Viktor G. Kravchenko

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

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.

A Quaternionic Generalization of the Riccati Differential Equation

A quaternionic partial differential equation is shown to be a generalisation of the traditional Riccati equation and its relationship with the Schrodinger equation is established. Various approaches to the problem of finding particular solutions to this equation are explored, and the generalisations of two theorems of Euler on the Riccati equation, which correspond to this partial differential equation, are stated and proved.

Spectrum Problems for Singular Integral Operators with Carleman Shift

In this paper we are concerned with the complete spectral analysis for the operator = in the space Lp() ( denoting the unit circle), where is the characteristic function of some arc of , is the singular integral operator with Cauchy kernel and is a Carleman shift operator which satisfies the relations 2 = I and = ±, where the sign + or — is taken in dependence on whether is a shift operator on Lp() preserving or changing the orientation of . This includes the identification of the Fredholm and essential parts of the spectrum of the operator , the determination of the defect numbers of — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.

Factorization of Singular Integral Operators with a Carleman Shift via Factorization of Matrix Functions

This paper is devoted to singular integral operators with a linear fractional Carleman shift of arbitrary order preserving the orientation on the unit circle. The main goal is to obtain a special factorization of the operator with the help of a factorization of a related matrix function in a suitable algebra. This factorization allows us to characterize the kernel and the range of the operator under consideration, similarly to the case of singular integral operators without shift.

Factorization of Some Classes of Matrix Functions and the Resolvent of a Hankel Operator

The factorization of some classes of matrix-valued functions is obtained, which yields some new results for a special class of Hankel integral operators in L 2 + . For each of its elements, it is shown that the resolvent operator can be explicitly determined through a matrix factorization obtained by solving two non-homogeneous equations.

On some Nonlinear Equations Generated by Fueter Type Operators

. Let JH(C) be the set of complex quaternions, ii, the standart basic quaternions and, for q E 111(C), denote j = E3=1 qkik and 4 = qo j. In the present work some procedure of factorization with reference to the Fueter type equations (8 aD)u = 0 for u = u(t,x) where D = >, I ik3 is discussed.

Factorization Algorithm for Some Special Non-rational Matrix Functions

We construct an algorithm that allows us to determine an effective generalized factorization of a special class of matrix functions. We use the same algorithm to analyze the spectrum of a self-adjoint operator which is related to the obtained generalized factorization.

Factorization of Matrix Functions and the Resolvents of Certain Operators

The explicit factorization of matrix functions of the form

Zakharov-Shabat system and hyperbolic pseudoanalytic function theory

{A_\gamma }(b) = \left( {\begin{array}{*{20}{c}} e&b \\ {b*}&{b*b + \gamma e} \end{array}} \right),