Exact solution of Klein–Gordon equation in fractional-dimensional space

In this paper, we present an exact solution of the Klein–Gordon equation in the framework of the fractional-dimensional space, in which the momentum and position operators satisfying the R-deformed Heisenberg algebras. Accordingly, three essential problems have been solved such as: the free Klein–Gordon equation, the Klein–Gordon equation with mixed scalar and vector linear potentials and with mixed scalar and vector inversely linear potentials of Coulomb-type. For all these considered cases, the expressions of the eigenfunctions are determined and expressed in terms of the special functions: the Bessel functions of the first kind for the free case, the biconfluent Heun functions for the second case and the confluent hypergeometric functions for the end case, and the corresponding eigenvalues are exactly obtained.

Dirac–Kratzer problem with spin and pseudo-spin symmetries in curved space–time

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by [Formula: see text], where [Formula: see text] is a scalar potential, coupled to vector [Formula: see text] and tensor [Formula: see text] potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

Aharonov–Bohm effect in a Coulomb-type potential under the influence of a cutoff point

We analyze the influence of a cutoff point on a Coulomb-type potential that stems from the interaction of an electron with electric fields. This cutoff point establishes a forbidden region for the electron. Then, we search for bound state solutions to the Schrödinger equation. In addition, we consider a rotating reference frame. We show that the effects of rotation break the degeneracy of the energy levels. Further, we discuss the Aharonov–Bohm effect for bound states.

Linear Central Potential Effects Induced by Lorentz Symmetry Violation on Spin-0 Particle Under Coulomb-type scalar Potential

In this work, quantum dynamics of a spin-0 particle under the effects of Lorentz symmetry violation in the presence of Coulombtype non-electromagnetic potential $(S(r) ∝ \frac{1}{r})$ is investigated. The non-electromagnetic (or scalar) potential is introduced by modifying the mass term via transformation $M → M + \frac{η_c}{r}$ in the relativistic wave equation. The linear central potential induced by the Lorentz symmetry violation is a linear radial electric and constant magnetic field and, analyze the effects on the spectrum of energy and the wave function

Evolution of Strain Gradient Formulation in Capturing Localization Phenomena in Granular Materials

This paper highlights the use of incorporating strain gradient into flow stress to study localization behavior in materials. Pioneered by Zbib and Aifantis in the late 1980s, the formulation enabled incorporation of length scales into continuum formulations naturally. The formulation has also evolved into being able to study the effects of microstructure and heterogeneity on localization in granular materials. A multi-slip Mohr-Coulomb type plasticity model with the flow stress in the constitutive equation modified with a higher order gradient term of the effective plastic strain is used for this purpose. The possibility of abrupt changes of mobilized friction caused by intense shearing rate often leads to particle breakage. Its effects on localization is accounted for by modifying the material properties such as mobilized friction using a scaling parameter averaged over a representative elementary area. The change of shearing rate in the integration points was monitored through quasi-statistically measure parameter called inertia number. The inertia number was set to be all the time to consider quasi static less than l.0E-3. The formulation was implemented into a finite element code and used to simulate plane strain compression tests on dry sand. The model highlights effects of confining pressure, anisotropic microstructure, the non-coaxial angle between the direction of principal stress and principal plastic strain rate directions on shear band characteristics.

Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions

The equation of motion in ℝ d of n generalized point charges interacting via the s -dimensional Coulomb potential, which contains for d = 2 a constant magnetic field, is considered. Planar exact solutions of the equation are found if either negative n − 1 > 2 charges and their masses are equal or n = 3 and the charges are different. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the infinite time for the considered equation without a magnetic field are also obtained. A condition permitting the existence of such solutions is proposed.