Narrow Results

An approach to the description of the Gibbs states of lattice models of interacting quantum anharmonic oscillators, based on integration in infinite dimensional spaces, is described in a systematic way. Its main feature is the representation of the local Gibbs states by means of certain probability measures (local Euclidean Gibbs measures). This makes it possible to employ the machinery of conditional probability distributions, known in classical statistical physics, and to define the Gibbs state of the whole system as a solution of the equilibrium (Dobrushin–Lanford–Ruelle) equation. With the help of this representation the Gibbs states are extended to a certain class of unbounded multiplication operators, which includes the order parameter and the fluctuation operators describing the long range ordering and the critical point respectively. It is shown that the local Gibbs states converge, when the mass of the particle tends to infinity, to the states of the corresponding classical model. A lattice approximation technique, which allows one to prove for the local Gibbs states analogs of known correlation inequalities, is developed. As a result, certain new inequalities are derived. By means of them, a number of statements describing physical properties of the model are proved. Among them are: the existence of the long-range order for low temperatures and large values of the particle mass; the suppression of the critical point behavior for small values of the mass and for all temperatures; the uniqueness of the Euclidean Gibbs states for all temperatures and for the values of the mass less than a certain threshold value, dependent on the temperature.

Let 0 < ε ≤ ½ be fixed. We prove that on a bounded pseudoconvex domain D ⋐ ℂⁿ the Bergman metric grows at least like times the euclidean metric, provided that on D there exists a family (φ_δ)_δ of smooth plurisubharmonic functions with a self-bounded complex gradient (uniformly in δ), such that for any δ the Levi form of φ_δ has eigenvalues ≥ δ^-2ε on the set {z ∈ D | δ_D(z) < δ}. Here, δ_D denotes the boundary-distance function on D.

Consider the following Dirichlet problem involving the fractional Hénon–Lane–Emden Laplacian:

In this paper, we consider the matrix of the Green function of the Stokes equations for a half space. To the best of author’s knowledge, the matrix of the Green function has been obtained for a half space in ${ℝ}^n(n=2,3)$. But the matrix of the Green function for a half space in ${ℝ}^n(n\ge 4)$ has not been obtained. Moreover, the author considers that the form of the matrix of the Green function in ${ℝ}^n(n=2,3)$ is irregular.

In this paper, we obtain the regular form of the matrix of the Green function of the Stokes equations for a half space in ${ℝ}^n(n\ge 2)$. Moreover we obtain the Green representation formula of the Stokes equations for a half space in ${ℝ}^n(n\ge 2)$.

The operator expansion of free Green function of Helmholtz equation for arbitrary N-dimensional space leads to asymptotic extension of three dimensions Grimus–Stockinger formula closely related to multipole expansion. Analytical examples inspired by neutrino oscillation and neutrino deficit problems are considered for relevant class of wave packets.

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach.

Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to gauge field theory.

We explicitly calculate the Green functions describing quantum changes of topology in Friedmann-Lemaître-Robertson-Walker Universes whose spacelike sections are compact but endowed with distinct topologies. The calculations are performed using the long wavelength approximation at second order in the gradient expansion. We argue that complex metrics are necessary in order to obtain a non-vanishing Green functions and interpret this fact as demonstrating that a quantum topology change can be viewed as a quantum tunneling effect. The result shows that the quantum topology changes in the direction of negatively curved hypersurfaces are strongly enhanced as time goes on, while transitions in the opposite direction are suppressed.

We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher-order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory.

We consider the Darboux transformation of the Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We obtained the Green functions of the transformed Dirac equation with the initial regular boundary conditions. We also obtain the formula for the full trace of the difference of the transformed and initial Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We illustrate our findings by the consideration of the Darboux transformation of the Green function on an interval.

Following the path integral approach and within in the context of nonrelativistic Snyder–de Sitter algebra, we formulate the Green function for Dirac oscillator particle in ($1+1$) space-time dimension. Using the momentum space representation $|p,p_0〉$, we determine the energy eigenvalues and the eigenfunctions, where the wave functions can be given in terms of Gegenbauer polynomials. In addition, the high-temperature thermodynamic properties of the relativistic harmonic oscillators are analyzed, which we found that the heat capacity is not equal two times greater than the heat capacity of the one-dimensional harmonic oscillator for higher temperatures in SdS model. The special cases are found and discussed.

In this paper, the Feynman path integral approach within three parameters Dunkl derivative is investigated in the one-dimensional case of nonrelativistic quantum mechanics. We successfully derive the propagator in Cartesian coordinates and give its exact expression for the free particle, inverse square, harmonic oscillator and pure Coulomb potentials. The energy eigenvalues and their corresponding wave functions are determined.

Green function metheod is introduced to investigate ultracold dilute gas of bosonic atoms in an optical lattice which can be described by a Bose–Hubbard model. The superfluid–Mott insulator phase transition condition is determined by the related energy-band structure with an obvious interpretation of the transition mechanism.

