Dual solutions Schrödinger type for Poisson equation in Dielectric and Magnetic linear media

Solutions are obtained for the dual form of the Schrödinger equation got from the transformation of Poisson equation for the vector and the scalar potential, in dielectric and magnetic materials, having into account homogeneous isotropic linear mechanisms.We study and apply these dual equation solutions in some specific potentials.

Some Bernstein processes similar to Cox–Ingersoll–Ross ones

We present some results on Bernstein processes, which are Brownian diffusions that appear in Euclidean Quantum Mechanics. We express the distributions of these processes with the help of those of Bessel processes. We then determine two solutions of the dual equation of the heat equation with potential. These results first appeared in the first author’s PhD thesis (Rouen, 2013).

DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

VORTICES AS INSTANTONS IN NONCOMMUTATIVE DISCRETE SPACE: USE OF Z2 COORDINATES

We show that vortices of Yang–Mills–Higgs model in R2 space can be regarded as instantons of Yang–Mills model in R2 × Z2 space. For this, we construct the noncommutative Z2 space by explicitly fixing the Z2 coordinates and then show, by using the Z2 coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation.

SELF-DUAL VORTICES IN THE FRACTIONAL QUANTUM HALL SYSTEM

Based on the ϕ-mapping theory, we obtain an exact Bogomol'nyi self-dual equation with a topological term, which is ignored in traditional self-dual equation, in the fractional quantum Hall system. It is revealed that there exist self-dual vortices in the system. We investigate the inner topological structure of the self-dual vortices and show that the topological charges of the vortices are quantized by Hopf indices and Brouwer degrees. Furthermore, we study the branch processes in detail. The vortices are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of the vector field ϕ.

TOPOLOGICAL STRUCTURE OF THE MANY VORTICES SOLUTION IN JACKIW–PI MODEL

By using the gauge potential decomposition, we discuss the self-dual equation and its solution in Jackiw–Pi model. We obtain a new concrete self-dual equation and find relationship between Chern–Simons vortices solution and topological number which is determined by Hopf indices and Brouwer degrees of Ψ-mapping. To show the meaning of topological number we give several figures with different topological numbers. In order to investigate the topological properties of many vortices, we use five parameters (two positions, one scale, one phase per vortex and one charge of each vortex) to describe each vortex in many vortices solutions in Jackiw–Pi model. For many vortices, we give three figures with different topological numbers to show the effect of the charge on the many vortices solutions. We also study the quantization of flux of those vortices related to the topological numbers in this case.