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Anton A. Kutsenko

Publications and source records attributed to Anton A. Kutsenko.

27 records · Page 2Linked to original sources
arXiv2016

Application of matrix-valued integral continued fractions to spectral problems on periodic graphs

We show that spectral problems for periodic operators on lattices with embedded defects of lower dimensions can be solved with the help of matrix-valued integral continued fractions. While these continued fractions are usual in the approximation theory, they are less known in the context of spectral problems. We show that the spectral points can be expressed as zeroes of determinants of the continued fractions. They are also useful in the study of inverse problems (one-to-one correspondence between spectral data and defects). Finally, the explicit formula for the resolvent in terms of the continued fractions is also provided. We apply some of our results to the Schrödinger operator acting on the graphene with line and point defects.

math.SP↗
arXiv2016

Recovery of defects from the information at detectors

The discrete wave equation in a multidimensional uniform space with local defects and sources is considered. The characterization of all possible defect configurations corresponding to given amplitudes of waves at the receivers (detectors) is provided.

math-ph↗
arXiv2016

Algebra of 2D periodic operators with local and perpendicular defects

We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations on some matrix-valued functions and few number of integrations, the discrete part is much more complicated.

math.SP↗
arXiv2015

Determinants and traces of multidimensional discrete periodic operators with defects

As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $$ {\mathcal A}{\bf u}={\bf A}_0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf A}_N\int_0^1dk_1...\int_0^1dk_N{\bf B}_N{\bf u},\ \ {\bf u}\in L^2([0,1]^N,\mathbb{C}^M), $$ where $({\bf A},{\bf B})(k_1,...,k_N)$ are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra ${\mathscr H}_{N,M}$. In this article we show that there exist a trace $\pmbτ$ and a determinant $\pmbπ$ defined for operators from ${\mathscr H}_{N,M}$ with the properties $$ \pmbτ(α{\mathcal A}+β{\mathcal B})=α\pmbτ({\mathcal A})+β\pmbτ({\mathcal B}),\ \ \pmbτ({\mathcal A}{\mathcal B})=\pmbτ({\mathcal B}{\mathcal A}),\ \ \pmbπ({\mathcal A}{\mathcal B})=\pmbπ({\mathcal A})\pmbπ({\mathcal B}),\ \ \pmbπ(e^{\mathcal A})=e^{\pmbτ({\mathcal A})}. $$ The mappings $\pmbπ$, $\pmbτ$ are vector-valued functions. While $\pmbπ$ has a complex structure, $\pmbτ$ is simple $$ \pmbτ({\mathcal A})=\left({\rm Tr}{\bf A}_0,\int_0^1dk_1{\rm Tr}{\bf B}_1{\bf A}_1,...,\int_0^1dk_1...\int_0^1dk_N{\rm Tr}{\bf B}_N{\bf A}_N\right). $$ There exists the norm under which the closure $\overline{\mathscr H}_{N,M}$ is a Banach algebra, and $\pmbπ$, $\pmbτ$ are continuous (analytic) mappings. This algebra contains simultaneously all operators of multiplication by matrix-valued functions and all operators from the trace class. Thus, it generalizes the other algebras for which determinants and traces was previously defined.

math-ph↗
arXiv2013

Periodic lattice with defects

The discrete periodic lattice of masses and springs with line and point defects is considered. The dispersion equations for propagative, guided and localised waves are obtained. The detailed analysis of example with three masses is provided.

math-ph↗
arXiv2012

On the measure of the spectrum of direct integrals

We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example: these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.

math.FA↗
arXiv2010

Zigzag nanoribbons in external electric and magnetic fields

We consider the Schrödinger operators on zigzag nanoribbons (quasi-1D tight-binding models) in external magnetic fields and an electric potential $V$. The magnetic field is perpendicular to the plane of the ribbon and the electric field is perpendicular to the axis of the nanoribbon and the magnetic field. If the magnetic and electric fields are absent, then the spectrum of the Schrödinger (Laplace) operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on the magnetic field, then the spectrum of the magnetic Schrödinger operator consists of some non-flat bands and one flat band between them. Thus the magnetic field changes the continuous spectrum but does not the flat band. If we switch on a weak electric potential $V\to 0$, then there are two cases: (1) the flat band splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed flat band remains the flat band. We describe all potentials when the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Moreover, we solve inverse spectral problems for small potentials.

math.FA↗