arXiv subjects

Volodymyr Molyboga

Publications and source records attributed to Volodymyr Molyboga.

12 recordsLinked to original sources

arXiv2026

Uniqueness problem for accretive Schrödinger operators with complex singular coefficients

This paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space $L^{2}(\mathbb{R})$. The coefficients of the expression are complex-valued and satisfy \begin{equation*} q=s+Q', \quad s \in L^1_{loc}\left(\mathbb{R}\right) \quad\text{and}\quad Q, r \in L^2_{loc}\left(\mathbb{R}\right), \end{equation*} where the derivative is understood in the sense of distributions. In particular, the potential $q$ can be a Radon measure on the line. With the help of specially selected quasi-derivatives, the expression $l$ is treated as a quasi-differential expression. The domains of the minimal $\mathrm{L}_{0}$ and maximal $\mathrm{L}$ operators associated with the expression $l$ in the space $L^{2}(\mathbb{R})$ are described. We find constructive conditions on the behaviour of $\mathrm{Im}\,r$ near $\pm \infty$ that guarantee that $\mathrm{L}_{0}=\mathrm{L}$ if the operator $\mathrm{L}_{0}$ is accretive. We prove that these conditions are sharp even in the class of differential operators with smooth real-valued coefficients. Examples are given to illustrate the main results of the paper.

math.SP↗

arXiv2021

Povzner-Wienholtz-type theorems for Sturm-Liouville operators with singular coefficients

We introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the coefficients. They are assumed to satisfy conditions \[q=Q'+s;\quad \frac{1}{\sqrt{|p|}}, \frac{Q}{\sqrt{|p|}}, \frac{r}{\sqrt{|p|}} \in L^2_{loc}\left(\mathbb{R}\right), \quad s \in L^1_{loc}\left(\mathbb{R}\right), \quad\frac{1}{p}\neq 0\,\,\text{a.e.,} \] where the derivative of the function $Q$ is understood in the sense of distributions, and all functions $p$, $Q$, $r$, $s$ are real-valued. In particular, the coefficients $q$ and $r'$ may be Radon measures on $\mathbb{R}$, while function $p$ may be discontinuous. The main result of the paper are constructive sufficient conditions on the coefficient $p$ which provide that the operator $L_0$ being semi-bounded implies it being self-adjoint.

math.SP↗

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↗

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

Estimates for Periodic Eigenvalues of the Differential Operator $\mathbf{(-1)^{m}d^{2m}/dx^{2m}+V}$ with V -- Distribution

The periodic eigenvalue problem for the differential operator $(-1)^{m}d^{2m}/dx^{2m}+V$ is studied for complex-valued distribution V in the Sobolev space $H^{-mα}_{per}[-1,1]\;(m\in\mathbb{N},\; 0\leqα<1)$. The following result is shown: The periodic spectrum consists of a sequence $(λ_{k})_{k\geq0}$ of complex eigenvalues satisfying the asymptotics (for any $\varepsilon>0$) $$ λ_{2n-1},λ_{2n}=n^{2m}π^{2m}+\hat{V}(0)\pm \sqrt{\hat{V}(-2n)\hat{V}(2n)}+o(n^{m(2α-1+\varepsilon)}), $$ where $\hat{V}(k)$ denote the Fourier coefficients of V.

math.FA↗

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↗

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↗

arXiv2009

Hill's potentials in Hörmander spaces and their spectral gaps

In the paper we study the behaviour of the lengths of spectral gaps $\{γ_{q}(n)\}_{n\in \mathbb{N}}$ 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 distribution potentials $$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2πx}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$ in dependence on the weight $ω$ of the Hörmander spaces $H^ω(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^ω(\mathbb{N})$ be a Hilbert space of weighted sequences. It is proved that $$ \{\hat{q}(\cdot)\}\in h^ω(\mathbb{N})\Leftrightarrow\{γ_{q}(\cdot)\}\in h^ω(\mathbb{N}) \leqno(\ast) $$ if a positive, in general non-monotonic, weight $ω=\{ω(k)\}_{k\in \mathbb{N}}$ is inter-power one. In the case $q\in L^{2}(\mathbb{T})$, and $ω(k)=(1+2k)^{s}$, $s\in \mathbb{Z}_{+}$, the statement $(\ast)$ is due to Marchenko and Ostrovskii (1975).

math.SP↗

arXiv2009

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

The behaviour of the lengths of spectral gaps $\{γ_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,φ}$, $s>-1$, $φ\in \mathrm{SV}$. The case $q(x)\in L_{1{-}per}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $φ\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.

math.SP↗