Abstract:
In our previous work [1] we calculated RKKY interaction between two magnetic impurities in pristine graphene using the Green’s functions (GF) in the coordinate-imaginary time representation. Now we show that the calculations of the GF in this representation can be simplified by using the Feynman’s trick, which allows to easily calculate RKKY interaction in gapped graphene. We also present calculations of the RKKY interaction in gapped or doped graphene using the coordinate-imaginary frequency representation. Both representations, corresponding to calculation of the bubble diagram in Euclidean space, have an important advantage over those corresponding to calculation in Minkowskii space, which are very briefly reviewed in the Appendix to the present work. The former, in distinction to the latter, operate only with the convergent integrals from the start to the end of the calculation.

The
spheroidal wave functions are investigated in the case $m = 1$. The integral equation is obtained for them. There are two
kinds of eigenvalues in the differential and corresponding integral equations, and
the relation between them is given explicitly. This is the great advantage of
our integral equation, which will provide useful information through the study
of the integral equation. Also an example is given for the special case, which
shows another way to study the eigenvalue problem.

Abstract:
a problem related to the quantum interference phenomenon in a nanostructured system is investigated by studying the segregation of substitutional impurities. such systems are formed by the juxtaposition of transition metals atomic layers. the substitutional impurities segregation is treated calculating the total electronic energy variation of the system considering di←erent positions of the impurity relatively to the surface. a single orbital tight-binding model is employed. the potential impurities are determined imposing local or global charge neutrality and satisfying friedel sum rule. we consider a simple cubic lattice with (100) and (110) directions.

Temporal variations of atmospheric density distribution induce changes in the gravitational air mass attraction at a specific observation site. Additionally, the load of the atmospheric masses deforms the Earth’s crust and the sea surface. Variations in the local gravity acceleration and atmospheric pressure are known to be corrected with an admittance of about 3nm/s^{2} per hPa as a standard factor, which is in accordance with the IAG Resolution No. 9, 1983. A more accurate admittance factor for a gravity station is varying with time and depends on the total global mass distribution within the atmosphere.The Institut für Erdmessung (IfE) performed absolute gravity observations in the Fennoscandian land uplift area nearly every year from 2003 to 2008. The objective is to ensure a reduction with 3 nm/s^{2} accuracy. Therefore, atmospheric gravity changes are modeled using globally distributed ECMWF data. The attraction effect from the local zone around the gravity station is calculated with ECMWF 3D weather data describing different pressure levels up to a height of 50km. To model the regional and global attraction, and all deformation components the Green’s functions method and surface ECMWF 2D weather data are used.For the annually performed absolute gravimetry determinations, this approach improved the reductions by 8 nm/s^{2} (-19 nm/s^{2} to +4 nm/s^{2}). The gravity modeling was verified using superconducting gravimeter data at station Membach inBelgiumimproving the residuals by about

Abstract:
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later. In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform. The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.

Abstract:
the authors, together with a. del campo, developed an alternative method for determining the feynman propagator [1] for a non-relativistic problem. one started with the time dependent schroedinger equation for the problem. carried out a laplace transform with respect to time to get the equation for the energy dependent green function and derived it explicitly. we then carried out the inverse laplace transform in the energy to get feynman propagator. in this paper we carry out the same programme for a relativistic problem associated with the one dimensional dirac equation of a free particle and the dirac oscillator proposed by moshinsky and szczepaniak [2] twenty years ago.

Abstract:
We attempt to describe the magnetic properties of parent pnictide compounds by using the J1 - J2 - Jc Heisenberg model Hamiltonian. In order to obtain the ground state magnetization and spin wave dispersion we use the Green's functions method for spin operators. The equations of motion for Green's functions are decoupled by employing the random phase approximation. We analyze the results numerically and after comparison with experimental data we conclude that the model is to be modified to make it more relevant to iron pnictides.

Abstract:
We analyze the J1 - J2 - Jc Heisenberg model Hamiltonian by using the Dyson-Maleev representation for spin operators and keeping the terms quadratic in Bose operators (linear spin wave theory). From the resulting Hamiltonian we find the ground state magnetisation and spin wave dispersion by employing the Green's function method. We compare the results with those of random phase approximation analysis from Part I and with experimental data for parent pnictide compounds. Neither of the two approaches gives a completely satisfactory description of the magnetic properties of iron pnictides. We conclude that alterations of the model Hamiltonian are needed to get a better agreement between the theory and experiments.

Abstract:
. We consider equity-linked debt where the holder receives both interest payments and payments linked to the performance of an equity index. We use a Green's function approach to value such instruments under the assumption that the equity index obeys a lognormal random walk and the risk-free interest rate is given by the Vasicek model.