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Vladimir Mikhailets

Publications and source records attributed to Vladimir Mikhailets.

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

Parabolic problems in generalized Sobolev spaces

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $φ:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $φ$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.

math.AP↗
arXiv2018

Schrödinger operators with measure-valued potentials: semiboundedness and spectrum

We study the 1-D Schrödinger operators in Hilbert space $L^{2}(\mathbb{R})$ with real-valued Radon measure $q'(x)$, $q\in \mathrm{BV}_{loc}(\mathbb{R})$ as potentials. New sufficient conditions for minimal operators to be bounded below and selfadjoint are found. For such operators a criterion for the discreteness of the spectrum is proved, which generalizes Molchanov's, Brinck's, and the Albeverio-Kostenko-Malamud criteria. The quadratic forms corresponding to the investigated operators are described.

math.SP↗
arXiv2018

Sobolev-like Hilbert spaces induced by elliptic operators

We investigate properties of function spaces induced by the inner product Sobolev spaces $H^{s}(Ω)$ over a bounded Euclidean domain $Ω$ and by an elliptic differential operator $A$ on $\overlineΩ$. The domain and the coefficients of $A$ are of the class $C^{\infty}$. These spaces consist of all distributions $u\in H^{s}(Ω)$ such that $Au\in H^λ(Ω)$ and are endowed with the corresponding graph norm, with $s,λ\in\mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.

math.AP↗
arXiv2017

On Kiguradze theorem for linear boundary value problems

We investigate the limiting behavior of solutions of nonhomogeneous boundary value problems for the systems of linear ordinary differential equations. The generalization of Kiguradze theorem (1987) on passage to the limit is obtained.

math.CA↗
arXiv2017

Localization principles for Schrödinger operator with a singular matrix potential

We study the spectrum of the one-dimensional Schrödinger operator $H_0$ with a matrix singular distributional potential $q=Q'$ where $Q\in L^{2}_{\mathrm{loc}}(\mathbb{R},\mathbb{C}^{m})$. We obtain generalizations of Ismagilov's localization principles, which give necessary and sufficient conditions for the spectrum of $H_0$ to be bounded below and discrete.

math.AP↗
arXiv2017

Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces

We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.

math.CA↗
arXiv2017

A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems

We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}$. The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.

math.CA↗
arXiv2015

Spectral gaps of the Hill--Schrödinger operators with distributional potentials

The paper studies the Hill--Schrödinger operators with potentials in the space $H^ω\subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of these operators. The space $H^ω$ coincides with the Hörmander space $H^ω_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $ω(\sqrt{1+ξ^{2}})$ if $ω$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $ω$ are power, then these spaces coincide with the Sobolev spaces. The functions $ω$ may be nonmonotonic.

math.SP↗
arXiv2014

Singular eigenvalue problems on the circle

The eigenvalue problem on the circle for the non-self-adjoint operators $L_{m}(V)=(-1)^{m}\frac{d^{2m}}{dx^{2m}}+V$, $m\in \mathbb{N}$ with singular complex-valued 2-periodic distributions $V\in H_{per}^{-m}[-1,1]$ is studied. Asymptotic formulae for the eigenvalues uniformly in $V$ in the space $H_{per}^{m}[-1,1]$ and local uniformly in $V$ in the space $H_{per}^{-m}[-1,1]$ are found.

math.FA↗
arXiv2014

Uniform estimates for the semi-periodic eigenvalues of the singular differential operators

Let $m\in \mathbb{N}$, $α\in[0,1]$, and $V$ be a 1-periodic complex-valued distribution in the negative Sobolev space $H^{-mα}[0,1]$. The singular non-self-adjoint eigenvalue problem $D^{2m}u+V u=λu$, $D=-i d/dx$, with semi-periodic boundary conditions is investigated. The uniform in $V$ asymptotic and non-asymptotic eigenvalue estimates are found and proved. The case of periodic boundary conditions was studied by authors earlier.

math.FA↗
arXiv2013

Schrödinger operators with complex singular potentials

We study one-dimensional Schrödinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class $H_{unif}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{loc}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{unif}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.

