Non-singular Potential Research Articles

Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.

Gravastars with Kuchowicz Metric in Energy-Momentum Squared Gravity

This paper investigates the geometry of a gravitational vacuum star (also known as a gravastar) from the perspective of f(R,T2) gravity. The gravastar can be treated as a black hole substitute with three domains: (i) the inner domain, (ii) the intrinsic shell, and (iii) the outer domain. We examine these geometries using Kuchowicz ansatz for temporal metric function corresponding to a specific f(R,T2) model. We compute a nonsingular radial metric potential for both the interior and intermediate domains. The matching of these domains with exterior Schwarzschild vacuum results in boundary conditions that assist in the evaluation of unknown constants. Finally, we examine various attributes of gravastar domains, such as the equation of state parameter, proper length, energy, and surface redshift. We conclude that the gravastar model is a viable alternative to the black hole in the background of this gravity.

Self-gravitating anisotropic star using gravitational decoupling

This paper addresses a new gravitationally decoupled anisotropic solution for the compact star model via the minimal geometric deformation (MGD) approach. We consider a non-singular well-behaved gravitational potential corresponding to the radial component of the seed spacetime and embedding Class I condition that determines the temporal metric function to solve the seed system completely. However, two different well-known mimic approaches such as and have been employed to determine the deformation function which gives the solution of the second system corresponding to the extra source. In order to test the physical viability of the solution, we have checked several conditions such as regularity conditions, energy conditions, causality conditions, hydrostatic equilibrium, etc. Moreover, the stability of the solutions has been also discussed by the adiabatic index and its critical value. We find that the solutions set seems viable as far as observational data are concerned.

Discrete phase space, relativistic quantum electrodynamics, and a non-singular Coulomb potential

This paper deals with the relativistic, quantized electromagnetic and Dirac field equations in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. In the consequent relativistic quantum electrodynamics, the corresponding Feynman diagrams and [Formula: see text]-matrix elements are derived. In the special case of electron–electron scattering (Møller scattering), the explicit second-order element [Formula: see text] is deduced. Moreover, assuming the slow motions for two external electrons, the approximation of [Formula: see text] yields a divergence-free Coulomb potential.

Revealing infinite derivative gravity’s true potential: The weak-field limit around de Sitter backgrounds

General Relativity is known to produce singularities in the potential generated by a point source. Our universe can be modelled as a de Sitter (dS) metric and we show that ghost-free Infinite Derivative Gravity (IDG) produces a non-singular potential around a dS background, while returning to the GR prediction at large distances. We also show that although there are an apparently infinite number of coefficients in the theory, only a finite number actually affect the predictions. By writing the linearised equations of motion in a simplified form, we find that at distances below the Hubble length scale, the difference between the IDG potential around a flat background and around a de Sitter background is negligible.

Newtonian potential and geodesic completeness in infinite derivative gravity

Recent study has shown that a non-singular oscillating potential, a feature of Infinite Derivative Gravity (IDG) theories, matches current experimental data better than the standard GR potential. In this work we show that this non-singular oscillating potential can be given by a wider class of theories which allows the defocusing of null rays, and therefore geodesic completeness. We consolidate the conditions whereby null geodesic congruences may be made past-complete, via the Raychaudhuri Equation, with the requirement of a non-singular Newtonian potential in an IDG theory. In so doing, we examine a class of Newtonian potentials characterised by an additional degree of freedom in the scalar propagator, which returns the familiar potential of General Relativity at large distances.

Semirelativistic Bound-State Equations: Trivial Considerations

Observing renewed interest in long-standing (semi-) relativistic descriptions of bound states, we would like to make a few comments on the eigenvalue problem posed by the spinless Salpeter equation and, illustrated by the examples of the nonsingular Woods-Saxon potential and the singular Hulth\'en potential, recall elementary tools that practitioners looking for analytic albeit approximate solutions might find useful in their quest.

Brane world regularization of point particle classical self-energy

Physical effects in brane worlds models emerge by the incorporation of field modes coming from extra dimensions with the usual four dimensional ones. Such effects can be tested with well established experiments to set bounds on the parameters of the brane models. In this work we extend a previous result which gave finite electromagnetic potentials and self energies for a source looking pointlike to an observer sitting in a 4D Minkowski subspace of a single brane of a Randall-Sundrum spacetime including compact dimensions, and along which the source stretches uniformly. We show that a scalar particle produces a nonsingular static potential, possess a finite self-energy and that technically its analysis is very similar to the electrostatic case. As for the latter, we use the deviations from the Coulomb potential to set bounds on the anti de Sitter radius of the brane model on the basis of two experiments, namely, one of the Cavendish type and other being the scattering of electrons by Helium atoms. We found these are less stringent than others previously obtained using the Lamb shift in Hydrogen.

Geometric efficiency for a circular detector and a linear source of arbitrary orientation and position

A new axisymmetric radiation vector potential which is singular along its entire axis of symmetry is derived for a spherically symmetric point radiation source. This potential and a previously given non-singular point source potential are integrated to give radiation vector potentials for a straight linear source of constant strength. Analytical solutions are given for the geometric efficiency G of a line source and a circular disk detector when the line source is parallel to the detector axis. The analytical solution is also given for the case where the line source is parallel to the disk surface, such that the source axis and the detector axis intersect. All other cases are given as simple one-dimensional trigonometric integrals. Numerical results for G are given for all cases considered, and results given previously by Pommé for a line source parallel to the detector plane have been verified. The methods and vector potentials presented here can be adapted for calculations with many different geometries, and many results are applicable in other fields such as electromagnetism, gravity and fluid mechanics.

