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In this paper we consider solutions of the Dirac equation at ultra high energies. The study provides new insights including features overlooked thus far and also new ramifications.
We provide elementary and accurate numerical solutions to the differential-difference equation, which improves an explicit version of the linear sieve given by Nathanson.
The cosmological Friedmann equation for the universe filled with a scalar field is reduced to a system of two equations of the first order, one of which is an equation with separable variables. For the second equation the exact solutions…
The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of…
We study the distribution of solutions of equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$. We give new upper bounds for these solutions.
It has been shown recently that Dirac operators satisfying the Ginsparg-Wilson relation provide a solution of the chirality problem in QCD at finite lattice spacing. We discuss different ways to construct these operators and their…
In this paper we improve upon in terms of S the best known effective upper bounds for the solutions of S-unit equations and decomposable form equations.
One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the…
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a…
An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by…
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated…
Starting with the Dirac equation in the extreme Kerr metric we derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the class of smooth compactly supported functions.
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
It has recently been established [Naghshineh et al., IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the…
Exact boundary conditions at finite distance for the solutions of the time-dependent Schrodinger equation are derived. A numerical scheme based on Crank-Nicholson method is proposed to illustrate its applicability in several examples.
A symmetry reduction of the Dirac equation is shown to yield the system of ordinary differential equations whose integrability by quadratures is closely connected to the stationary mKdV hierarchy.
In this paper, we provide a procedure to solve the eigen solutions of Dirac equation with complicated potential approximately. At first, we solve the eigen solutions of a linear Dirac equation with complete eigen system, which approximately…