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Mikhail Kudryavtsev

Publications and source records attributed to Mikhail Kudryavtsev.

12 recordsLinked to original sources
arXiv2017

Inverse spectral analysis for a class of finite band symmetric matrices

In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a matrix in our class and give an algorithm for recovering this matrix from the spectral function. The reconstructive algorithm is applicable to matrices which cannot be treated by known inverse block matrix methods. Our approach to the inverse problem is based on the rational interpolation theory developed in a previous paper.

math-ph↗
arXiv2016

Inverse problems for Jacobi operators IV: Interior mass-spring perturbations of semi-infinite systems

This work gives results on the interplay of the spectra of two Jacobi operators corresponding to an infinite mass-spring system and a modification of it obtained by changing one mass and one spring of the system. It is shown that the system can be recovered from these two spectra. Necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one are provided.

math-ph↗
arXiv2015

Inverse spectral analysis for a class of infinite band symmetric matrices

This note deals with the direct and inverse spectral analysis for a class of infinite band symmetric matrices. This class corresponds to operators arising from difference quations with usual and inner boundary conditions. We give a characterization of the spectral functions for the operators and provide necessary and sufficient conditions for a matrix-valued function to be a spectral function of the operators. Additionally, we give an algorithm for recovering the matrix from the spectral function. The approach to the inverse problem is based on the rational interpolation theory.

math-ph↗
arXiv2015

On a linear interpolation problem for n-dimensional vector polynomials

This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear combination of them is satisfied for each one of the N interpolation nodes. The results of this work generalize previous results on the so-called rational interpolation and have applications to direct and inverse spectral analysis of band matrices.

math.CA↗
arXiv2013

Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems

Consider an infinite linear mass-spring system and a modification of it obtained by changing the first mass and spring of the system. We give results on the interplay of the spectra of such systems and on the reconstruction of the system from its spectrum and the one of the modified system. Furthermore, we provide necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one.

math-ph↗
arXiv2011

Inverse problems for Jacobi operators I: Interior mass-spring perturbations in finite systems

We consider a linear finite spring mass system which is perturbed by modifying one mass and adding one spring. From knowledge of the natural frequencies of the original and the perturbed systems we study when masses and springs can be reconstructed. This is a problem about rank two or rank three type perturbations of finite Jacobi matrices where we are able to describe quite explicitly the associated Green's functions. We give necessary and sufficient conditions for two given sets of points to be eigenvalues of the original and modified system respectively.

math.SP↗
arXiv2007

Rational interpolation and mixed inverse spectral problem for finite CMV matrices

For finite dimensional CMV matrices the mixed inverse spectral problem of reconstruction the matrix by its submatrix and a part of its spectrum is considered. A general rational interpolation problem which arises in solving the mixed inverse spectral problem is studied, and the description of the space of its solutions is given. We apply the developed technique to give sufficient conditions for the uniqueness of the solution of the mixed inverse spectral problem.

math.SP↗
arXiv2007

An inverse spectral theory for finite CMV matrices

For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV matrices with different "boundary conditions", and the problem of reconstructing the CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by truncation.

math.SP↗
arXiv2001

Resolution of the Cauchy problem for the Toda lattice with non-stabilized initial data

This paper is the continuation of the work "On an inverse problem for finite-difference operators of second order". We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with bounded elements, whose spectrum of multipliciy 2 is separated from its simple spectrum and contains an interval of asolutely continuous spectrum. Using the integral equation of the inverse problem for this matrix, obtained in the previous work, we solve the Cauchy problem for the Toda lattice with non-stabilized initial data.

math.SP↗
arXiv2001

On an inverse problem for finite-difference operators of second order

The Jacobi matrices with bounded elements whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum are considered. A new type of spectral data, which are analogous for scattering data, is introduced for this matrix. An integral equation that allows us to reconstruct the matrix from this spectral data is obtained. We use this equation to solve the Cauchy problem for the Toda lattice with the initial data that are not stabilized.

math.SP↗
arXiv2001

The Riemann problem with additional singularities

The Riemann problem is studied in the case when the unknown function has nonisolated singularities, concentrated on the real axis. The problem is used for the factorization of functions, holomorphic outside of the unit circle and the real axis, in the form of the product of two functions which have singulariries on the given set of the real axis.

math.SP↗