Related papers: On solvable Dirac equation with polynomial potenti…
Considering a four dimensional parallelisable manifold, we develop a concept of Dirac-type tensor equations with wave functions that belong to left ideals of the set of nonhomogeneous complex valued differential forms.
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
The Dirac oscillator coupled to an external two-component field can retain its solvability, if couplings are appropriately chosen. This provides a new class of integrable systems. A simplified way of solution is given, by recasting the…
We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated…
Classes of relativistic symmetries accommodating supersymmetric patterns are considered for the Dirac Hamiltonian with axially-deformed scalar and vector potentials.
In this note a simple extension of the complex algebra to higher dimension is proposed. Using the postulated algebra a two dimensional Dirac equation is formulated and its solution is calculated. It is found that there is a sub-algebra…
We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions…
An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation which can be solved by iterative procedure to find the wave functions is…
The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is discussed. For weak magnetic fields, the approximate energy values are obtained by semiclassical method. In the case with strong…
In this paper, the quaternionic Dirac equation is solved for quaternionic potentials, iV0+jW0. The study shows two different solutions. The first solution contains particles and anti-particles and leads to the diffusion, tunneling and Klein…
We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac…
We introduce a Dirac equation which reproduces the usual radial sextic oscillator potential in the non-relativistic limit. We determine its energy spectrum in the presence of the magnetic field. It is shown that the equation is solved in…
We consider the nonlinear Dirac equations in one dimension and review various results on global existence of solutions in H1. Depending on the character of the nonlinear terms, existence of the large-norm solutions can be extended for all…
We prove the existence of a family of non-trivial solutions of the Liouville equation in dimensions two and four with infinite volume. These solutions are perturbations of a finite-volume solution of the same equation in one dimension less.…
We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support a tridiagonal matrix representation of the wave operator. Doing so results in exactly solvable problems with a…
We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of…
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solution by means of the rotation number. We then give a global bifurcation result for a planar nonlinear…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthen potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the…