Searcharxiv⌕ Search

arXiv subjects

Alexander Sakhnovich

Publications and source records attributed to Alexander Sakhnovich.

At least 37 records · Page 2Linked to original sources
arXiv2019

The inverse approach to Dirac-type systems based on the $A$-function concept

The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schrödinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove that this integro-differential equation for A is uniquely solvable. We show how to recover the matrix-valued potential coefficient from A. This approach is also applicable to matrix-valued Schrödinger operators on a half-line.

math.SP↗
arXiv2018

Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions

We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.

math.SP↗
arXiv2017

GBDT of discrete skew-selfadjoint Dirac systems and explicit solutions of the corresponding non-stationary problems

Generalized Bäcklund-Darboux transformations (GBDTs) of discrete skew-selfadjoint Dirac systems have been successfully used for explicit solving of direct and inverse problems of Weyl-Titchmarsh theory. During explicit solving of the direct and inverse problems, we considered GBDTs of the trivial initial systems. However, GBDTs of arbitrary discrete skew-selfadjoint Dirac systems are important as well and we introduce these transformations in the present paper. The obtained results are applied to the construction of explicit solutions of the interesting related non-stationary systems.

math.CA↗
arXiv2017

Verblunsky-type coefficients for Dirac and canonical systems generated by Toeplitz and Hankel matrices, respectively

We introduce Verblunsky-type coefficients of Toeplitz and Hankel matrices, which correspond to the discrete Dirac and canonical systems generated by Toeplitz and Hankel matrices, respectively. We prove one to one correspondences between positive-definite Toeplitz (Hankel) matrices and their Verblunsky-type coefficients as analogs of the well-known Verblunsky's theorem. Several interconnections with the spectral theory are described as well.

math.SP↗
arXiv2017

Inversion of the convolution operators on a rectangular

Using simultaneously two operator identities, we consider the inversion of the convolution operators on a rectangular. The structure of the inverse operators and of some corresponding forms, which are important in signal processing, is derived.

math.CA↗
arXiv2016

On accelerants and their analogs, and on the characterization of the rectangular Weyl functions for Dirac systems with locally square-integrable potentials on a semi-axis

We characterize the set of rectangular Weyl matrix functions corresponding to Dirac systems with locally square-integrable potentials on a semi-axis and demonstrate a new way to recover the locally square-integrable potential from the Weyl function. Important interconnections between our approach and accelerants of convolution operators are discussed as well.

math.SP↗
arXiv2016

Dynamics of electrons and explicit solutions of Dirac-Weyl systems

Explicit solutions of Dirac-Weyl system, which is essential in graphene studies, are constructed using our recent approach to the construction of solutions of dynamical systems. The obtained classes of solutions are much wider than the ones, which were considered before. It is proved that neither constructed potentials nor corresponding solutions have singularities. Various examples are provided.

math-ph↗
arXiv2016

Hamiltonian systems and Sturm-Liouville equations: Darboux transformation and applications

We introduce GBDT version of Darboux transformation for symplectic and Hamiltonian systems as well as for Shin-Zettl systems and Sturm-Liouville equations. These are the first results on Darboux transformation for general-type Hamiltonian and for Shin-Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl-Titchmarsh functions and to the construction of explicit solutions of dynamical symplectic systems, of two-way diffusion equations and of indefinite Sturm-Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT.

math.CA↗
arXiv2016

Dynamical canonical systems and their explicit solutions

Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct Bäcklund-Darboux transformation and explicit solutions of the dynamical canonical systems. We study also those properties of the solutions, which are of interest in evolution and control theories.

math.DS↗
arXiv2015

Dynamical and spectral Dirac systems: response function and inverse problems

We establish simple connections between response functions of the dynamical Dirac systems and $A$-amplitudes and Weyl functions of the spectral Dirac systems. Using these connections we propose a new and rigorous procedure to recover a general-type dynamical Dirac system from its response function as well as a procedure to construct explicit solutions of this problem.

math.SP↗
arXiv2015

Evolution of Weyl functions and initial-boundary value problems

This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well.

math.SP↗
arXiv2015

Nonlinear Fokker-Planck equation: stability, distance and corresponding extremal problem in the spatially inhomogeneous case

We start with a global Maxwellian $M_{k}$, which is a stationary solution, with the constant total density ($ρ(t)\equiv \wt ρ$), of the Fokker-Planck equation. The notion of distance between the function $M_{k}$ and an arbitrary solution $f$ (with the same total density $\wt ρ$ at the fixed moment $t$) of the Fokker-Planck equation is introduced. In this way, we essentially generalize the important Kullback-Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.

math.AP↗