formula for the transmission. The main analytical tool in the multidimensional case is the Calderón boundary integral projectors for the Helmholtz equation.

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions for the stability mentioned above by two methods. Also, an example is given to illustrate our theoretical results. The proposed problem is the generalization of third-order ordinary differential equations with classical, initial and anti-periodic boundary conditions.

Gravitational waves are considered as metric perturbations about a curved background metric, rather than the flat Minkowski metric since several situations of physical interest can be discussed by this generalization. In this case, when the de Donder gauge is imposed, its preservation under infinitesimal spacetime diffeomorphisms is guaranteed if and only if the associated covector is ruled by a second-order hyperbolic operator which is the classical counterpart of the ghost operator in quantum gravity. In such a wave equation, the Ricci term has opposite sign with respect to the wave equation for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate the solutions of the two problems, and the algorithm is applied to the case when the curved background geometry is the de Sitter spacetime. Such vector wave equations are studied in two different ways: (i) an integral representation, (ii) through a solution by factorization of the hyperbolic equation. The latter method is extended to the wave equation of metric perturbations in the de Sitter spacetime. This approach is a step towards a general discussion of gravitational waves in the de Sitter spacetime and might assume relevance in cosmology in order to study the stochastic background emerging from inflation.

We consider the multivalued problem under four boundary conditions u(0) = x₀, u(η) = u(θ) = u(T) where 0 < η < θ < T and for F is a multifunctions from [0, T] × ℝⁿ × ℝⁿ to the nonempty compact subsets of ℝⁿ not necessary convex. We give a lemma which is useful in the study of four boundary problems for the differential equations and the differential inclusions. Further we have results that improve earlier theorems.

The alternating Fe–Mn layered structures of the compounds FeMnAs_xP${}_{1-x}$ display properties which have been demonstrated experimentally as very promising as far as commercial applications of the magnetocaloric effect are concerned. However, the theoretical literature on this and other families of magnetocaloric compounds still adopts simple molecular-field models in the description of important statistical mechanical properties like the entropy variation that accompanies applied isothermal magnetic field cycling, as well as the temperature variation following adiabatic magnetic field cycles. In the present paper, a random phase approximation Green function theoretical treatment is applied to such structures. The advantages of such approach are well known since the details of the crystal structure are easily incorporated in the model, as well as a precise description of correlations between neighbor spins can be obtained. We focus on a simple one-exchange parameter Heisenberg model, and the observed first-order phase transitions are reproduced by the introduction of a biquadratic term in the Hamiltonian whose origin is related both to the magnetoelastic coupling with the phonon spectrum in these compounds as well as with the values of spins in the Fe and Mn ions. The calculations are compared with experimental magnetocaloric data for the FeMnAs_xP${}_{1-x}$ compounds. In particular, the magnetic field dependence for the entropy variation at the transition temperature predicted from the Landau theory of continuous phase transitions is reproduced even in the case of discontinuous transitions.

Following our previous work on the magnetocaloric compounds FeMnAs_xP${}_{1-x}$, we obtain thermomagnetic properties for the barocaloric antiperovskite Mn₃GaN, which has been investigated in recent years in view of its thermal response under pressure cycling. Its structure displays canting of spins and a volume dilation in the antiferromagnetic (AFM) transition, related to magnetoelastic coupling. A random-phase-approximation (RPA) Green function theoretical treatment is applied to a two-exchange parameter Heisenberg model, and the canting of the spins orientation in the (111) planes of the structure is considered in the model. The observed first-order AFM phase transitions are reproduced by the introduction of a biquadratic spin term in the Hamiltonian, coupled to the volume dilation in the transition. The model is able to predict a greater stability for the canted spin structure as compared with conventional AFM, if the magnitudes of Mn magnetic moments in these structures are taken from the literature. The calculations of quantities of interest for barocaloric applications reproduce quite well the available data on temperature changes and latent heat under pressure cycling, at adiabatic and isothermal conditions, respectively.

Suppose D is a bounded C^1,1 domain in Rⁿ and G_D is the Green function of the killed symmetric α-stable process in D. In this note we establish a sharp upper bound estimate for G_D in such a way that the explicit dependence of the constant in the estimate on α is also given, This sharp estimate allows us to recover the sharp upper bound estimate for the Green function of killed Brownian motion in D by letting α ↑ 2.

Interacting Bose gas confined by an external potential is studied using Green functions in spectral representation. The calculation is presented transparently using the equation of motion method. With this, the interplay between the condensed and the non-condensed atoms is inevitably seen. An expression for the condensate number at finite temperature is obtained in the lowest and first orders, from which depletion of bosons from the ground state is qualitatively analyzed. Finally, we discuss the behaviour of the specific heat of a trapped interacting Bose gas in the quasi-continuum limit.

The consecutive convolution of the harmonic Green function with itself provides a polyharmonic Green function of any order. Convoluting such a polyharmonic Green function with a polyharmonic Green-Almansi function gives a polyharmonic Green function of new type. It leads to a certain Dirichlet problem for a Poisson equation of certain order. This problem is well-posed. Its solution is given here.