Related papers: On integrable rational potentials of the Dirac equ…
We establish soliton-like asymptotics for finite energy solutions to the Dirac equation coupled to a relativistic particle. Any solution with initial state close to the solitary manifold, converges in long time limit to a sum of traveling…
Spectrum of the Dirac Equation is obtained algebraically for arbitrary combination of Lorentz-scalar and Lorentz-vector Coulomb potentials using the Witten's Superalgebra approach. The result coincides with that, known from the explicit…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand-Levitan equation and the Fourier-Legendre series expansion of the transmutation kernel. A linear…
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…
By using two-component approach to the one-dimensional effective mass Dirac equation bound states are investigated under the effect of two new non-PT-symmetric, and non-Hermitian, exponential type potentials. It is observed that the Dirac…
Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of two different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The…
An elementary treatment of the Dirac Equation in the presence of a three-dimensional spherically symmetric $\delta (r-r_0)$-potential is presented. We show how to handle the matching conditions in the configuration space, and discuss the…
A general Riccati equation is integrated in quadratures in case one of its coefficients is an arbitrary function and two others are expressed through it.
We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal…
In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a…
The model of a classical particle with the weak linear AAD potential is subjected to path integral quantization. The light cone constraints and peculiar properies of its internal variables permit to use in calculations commutative dynamics…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
Solutions of the one dimensional Dirac equation with piece-wise constant potentials are presented using standard methods. These solutions show that the Klein Paradox is non-existent and represents a failure to correctly match solutions…
The usual method of solving the free particle Dirac equation results in the so called continuum energy solutions. Here, we take a different approach and find a set of solutions with quantized energies which are proportional to the total…
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal…
A new class of integrable theories of 0+1 and 1+1 dimensional dilaton gravity coupled to any number of scalar fields is introduced. These models are reducible to systems of independent Liouville equations whose solutions must satisfy the…
We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $-1$ which are real analytic on $\mathbb{R}^2\setminus \{0\}$. In the more general case when $V$ is only meromorphic on an open set of an…
It is argued from geometrical, group-theoretical and physical points of view that in the framework of QCD it is not only necessary but also possible to modify the Dirac equation so that correspondence principle holds valid. The Dirac wave…
We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space…