math.SP↗
arXiv2013

Remarks on Schrödinger operators with singular matrix potentials

In this paper the asymmetric generalization of the Glazman-Povzner-Wienholtz theorem is proved for one-dimensional Schrödinger operators with strongly singular matrix potentials from the space $H_{loc}^{-1}(\mathbb{R}, \mathbb{C}^{m\times m})$. This result is new in the scalar case as well.

math.SP↗
arXiv2013

Formally self-adjoint quasi-differential operators and boundary value problems

We develop the machinery of boundary triplets for one-dimensional operators generated by formally self-adjoint quasi-differential expression of arbitrary order on a finite interval. The technique are then used to describe all maximal dissipative, accumulative and self-adjoint extensions of the associated minimal operator and its generalized resolvents in terms of the boundary conditions. Some specific classes are considered in greater detail.

math.FA↗
arXiv2011

Continuous parameter dependence for solutions of one-dimensional boundary value problems

We investigate continuous parameter dependence for solutions of general boundary value problems for ordinary linear differential systems. The generalization of Kiguradze's theorem (1987) on correctness of such problems was obtained. Also sufficient conditions for the Green matrices of such problems to converge uniformly to the Green matrix of the limiting problem were found.

math.AP↗
arXiv2011

Regularization of binomial differential equations with singular coefficients

We propose a regularization of the formal differential expression of order $m \geqslant 3$ $$ l(y) = i^my^{(m)}(t) + q(t)y(t), \,t \in (a, b), $$ applying quasi-derivatives. The distribution coefficient $q$ is supposed to have an antiderivative $Q \in L([a,b];\mathbb{C})$. For the symmetric case ($Q = \bar{Q}$) self-adjoint and maximal dissipative extensions of the minimal operator and its generalized resolvents are described. The resolvent approximation with resrect to the norm of the considered operators is also investigated. The case $m = 2$ for $Q \in L_2([a, b];\mathbb{C})$ was investigated earlier.

math.FA↗
arXiv2010

Regularization of singular Sturm-Liouville equations

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization corresponding operators are correctly defined as quasi-differential. Their resolvent approximation is investigated and all self-adjoint and maximal dissipative extensions and generalized resolvents are described in terms of homogeneous boundary conditions of the canonic form. Some results are new for the case $p(t)\equiv 1$ as well.

math.FA↗
arXiv2010

Smoothness of Hill's potential and lengths of spectral gaps

Let ${γ_q(n)}_{n \in \mathbb{N}}$ be the lengths of spectral gaps in a continuous spectrum of the Hill-Schrödinger operators S(q)u=-u''+q(x)u,\quad x\in \mathbb{R}, with 1-periodic real-valued potentials $q \in L^{2}(\mathbb{T})$. Let weight function $ω:\;[1,\infty)\to (0,\infty)$. We prove that under the condition \exists s\in [0,\infty):\quad k^{s}\llω(k)\ll k^{s+1},\; k\in \mathbb{N}, the map $γ:\, q \mapsto \{γ_{q}(n)}_{n \in \mathbb{N}}$ satisfies the equalities: \verb"i")\quad γ(H^ω) = h_{+}^ω, \verb"ii")\quad γ^{-1}(h_{+}^ω) = H^ω, where the real function space H^ω & ={f=\sum_{k\in \mathbb{Z}}\hat{f}\,(k)e^{i k2πx}\in L^{2}(\mathbb{T})| \sum_{k\in \mathbb{N}} ω^{2}(k)|\hat{f}(k)|^{2}<\infty,\; \hat{f}(k)=\bar{\hat{f}(-k)},\;k\in \mathbb{Z}.}, and h^ω = {a=\{a(k)\}_{k\in \mathbb{N}}|\sum_{k\in \mathbb{N}}ω^{2}(k)|a(k)|^{2}<\infty.}, h_{+}^ω = {a=\{a(k)\}_{k\in \mathbb{N}}\in h^ω| a(k)\geq 0.}. If the weight $ω$ is such that \exists a>1,c>1:\qquad c^{-1}\leq \frac{ω(λt)}{ω(t)}\leq c\quad\forall t\geq 1,\;λ\in [1,a] then the function class $H^ω$ is a real Hörmander space $H_{2}^ω(\mathbb{T},\mathbb{R})$ with the weight $ω(\sqrt{1+ξ^{2}})$.

math.SP↗