On a factorization of second order elliptic operators and applications

We show that given a nonvanishing particular solution of the equation (divpgrad+q)u=0 (1) the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers of the variable z). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(divpgrad+q) in the sense that any solution of (1) in a simply connected domain can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of . Finally we give a similar factorization of the operator (divpgrad+q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does

On the reduction of the multidimensional Schroedinger equation to a first order equation and its relation to the pseudoanalytic function theory

Given a particular solution of a one-dimensional stationary Schroedinger equation (SE) this equation of second order can be reduced to a first order linear differential equation. This is done with the aid of an auxiliary Riccati equation. We show that a similar fact is true in a multidimensional situation also. We consider the case of two or three independent variables. One particular solution of (SE) allows us to reduce this second order equation to a linear first order quaternionic differential equation. As in one-dimensional case this is done with the aid of an auxiliary Riccati equation. The resulting first order quaternionic equation is equivalent to the static Maxwell system. In the case of two independent variables it is the Vekua equation from theory of generalized analytic functions. We show that even in this case it is necessary to consider not complex valued functions only, solutions of the Vekua equation but complete quaternionic functions. Then the first order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of (SE) and the other can be considered as an auxiliary equation of a simpler structure. For the auxiliary equation we always have the corresponding Bers generating pair, the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of (SE). We obtain an analogue of the Cauchy integral theorem for solutions of (SE). For an ample class of potentials (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing to obtain an infinite sequence of solutions of (SE) from one known particular solution

On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure

On an inverse problem for anisotropic conductivity in the plane

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat \Omega$, we give an explicit procedure to find a unique domain $\Omega$, an isotropic conductivity $\sigma$ on $\Omega$ and the boundary values of a quasiconformal diffeomorphism $F:\hat \Omega \to \Omega$ which transforms $\hat \sigma$ into $\sigma$.Comment: 9 pages, no figur

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical "one-dimensional" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem

Spin relaxation in low-dimensional systems

We review some of the newest findings on the spin dynamics of carriers and excitons in GaAs/GaAlAs quantum wells. In intrinsic wells, where the optical properties are dominated by excitonic effects, we show that exciton-exciton interaction produces a breaking of the spin degeneracy in two-dimensional semiconductors. In doped wells, the two spin components of an optically created two-dimensional electron gas are well described by Fermi-Dirac distributions with a common temperature but different chemical potentials. The rate of the spin depolarization of the electron gas is found to be independent of the mean electron kinetic energy but accelerated by thermal spreading of the carriers.Comment: 1 PDF file, 13 eps figures, Proceedings of the 1998 International Workshop on Nanophysics and Electronics (NPE-98)- Lecce (Italy

Lattice distortions in a sawtooth chain with Heisenberg and Ising bonds

An exactly solvable model of the sawtooth chain with Ising and Heisenberg bonds and with coupling to lattice distortion for Heisenberg bonds is considered in the magnetic field. Using the direct transfer-matrix formalism an exact description of the thermodynamic functions is obtained. The ground state phase diagrams for all regions of parameters values containing phases corresponding to the magnetization plateaus at $M=0,1/4$ and 1/2 have been obtained. Exact formulas for bond distortions for various ground states are presented. A novel mechanism of magnetization plateau stabilization corresponding to $M=1/4$ state is reported.Comment: 16 pages, 12 figure

Integrable equations in nonlinear geometrical optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.Comment: 33 pages, 7 figure

Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section

Magnetic properties and concurrence for fluid 3He on kagome lattice

We present the results of magnetic properties and entanglement for kagome lattice using Heisenberg model with two-, and three-site exchange interactions in strong magnetic field. Kagome lattice correspond to the third layer of fluid 3He absorbed on the surface of graphite. The magnetic properties and concurrence as a measure of pairwise thermal entanglement are studied by means of variational mean-field like treatment based on Gibbs-Bogoliubov inequality. The system exhibits different magnetic behaviors, depending on the values of the exchange parameters (J2, J3). We have obtained the magnetization plateaus at low temperatures. The central theme of the paper is the comparing the entanglement and magnetic behavior for kagome lattice. We have found that in the antiferromagnetic region behaviour of the concurrence coincides with the magnetization one.Comment: Physics of Atomic Nuclei (accepted for publication) 201