V. M. Molyboga

orcid.org/0000-0002-5683-5694
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Articles: 9

Spectral gaps of the Hill-Schrödinger operators with distributional potentials

Vladimir Mikhailets, Volodymyr Molyboga

↓ Abstract   |   Article (.pdf)

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

↓ Abstract   |   Article (.pdf)

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

↓ Abstract   |   Article (.pdf)

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.

Hill's potentials in Hörmander spaces and their spectral gaps

V. A. Mikhailets, V. M. Molyboga

↓ Abstract   |   Article (.pdf)

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 $\mathrm{OR}$.

Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials

Vladimir Mikhailets, Volodymyr Molyboga

↓ Abstract   |   Article (.pdf)

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

↓ Abstract   |   Article (.pdf)

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.

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

Estimates for periodic eigenvalues of the differential operator $(-1)^md^2m/dx^2m+V$ with $V$-distribution

Volodymyr Molyboga

Methods Funct. Anal. Topology 9 (2003), no. 2, 163-178


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