Poynting–Robertson effect in non-singular gravitational potential

We use a non-singular potential that appears in the literature under the influence of which the Poynting- Robertson effect is studied. For that, dust particles originat- ing within the asteroid belt are used, in circular and elliptic orbits, and expressions for the semimajor axis as a function of time are obtained. The derived expressions are written in terms of the two basic dust particle parameters, namely the density and the diameter. In both cases, we obtain expres- sions for the time that the dust particles take to reach the orbit of Earth under the action of the non-singular poten- tial and solar radiation. For the non-singular potential, dust particles of diameter 10 −3 m in circular and elliptical orbits require times of the order of 4.058 × 10 7 and 2.823 × 10 7 y to reach the orbit of the Earth respectively. Finally, the de- rived expressions and numerical results are compared with those of the Newtonian potential.

Satellite motion in a non-singular gravitational potential

We study the effects of a non-singular gravitational potential on satellite orbits by deriving the corresponding time rates of change of its orbital elements. This is achieved by expanding the non-singular potential into power series up to second order. This series contains three terms, the first been the Newtonian potential and the other two, here R 1 (first order term) and R 2 (second order term), express deviations of the singular potential from the Newtonian. These deviations from the Newtonian potential are taken as disturbing potential terms in the Lagrange planetary equations that provide the time rates of change of the orbital elements of a satellite in a non-singular gravitational field. We split these effects into secular, low and high frequency components and we evaluate them numerically using the low Earth orbiting mission Gravity Recovery and Climate Experiment (GRACE). We show that the secular effect of the second-order disturbing term R 2 on the perigee and the mean anomaly are 4″.307×10−9/a, and −2″.533×10−15/a, respectively. These effects are far too small and most likely cannot easily be observed with today’s technology. Numerical evaluation of the low and high frequency effects of the disturbing term R 2 on low Earth orbiters like GRACE are very small and undetectable by current observational means.

Solutions to the 1d Klein–Gordon equation with cut-off Coulomb potentials

Abstract In a recent paper by Barton [G. Barton, J. Phys. A: Math. Gen. 40 (2007) 1011], the 1-dimensional Klein–Gordon equation was solved analytically for the non-singular Coulomb-like potential V 1 ( | x | ) = − α / ( | x | + a ) . In the present Letter, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cut-off Coulomb potential V 2 ( | x | ) = − α / | x | , | x | > a , and otherwise V 2 ( | x | ) = − α / a .

Propagator for finite range potentials

The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.

Bound states by a pseudoscalar Coulomb potential in one-plus-one dimensions

The Dirac equation is solved for a pseudoscalar Coulomb potential in a two-dimensional world. An infinite sequence of bounded solutions are obtained. These results are in sharp contrast with those ones obtained in 3+1 dimensions where no bound-state solutions are found. Next the general two-dimensional problem for pseudoscalar power-law potentials is addressed consenting us to conclude that a nonsingular potential leads to bounded solutions. The behaviour of the upper and lower components of the Dirac spinor for a confining linear potential nonconserving- as well as conserving-parity, even if the potential is unbounded from below, is discussed in some detail.

A non-singular potential for the Dirac monopole

We propose a new vector potential for the Abelian magnetic monopole. The potential is non-singular in the entire region around the monopole. We criticize Wu and Yang's derivation of the Dirac quantization condition, and argue how the condition can be derived for any choice of potential. Unlike the Wu–Yang case where two patches are needed to define the full potential, our potential needs only a single patch.

Semiclassical time-dependent propagation in three dimensions for a Coulomb potential

A unified semiclassical time propagator is used to calculate the semiclassical time-correlation function in three cartesian dimensions for a particle moving in an attractive Coulomb potential. It is demonstrated that under these conditions the singularity of the potential does not cause any difficulties and the Coulomb interaction can be treated as any other non-singular potential. Moreover, by virtue of our three-dimensional calculation, we can explain the discrepancies between previous semiclassical and quantum results obtained for the one-dimensional radial Coulomb problem.

New non-singular description of the Abelian monopole

A new Abelian non-singular monopole potential is constructed using Banderet's ansatz and a new way of covering the space R 3−{0} by two sets.

Green's function for the Dirac equation with nonsingular central potential and complex energy. Estimation algorithm

Calculation of the electron radial Green's function leads to solution of a system of ordinary differential equations for three functions: F, G, which represent the regular solution of the Dirac equation for r → 0, and W− (the antiWronskian) —a function regular everywhere: W− = 1 for r = 0; W− ≈ 1/r as r →∞. An algorithm is formulated for calculating boundary values of W− as r → 0 and for estimation purposes at asymptotically large values of r.

Collapsing systems.

The stability problem posed in statistical mechanics by self-gravitating systems is discussed for the simpler case of systems with a purely attractive nonsingular pair potential of short range. Molecular-dynamics simulations of such systems are reported and are found to agree remarkably well with a modified version of the Hertel-Thirring cell model. A first-order phase transition is observed between a homogeneous phase at high energies and a collapsing phase with a single, very compact cluster in a diluted homogeneous background at low energies. The latter is not an extensive or thermodynamic phase. According to the cell model, really large systems are most likely to be found in a critical state hesitating between these two phases.

Nonsingular quarkonium potential.

It is shown that the appearance of singular terms in the quasistatic potential models of quarkonium can be avoided with the use of an improved quasistatic approximation, which yields a nonsingular quark-antiquark potential.