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Orif O. Ibrogimov

Publications and source records attributed to Orif O. Ibrogimov.

13 recordsLinked to original sources
arXiv2020

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real-valued potentials. For $L^1$-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for $L^p$-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all $p\in[1,\infty)$. The sharpness of the results are demonstrated by means of explicit examples.

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arXiv2019

Spectral enclosures for non-self-adjoint discrete Schrödinger operators

We study location of eigenvalues of one-dimensional discrete Schrödinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two different spectral bounds are obtained. The method relies on the Birman-Schwinger principle and various techniques for estimations of the norm of the Birman-Schwinger operator.

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arXiv2019

Location of eigenvalues of non-self-adjoint discrete Dirac operators

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

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arXiv2019

Spectral analysis of the spin-boson Hamiltonian with two bosons for arbitrary coupling and bounded dispersion relation

We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling $α>0$ in the case when the dispersion relation of the free field is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of the coupling constant and the energy level of the atom (with respect to certain constants depending on the maximal and the minimal values of the boson energy) as well as the validity or the violation of the infrared regularity type conditions, the essential spectrum is either a single finite interval or a disjoint union of at most six finite intervals. The corresponding critical values of the coupling constant are determined explicitly and the asymptotic lengths of the possible gaps are given when $α$ approaches to the respective critical value. Under minimal smoothness and regularity conditions on the boson dispersion relation and the coupling function, we show that discrete eigenvalues can never accumulate at the edges of the two-particle branch. Moreover, we show the absence of the discrete eigenvalue accumulation at the edges of the three-particle branch in the infrared regular case.

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arXiv2018

Spectral analysis of the spin-boson Hamiltonian with two photons for arbitrary coupling

We study the spectrum of the spin-boson model with two photons in $\mathbb{R}^d$ for arbitrary coupling $α>0$. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under "minimal" regularity conditions on the photon dispersion and the coupling function.

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arXiv2018

On the spectrum of an operator in truncated Fock space

We study the spectrum of an operator matrix arising in the spectral analysis of the energy operator of the spin-boson model of radioactive decay with two bosons on the torus. An analytic description of the essential spectrum is established. Further, a criterion for the finiteness of the number of eigenvalues below the bottom of the essential spectrum is derived.

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arXiv2017

Essential spectrum of elliptic systems of pseudo-differential operators on $L^2(\mathbb{R}^N)\oplus L^2(\mathbb{R}^N)$

Inspired by a result of Wong (Commun. Partial Differ. Equ. 13(10):1209-1221, 1988), we establish an analytic description of the essential spectrum of non-self-adjoint mixed-order systems of pseudo-differential operators on $L^2(\mathbb{R}^N) \oplus L^2(\mathbb{R}^N)$ that are uniformly Douglis-Nirenberg elliptic with positive-order diagonal entries. We apply our result to a problem arising in the dynamics of falling liquid films.

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arXiv2017

Essential spectrum of non-self-adjoint singular matrix differential operators

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ induced by matrix differential expressions of the form \begin{align}\label{abstract:mdo} \left(\begin{array}{cc} τ_{11}(\,\cdot\,,D) & τ_{12}(\,\cdot\,,D)\\[3.5ex] τ_{21}(\,\cdot\,,D) & τ_{22}(\,\cdot\,,D) \end{array}\right), \end{align} where $τ_{11}$, $τ_{12}$, $τ_{21}$, $τ_{22}$ are respectively $m$-th, $n$-th, $k$-th and 0 order ordinary differential expressions with $m=n+k$ being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity.

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arXiv2015

Analysis of the essential spectrum of singular matrix differential operators

A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q & -\displaystyle{\frac{\rm d}{\rm d t}} b^* \! + c^* \\[2mm] \hspace{6mm} b \displaystyle{\frac{\rm d}{\rm d t}} + c & \hspace{4mm} D \end{pmatrix} \quad \text{in } \ L^2((α, β)) \oplus \bigl(L^2((α, β))\bigr)^n \label{mo} \end{align} singular at $β\in\mathbb R\cup\{\infty\}$ is given; the coefficient functions $p$, $q$ are scalar real-valued with $p>0$, $b$, $c$ are vector-valued, and $D$ is Hermitian matrix-valued. The so-called "singular part of the essential spectrum" $σ_{\rm ess}^{\rm \,s}(\mathcal A)$ is investigated systematically. Our main results include an explicit description of $σ_{\rm ess}^{\rm \,s}(\mathcal A)$, criteria for its absence and presence; an analysis of its topological structure and of the essential spectral radius. Our key tools are: the asymptotics of the leading coefficient $π(\cdot,λ)=p-b^*(D-λ)^{-1}b$ of the first Schur complement of $\mathcal A$, a scalar differential operator but non-linear in $λ$; the Nevanlinna behaviour in $λ$ of certain limits $t\!\nearrow\!β$ of functions formed out of the coefficients in $\mathcal A$. The efficacy of our results is demonstrated by several applications; in particular, we prove a conjecture on the essential spectrum of some symmetric stellar equilibrium models.

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arXiv2015

Functional Equations and the Cauchy Mean Value Theorem

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.

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