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Andrii Khrabustovskyi

Publications and source records attributed to Andrii Khrabustovskyi.

At least 19 recordsLinked to original sources
arXiv2026

Homogenization and operator estimates for Steklov problems in perforated domains

Let the set $Ω_\varepsilon$ be obtained from the bounded domain $Ω$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $Ω_\varepsilon$ one considers the Steklov spectral problem. It is known from [Girouard-Henrot-Lagacé, ARMA (2021)] that, if the radii of the holes shrink at a critical rate such that the surface area of a single hole is comparable to the volume of a periodicity cell, then, in the limit $\varepsilon \to 0$, the Steklov spectrum converges to the spectrum of the problem $-Δu=λQ u$ on $Ω$ with some weight $Q>0$. In the present work, we extend this result by proving, under fairly general assumptions on the locations and shapes of the holes, convergence of the associated resolvent operators in the operator norm topology, together with quantitative estimates for the Hausdorff distance between the spectra. The underlying domain $Ω$ is not assumed to be bounded.

math.AP↗
arXiv2026

Periodic discrete graphs with prescribed spectrum

We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely $n$ gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to $n$ additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.

math.SP↗
arXiv2025

A geometric approximation of non-local interface and boundary conditions

We analyze an approximation of a Laplacian subject to non-local interface conditions of a $δ'$-type by Neumann Laplacians on a family of Riemannian manifolds with a sieve-like structure. We establish a (kind of) resolvent convergence for such operators, which in turn implies the convergence of spectra and eigenspaces, and demonstrate convergence of the corresponding semigroups. Moreover, we provide an explicit example of a manifold allowing to realize any prescribed integral kernel appearing in that interface conditions. Finally, we extend the discussion to similar approximations for the Laplacian with non-local Robin-type boundary conditions.

math.AP↗
arXiv2025

Quasinormal Ringing and Shadows of Black Holes and Wormholes in Dark Matter-Inspired Weyl Gravity

Weyl gravity naturally generates effective dark matter and cosmological constant terms as integration constants, eliminating the need to explicitly introduce them into the theory. Additionally, the framework permits three intriguing solutions for compact objects: an asymptotically de Sitter Schwarzschild-like black hole described by the Mannheim-Kazanas solution, a non-Schwarzschild black hole, and a traversable wormhole that exists without exotic matter. In this work, we investigate the quasinormal spectra of all three solutions. We demonstrate that when the mass of the black hole corresponding to the Mannheim-Kazanas solution approaches zero, the perturbation equations yield an exact solution expressible through hypergeometric functions. The quasinormal modes of black holes in Weyl gravity can be classified into three distinct branches: Schwarzschild-like modes modified by effective dark matter and cosmological terms, and modes associated with empty spacetime (de Sitter and dark matter branches), which are further influenced by the black hole mass. Previous studies have shown that the dark matter term induces a secondary stage of quasinormal ringing following the initial Schwarzschild phase. Here, we compute the frequencies using convergent methods and elucidate how this unique time-domain behavior translates into the frequency domain. Furthermore, we demonstrate that the non-Schwarzschild black hole can be distinguished from both the Schwarzschild-like solution and the wormhole through their distinct quasinormal spectra. We also compute shadow radii for black holes and wormholes within Weyl gravity, revealing that wormholes with large throat radii can produce significantly smaller shadows compared to black holes of equivalent mass.

gr-qc↗
arXiv2024

Quantum lattice transport along an infinitely extended perturbation

We consider a periodic quantum graph in the form of a rectangular lattice with the $δ$-coupling of strength $γ$ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a $\widetildeγ\neγ$. We analyze the band spectrum of the system and show that it remains preserved as a set provided $\widetildeγ>γ>0$ while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals $\frac12$.

math.SP↗
arXiv2024

Homogenization of the Dirac operator with position-dependent mass

We address the homogenization of the two-dimensional Dirac operator with position-dependent mass. The mass is piecewise constant and supported on small pairwise disjoint inclusions evenly distributed along an $\varepsilon$-periodic square lattice. Under rather general assumptions on geometry of these inclusions we prove that the corresponding family of Dirac operators converges as $\varepsilon\to 0$ in the norm resolvent sense to the Dirac operator with a constant effective mass provided the masses in the inclusions are adjusted to the scaling of the geometry. We also estimate the speed of this convergence in terms of the scaling rates.

math.AP↗
arXiv2024

The Neumann sieve problem revisited

Let $Ω\subset\mathbb{R}^n$ be a domain, $Γ$ be a hyperplane intersecting it. Let $\varepsilon>0$, and $Ω_\varepsilon=Ω\setminus\overline{Σ_\varepsilon}$, where $Σ_\varepsilon$ ("sieve") is an $\varepsilon$-neighbourhood of $Γ$ punctured by many narrow passages. When $\varepsilon\to0$, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed and appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann Laplacian on $Ω_\varepsilon$ converges in a strong resolvent sense to the Laplacian on $Ω\setminusΓ$ subject to the so-called $δ'$-conditions on $Γ$. We will refine this result by deriving estimates on the rate of convergence in terms of various operator norms, and providing the estimate for the distance between the spectra. The assumptions we impose on the passages are rather general. For $n=2$ the results of T. Del Vecchio are not complete, some cases remain as open problems, and in this work we will fill these gaps.

