# V. A. Mikhailets

orcid.org/0000-0002-1332-1562

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### 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