V. A. Mikhailets
orcid.org/0000-0002-1332-1562
Search this author in Google Scholar
Schrödinger operators with measure-valued potentials: semiboundedness and spectrum
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 24 (2018), no. 3, 240-254
We study 1-D Schrödinger operators in the Hilbert space $L^{2}(\mathbb{R})$ a 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 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.
Localization principles for Schrödinger operator with a singular matrix potential
Vladimir Mikhailets, Aleksandr Murach, Viktor Novikov
Methods Funct. Anal. Topology 23 (2017), no. 4, 367-377
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.
A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems
Vladimir Mikhailets, Aleksandr Murach, Vitalii Soldatov
Methods Funct. Anal. Topology 22 (2016), no. 4, 375-386
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.
Spectral gaps of the Hill-Schrödinger operators with distributional potentials
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 20 (2014), no. 4, 321-327
The paper studies the Hill-Schrödinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences that arise as lengths of spectral gaps of these operators. The space $H^\omega$ coincides with the H\"{o}rmander space $H^{\omega}_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $\omega$ are power, then these spaces coincide with the Sobolev spaces. The functions $\omega$ may be nonmonotonic.
Remarks on Schrödinger operators with singular matrix potentials
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 19 (2013), no. 2, 161-167
In this paper, an 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.
Schrödinger operators with complex singular potentials
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 19 (2013), no. 1, 16-28
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 $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$. Particularly the class $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{{\operatorname{loc}}}^{-1}(\mathbb{R})$-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 $L_{{\operatorname{unif}}}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.
General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series
Vladimir A. Mikhailets, Aleksandr A. Murach
Methods Funct. Anal. Topology 17 (2011), no. 4, 330-340
We prove that the classical Menshov--Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces $L_{2}(X,d\mu;H)$ of vector-valued functions, where $(X,\mu)$ is an arbitrary measure space, and $H$ is a real or complex Hilbert space of an arbitrary dimension.
Hill's potentials in Hörmander spaces and their spectral gaps
V. A. Mikhailets, V. M. Molyboga
Methods Funct. Anal. Topology 17 (2011), no. 3, 235-243
The paper deals with the Hill-Schrödinger operators with singular periodic potentials in the space $H^{\omega}(\mathbb{T})\subset H^{-1}(\mathbb{T})$. The authors exactly describe the classes of sequences being the lengths of spectral gaps of these operators. The functions $\omega$ may be nonmonotonic. The space $H^{\omega}(\mathbb{T})$ coincides with the Hörmander space $H_{2}^{\omega}(\mathbb{T})$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ is in the Avakumovich class $\mathrm{OR}$.
Regularization of singular Sturm-Liouville equations
Andrii Goriunov, Vladimir Mikhailets
Methods Funct. Anal. Topology 16 (2010), no. 2, 120-130
The paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ with the coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1, $$ where the derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization, the corresponding operators are correctly defined as quasi-differentials. 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 canonical form.
Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 15 (2009), no. 1, 31-40
The behavior of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators $$ S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right), $$ with real-valued 1-periodic distributional potentials $q(x)\in H_{1\mbox{-}{\operatorname{per}}}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behavior as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1\mbox{-}{\operatorname{per}}}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$, corresponds to the Marchenko-Ostrovskii Theorem.
One-dimensional Schrödinger operators with singular periodic potentials
Vladimir Mikhailets, Volodymyr Molyboga
Methods Funct. Anal. Topology 14 (2008), no. 2, 184-200
We study the one-dimensional Schrödinger operators $$ S(q)u:=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the Hilbert space $L_{2}\left(\mathbb{R} \right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.
Interpolation with a function parameter and refined scale of spaces
Vladimir A. Mikhailets, Aleksandr A. Murach
Methods Funct. Anal. Topology 14 (2008), no. 1, 81-100
The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classical interpolation are proved. Some applications to the interpolation of isotropic Hörmander spaces over a closed manifold are given.
Uniform estimates for the semi-periodic eigenvalues of the singular differential operators
Volodymyr A. Mikhailets, Volodymyr M. Molyboga
Methods Funct. Anal. Topology 10 (2004), no. 4, 30-57
Singular eigenvalue problems on the circle
Volodymyr A. Mikhailets, Volodymyr M. Molyboga
Methods Funct. Anal. Topology 10 (2004), no. 3, 44-53
On abstract self-adjoint boundary conditions
Methods Funct. Anal. Topology 7 (2001), no. 2, 7-12