Network


Latest external collaboration on country level. Dive into details by clicking on the dots.

Hotspot


Dive into the research topics where Vladislav V. Kravchenko is active.

Publication


Featured researches published by Vladislav V. Kravchenko.


Complex Variables and Elliptic Equations | 2008

A representation for solutions of the Sturm-Liouville equation

Vladislav V. Kravchenko

A representation for the general solution of the equation (pu′)′ + qu = ω2 u in terms of a non-trivial solution of is given. This result is obtained with the aid of the theory of pseudoanalytic functions and their relationship to solutions of the stationary two-dimensional Schrödinger equation. The representation has a simple and easily verifiable form and lends itself to numerical computation. Its applications to spectral problems are discussed.


Journal of Physics A | 2004

On a quaternionic Maxwell equation for the time-dependent electromagnetic field in a chiral medium

Sergei M. Grudsky; Kira V. Khmelnytskaya; Vladislav V. Kravchenko

Maxwells equations for the time-dependent electromagnetic field in a homogeneous chiral medium are reduced to a single quaternionic equation. Its fundamental solution satisfying the causality principle is obtained which allows us to solve the time-dependent chiral Maxwell system with sources.


Journal of Physics A | 2005

New applications of pseudoanalytic function theory to the Dirac equation

Antonio Castañeda; Vladislav V. Kravchenko

In the present work, we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potential in a two-dimensional case and a pair of decoupled Vekua equations. In general, these Vekua equations are bicomplex. However, we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain nonrestrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general, this equation cannot be solved explicitly even if one looks for wavefunctions of one variable. Nevertheless, for such Dirac equation, we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers 1, z, z2, ... in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.


Journal of Physics A | 2006

On a factorization of second-order elliptic operators and applications

Vladislav V. Kravchenko

We show that given a nonvanishing particular solution of the equation the corresponding differential operator can be factorized into a product of two first-order operators. The factorization allows us to reduce the above equation to a first-order equation which in a two-dimensional case is a 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 positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z0)n, n = 0, 1, ...). This result means that under quite general conditions one can construct an infinite system of exact solutions of the above equation explicitly and, moreover, at least when p and q are real valued this system will be complete in ker(divp grad + q) in the sense that any solution of the above equation in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally, we give a similar factorization of the operator (divp grad + 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.


Journal of Mathematical Physics | 2011

Dispersion equation and eigenvalues for quantum wells using spectral parameter power series

Raúl Castillo-Pérez; Vladislav V. Kravchenko; Héctor Oviedo-Galdeano; Vladimir S. Rabinovich

We derive a dispersion equation for determining eigenvalues of inhomogeneous quantum wells in terms of spectral parameter power series and apply it for the numerical treatment of eigenvalue problems. The method is algorithmically simple and can be easily implemented using available routines of such environments for scientific computing as MATLAB.


Journal of Physics A | 2003

Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

Viktor G. Kravchenko; Vladislav V. Kravchenko

We show that an ample class of physically meaningful partial differential systems of first order such as the Dirac equation with different one-component potentials, static Maxwells system and the system describing the force-free magnetic fields are equivalent to a single quaternionic equation which in its turn reduces in general to a Schrodinger equation with quaternionic potential, and in some situations this last can be diagonalized. The rich variety of methods developed for different problems corresponding to the Schrodinger equation can be applied to the systems considered in the present work.


Journal of Physics A | 2005

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

Vladislav V. Kravchenko

Given a particular solution of a one-dimensional stationary Schrodinger equation this equation of second order can be reduced to a first-order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or three independent variables. One particular solution of the stationary Schrodinger equation allows us to reduce this second-order equation to a linear first-order quaternionic differential equation. As in the one-dimensional case this is done with the aid of an auxiliary quaternionic Riccati equation. The resulting first-order quaternionic equation is equivalent to the static Maxwell system and is closely related to the Dirac equation. In the case of two independent variables it is the well-known Vekua equation from theory of pseudoanalytic (or generalized analytic) functions. Nevertheless, we show that even in this case it is very useful to consider not only complex valued functions, solutions of the Vekua equation, but complete quaternionic functions. In this way the first-order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of the Schrodinger equation and the other one can be considered as an auxiliary equation of a simpler structure. Moreover for the auxiliary equation we always have the corresponding Bers generating pair (F, G), 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 the Schrodinger equation. Based on this fact we obtain an analogue of the Cauchy integral theorem for solutions of the stationary Schrodinger equation. Other results from theory of pseudoanalytic functions can be written for solutions of the Schrodinger equation. Moreover, for an ample class of potentials in the Schrodinger equation (which includes for instance all radial potentials), this new approach gives us a simple procedure allowing us to obtain an infinite sequence of solutions of the Schrodinger equation from one known particular solution.


Zeitschrift Fur Analysis Und Ihre Anwendungen | 2002

Quaternionic Reformulation of Maxwell Equations for Inhomogeneous Media and New Solutions

Vladislav V. Kravchenko

We propose a simple quaternionic reformulation of Maxwells equations for inhomogeneous media and use it in order to obtain new solutions in a static case.


Zeitschrift Fur Analysis Und Ihre Anwendungen | 1995

On a Biquaternionic Bag Model

Vladislav V. Kravchenko

There is considered a biquaternionic reformulation of a linear bag model describing the phenomenon of quarks confinement. The problem reduces to a boundary singular integral equation associated with the biquaternionic hyperholomorphic function theory which allows to study the invertibility and Fredholmness of the bag model.


Journal of Mathematical Analysis and Applications | 2012

Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane☆

Hugo M. Campos; Vladislav V. Kravchenko; Sergii M. Torba

An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {φk}k=0∞ such that Lφk=0 for k=0,1, Lφk=k(k−1)φk−2, for k=2,3,… and all φk satisfy certain prescribed initial conditions. We study the transmutation operators related to L in terms of the transformation of powers of the independent variable {(x−x0)k}k=0∞ to the elements of the L-basis and establish a precise form of the transmutation operator realizing this transformation. We use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary Schrodinger equation from a certain class. The system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result. Its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification. The obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory.

Collaboration


Dive into the Vladislav V. Kravchenko's collaboration.

Top Co-Authors

Avatar

Kira V. Khmelnytskaya

Instituto Politécnico Nacional

View shared research outputs
Top Co-Authors

Avatar

Héctor Oviedo

Instituto Politécnico Nacional

View shared research outputs
Top Co-Authors

Avatar
Top Co-Authors

Avatar

Raúl Castillo-Pérez

Instituto Politécnico Nacional

View shared research outputs
Top Co-Authors

Avatar
Top Co-Authors

Avatar
Top Co-Authors

Avatar

Vladimir S. Rabinovich

Instituto Politécnico Nacional

View shared research outputs
Top Co-Authors

Avatar

Raul Castillo Perez

Instituto Politécnico Nacional

View shared research outputs
Researchain Logo
Decentralizing Knowledge