arXiv subjects

Artem Savchuk

Publications and source records attributed to Artem Savchuk.

6 recordsLinked to original sources

arXiv2019

Asymptotic formulas for fundamental system of solutions of high order ordinary differential equations with coefficients -- distributions

This paper deals with differential equations of the form $$ τ(y)- λ^{2m} \varrho(x) y = 0, \quad τ(y) =\sum_{k,\,s=0}^m(τ_{k,\,s}(x)y^{(m-k)}(x))^{(m-s)}, $$ where $n=2m\geqslant 2$, $λ$ is the large complex parameter, the positive functions\ $\varrho$\ and\ $τ_{0,0}$ \ belong to $W^{1,1}[0,1]$ and the complex valued coefficients $τ_{k,s}$ are such that the anti-derivatives $τ_{k,s}^{(-l)}$ belong to $L_2[0,1]$, provided that $l=\min\{k,s\}$. Here the anti-derivatives are understood in the sense of distributions. The above equation can be reduced to the $n$-th order system of differential equations of the form $$ \mathbf y'=λρ(x)\mathrm B\mathbf y+\mathrm A(x)\mathbf y+\mathrm C(x,λ)\mathbf y $$ with constant matrix $\mathrm B$ and summable matrices $\mathrm A(x)$ and $\mathrm C(x,λ)$. The first objective of the paper is obtain new results on asymptotic representation for the matrix of fundamental solutions of the last equation with respect to $λ\to\infty$ in certain sectors of the complex plane. The second objective is to apply the obtained results for analyzing the asymptotic representation of fundamental solutions of the first scalar equation with distribution coefficients.

math.SP↗

arXiv2017

Spectral properties of complex Airy operator on the semi-axis

We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2/dx^2+p(x)$ on the semi-axis $\mathbb R_+$ with a complex--valued potential $p(x)$. It is assumed that the potential $p = q \pm ir$ is such that the real functions $q$ and $r$ are subject the conditions $$ q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^α, \quad α>0, $$ where the constants $c, \ c_0\in \mathbb R$, $c_1>0$ and $\arg(\pm i+c) < 2απ/(2+α)$. For the case of the Airy operator $L_c=-d^2/dx^2+cx$, $c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $|\arg c|<2π/3$. Using another technique based on the asymptotic behavior of the Airy functions we prove that the completeness theorem for the operator $L_c$ remains valid, provided that $|\arg c|<5π/6$.

math.SP↗

arXiv2015

Recovering of a potential of Sturm-Liouville operator from a finite sets of eigenvalues and norming constants

It is well known that a potential $q$ of the Sturm-Liouville operator $Ly= -y" +q(x)y$ on the finite interval $[0, π]$ can be uniquely recovered by the spectrum $\{λ_k\}_1^\infty$ and norming constants $\{α_k\}_1^\infty$ of this operator with Dirichlet boundary conditions. Given potential $q$ belonging to Sobolev space $W^θ_2[0, π]$ with $θ> -1$ we associate its $2N$-approximation $q_N$ constructed by the final sets $\{λ_k\}_1^N$ and $\{α_k\}_1^N$. The main result claims that for $-1\leqslantτ<θ$ the estimate $\|q -q_N\|_τ\leqslant CN^{θ-τ}$ holds, where $\|\cdot\|_τ$ is the norm in $W^τ_2$ and the constant $C$ depends on $R$ but does not depend on $q$ if $\|q\|_θ\leqslant R$.

math.SP↗

arXiv2015

Riesz basicity with parentheses for Dirac system with summable potential

We deal with the Dirac operator $\mathcal L_{P,U}$ generated in the space $\mathbb H=(L_2[0,π])^2$ by differential expression \begin{gather*} \ell_P(\mathbf y)=B\mathbf y'+P\mathbf y,\quad B = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}, \qquad P(x) = \begin{pmatrix} p_1(x) & p_2(x) \\ p_3(x) & p_4(x) \end{pmatrix}, \qquad \mathbf y(x)=\begin{pmatrix}y_1(x)\\ y_2(x)\end{pmatrix}, \end{gather*} and regular boundary conditions $$ U(\mathbf y)=\begin{pmatrix}u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}\begin{pmatrix}y_1(0)\\ y_2(0)\end{pmatrix}+\begin{pmatrix}u_{13} & u_{14}\\ u_{23} & u_{24}\end{pmatrix}\begin{pmatrix}y_1(π)\\ y_2(π)\end{pmatrix}=0. $$ The entries of a matrix $P$ suppose to be summable on the segment $[0,π]$ complex-valued functions. It is proved, that the operator $\mathcal L_{P,U}$ has purely discrete spectrum $\{λ_n\}_{n\in\mathbb Z}$ and $λ_n=λ_n^0+o(1)$ as $|n|\to\infty$. Here $\{λ_n^0\}_{n\in\mathbb Z}$ be the spectrum of operator $\mathcal L_{0,U}$ with zero potential and the same boundary conditions. In case this boundary conditions are strictly regular the spectrum of $\mathcal L_{P,U}$ is asymptotically simple. In this case the system of eigen and associated functions of operator $\mathcal L_{P,U}$ forms Riesz basis in $\mathbb H$. In case of regular but not strictly regular boundary conditions all eigenvalues of the operator $\mathcal L_{0,U}$ have multiplicity equal to $2$. In this case we give full proof of Riesz basicity of corresponding two-dimensional root subspaces of the operator $\mathcal L_{0,U}$.

math.SP↗

arXiv2014

The Dirac Operator with Complex-Valued Summable Potential

The paper deals with the Dirac operator generated on the finite interval $[0,π]$ by the differential expression $-B\mathbf{y}'+Q(x)\mathbf{y}$, where $$ B=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad Q(x)=\begin{pmatrix}q_1(x)&q_2(x)\\q_3(x)&q_4(x)\end{pmatrix}, $$ and the entries $q_j(x)$ belong to~$L_p[0,π]$ for some $p\geqslant 1$. The classes of regular and strongly regular operators of this form are defined, depending on the boundary conditions. The asymptotic formulas for the eigenvalues and eigenfunctions of such operators are obtained with remainders depending on~$p$. It it is proved that the system of eigen and associated functions of a regular operator forms a Riesz basis with parentheses in the space~$(L_2[0,π])^2$ and the usual Riesz basis, provided that the operator is strongly regular.

math.SP↗

arXiv2010

On the eigenfunctions of Sturm--Liouville operators with potentials --- distributions

In this paper we study a Sturm--Liouville operator $Ly=-y"+q(x)y$ in the space $L_2[0,π]$ with Direchlet boundary conditions. Here the potential $q$ is a first order distribution: $q\in W_2^{-1}[0,π]$. Such operators were defined in our previous papers. Here we clerify two leading terms in asymptotic formulae for eigenfunctions of such operators and for functions of biorthogonal system.

math.SP↗