The behavior of the compensation of a mixed spin-1 and spin-1/2 Heisenberg ferrimagnetic system on a square lattice is investigated theoretically by a two-time Green-function technique which takes into account the quantum nature of Heisenberg spins. The model includes the nearest and next-nearest neighbor interactions between spins, a crystal field and an external magnetic field. This model can be relevant for understanding the magnetic behavior of bimetallic molecular ferrimagnets that are currently being synthesized by several experimental groups. We study the spin-wave spectra of the ground state and investigate the effects of the next-nearest neighbor interactions, a crystal field and an external magnetic field on the compensation temperature. It is found that a compensation point appears and it is basically unchanged when the next-nearest-neighbor interaction between spin-1/2 is taken into account and exceeds a minimum value for other values in Hamiltonian fixed. The compensation temperature is influenced by the next-nearest-neighbor interaction between spin-1 and the external magnetic field and it is disappearing as these parameters exceed the trivial values.

Using the kinetic equation of subdynamics and the suitable time ordering rule we present a type of irreversible Liouville equation and nonlinear Liouville equation for open quantum systems. We also found a spectral relation for obtaining the non-equilibrium statistical distribution based on the boundary conditions which consistent with H theorem. Furthermore, a formalism of non-equilibrium project Green functions is given, and a non-perturbative method to solve the irreversible Liouville equation is studied. Finally, we give a model of spin-based quantum computing to show that the decoherence free states can be constructed using the non-perturbative approach.

The magnetic properties of a mixed spin-2 and spin-5 / 2 Heisenberg ferrimagnetic system on a layered honeycomb lattice are investigated theoretically by a multisublattice Green-function technique which takes into account the quantum nature of Heisenberg spins. We calculate the magnetization and the compensation temperature and transition temperature of the system in an external magnetic field and in a zero external magnetic field, and find that the transition temperature of the system increases on the effect of an external magnetic field, the compensation point disappears when the single-ion anisotropy is not large and there are two compensation points when the anisotropy is large. We also calculate the initial susceptibilities of the system.

We report a theoretical study on the bias dependence of the current-induced torque in the noncollinear ferromagnet/barrier/ferromagnet (FM) tunnel junction. By using the Keldysh–Green function technique, we derived an analytic expression of the spin transfer torque (STT) based on a continuum model. It was found that for finite bias, the torque induced by current increases monotonically with bias and is nearly antisymmetric with regard to the current direction, which agrees with the experimental observation on the STT in the spin valve system.

We obtain the numerical solutions for a single electron energy level of a hydrogen impurity confined inside an off-center donor in a spherical quantum dot. The energy spectrum of single electron impurity for off-center donor atom subject to screened Coulomb's attractive potential barrier well is presented. Using B-spline finite element method, we generate a complete set of wavefunctions within the central unit cell, and through the perturbation theory, the whole energy spectrum of an impurity at low temperature can be explored. We find that the energy splitting and different related magnetic quantum number m_l, i.e., (n, l, m_l) varies with quantum dot size. This is due to the fact that quantum confinement effect lifts the degeneracy which comes from the Coulomb interaction. On the other hand, the phenomena of energy spectrum splitting under this model is an analogous case, considering that degeneracy of hydrogen energy levels in free space has been lifted by an external magnetic field (Zeeman effect).

Double permutation (DP) method is developed here for designing Feynman diagrams for mass operator (MO) of interacting electrons in high orders of perturbation theory (PT). The derived expression allows finding the Young diagrams for the class of permutations corresponding to disconnect Feynman diagrams. The classification of DPs, carried out before, allows to identify the permutations corresponding to disconnected, singly connected (improper) diagrams and to derive expressions for intolerant cycles of permutations. Ordering the nonprimed digits in natural order in the cycles of DP, we avoid the permutations, corresponding to the Feynman diagrams of the same topology because of other numbering of nodes. Thus, the numbers of considered permutations is sufficiently reduced: (from 24 to 6 and from 720 to 42) in the second and the third orders of PT. All 414 expressions (diagrams) for MO in the fourth order of PT were designed using this method. The use of group theory allows us to conclude that no more Feynman diagrams can be designed. The developed method can be used as algorithm for Feynman diagrams designing for MO of interacting electrons (one sort fermions) in higher orders of PT.

In this paper, the ferroelectric transition and dielectric properties of the lead monohydrogen phosphate (PbHPO₄) crystal are theoretically investigated. By using Green’s function method and modified two sublattice pseudospin–lattice coupled mode model, formulae for PbHPO₄ type crystals are derived. Temperature variations of ferroelectric mode frequency, dielectric constant and loss tangent are calculated for the PbHPO₄ crystal. Our results agree well with the experimental data.