math.AP↗
arXiv2023

Spectrum of the Laplacian on a domain perturbed by small resonators

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an example of such a domain perturbation. Let $Ω$ be a (not {necessarily} bounded) domain in $\mathbb{R}^n$. We perturb it to $ Ω_\varepsilon=Ω\setminus \cup_{k=1}^m S_{k,\varepsilon},$ where $S_{k,\varepsilon}$ are closed surfaces with small suitably scaled holes (``windows'') through which the bounded domains enclosed by these surfaces (``resonators'') are connected to the outer domain. When $\varepsilon$ goes to zero, the resonators shrink to points. We prove that in the limit $\varepsilon\to 0$ the spectrum of the Laplacian on $Ω_\varepsilon$ with the Neumann boundary conditions on $S_{k,\varepsilon}$ and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on $Ω$ and the numbers $γ_k$, $k=1,\dots,m$, being equal $1/4$ times the limit of the ratio between the capacity of the $k$th window and the volume of the $k$th resonator. We obtain an estimate on the rate of this convergence with respect to the Hausdorff-type metrics. Also, an application of this result is presented: we construct an unbounded waveguide-like domain with inserted resonators such that the eigenvalues of the Laplacian on this domain lying below the essential spectrum threshold do coincide with prescribed numbers.

math.SP↗
arXiv2023

Creating and controlling band gaps in periodic media with small resonators

We investigate spectral properties of the Neumann Laplacian $\mathscr{A}_\varepsilon$ on a periodic unbounded domain $Ω_\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain $Ω_\varepsilon$ is obtained by removing from $\mathbb{R}^n$ $m\in\mathbb{N}$ families of $\varepsilon$-periodically distributed small resonators. We prove that the spectrum of $\mathscr{A}_\varepsilon$ has at least $m$ gaps. The first $m$ gaps converge as $\varepsilon\to 0$ to some intervals whose location and lengths can be controlled by a suitable choice the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.

math.SP↗
arXiv2022

Operator estimates for Neumann sieve problem

Let $Ω$ be a domain in $\mathbb{R}^n$, $Γ$ be a hyperplane intersecting $Ω$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $Γ\capΩ$; when $\varepsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\mathscr{A}_\varepsilon$ be the Neumann Laplacian in the perforated domain $Ω_\varepsilon=Ω\setminusΓ_\varepsilon$, where $Γ_\varepsilon=Γ\setminus (\cup_k D_{k,\varepsilon})$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, $\mathscr{A}_\varepsilon$ converges in the strong resolvent sense to the Laplacian on $Ω\setminusΓ$ subject to the so-called $δ'$-conditions on $Γ$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms; in the latter case a special corrector is required.

math.AP↗
arXiv2021

Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Let $\varepsilon>0$ be a small parameter. We consider the domain $Ω_\varepsilon:=Ω\setminus D_\varepsilon$, where $Ω$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $Δ_\varepsilon$ be the Laplace operator in $Ω_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+γ_\varepsilon u = 0$ with $γ_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on $d_\varepsilon$ and $γ_\varepsilon$, the operator $Δ_\varepsilon$ converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in $Ω$ and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.

math.AP↗
arXiv2021

Singular Schrödinger operators with prescribed spectral properties

The paper deals with singular Schrödinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }γ_k δ(\cdot-z_k),\quad γ_k\in\mathbb{R}, \end{gather*} in $\mathsf{L}^2(\ell_-,\ell_+)$, where $(\ell_-,\ell_+)$ is a bounded interval, and $ δ(\cdot-z_k)$ is the Dirac delta-function supported at $z_k\in (\ell_-,\ell_+)$. It will be shown that the interaction strengths $γ_k$ and the points $z_k$ can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets on a real line.

math.SP↗
arXiv2021

A geometric approximation of $δ$-interactions by Neumann Laplacians

We demonstrate how to approximate one-dimensional Schrödinger operators with $δ$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with "room-and-passage" geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense to the above singular Schrödinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.

math.SP↗
arXiv2020

Periodic quantum graphs with predefined spectral gaps

Let $Γ$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $Γ$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on its edges; here $\varepsilon>0$ is a small parameter. Let $m\in\mathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $σ(\mathcal{H}^\varepsilon)$ of $\mathcal{H}^\varepsilon$ has at least $m$ gaps as $\varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $σ(\mathcal{H}^\varepsilon)$ as $\varepsilon\to 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $\varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.

math.SP↗
arXiv2019

Construction of self-adjoint differential operators with prescribed spectral properties

In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

math.SP↗
arXiv2019

Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D.Krejcirik, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.

math.SP↗
arXiv2018

Retrieving effective material parameters of metamaterials characterized by nonlocal constitutive relations

The parameter retrieval is a procedure in which effective material properties are assigned to a given metamaterial. A widely used technique bases on the inversion of reflection and transmission from a metamaterial slab. Thus far, local constitutive relations have been frequently considered in this retrieval procedure to describe the metamaterial at the effective level. This, however, is insufficient. The retrieved local material properties frequently fail to predict reliably the optical response from the slab in situations that deviate from those that have been considered in the retrieval, e.g. when illuminating the slab at a different incidence angle. To significantly improve the situation, we describe here a parameter retrieval, also based on the inversion of reflection and transmission from a slab, that describes the metamaterial at the effective level with nonlocal constitutive relations. We retrieve the effective material parameters at the example of a fishnet metamaterial. We demonstrate that the nonlocal constitutive relation can describe the optical response much better than local constitutive relation would do. Our approach is widely applicable to a large class of metamaterials.

physics.optics↗