A. Khrabustovskyi

activity

20102026

most citedBeyond local effective material properties for metamaterials

33 citations · 146 across the 30 of their papers we have counts for

collaborators

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23 papers · 1 filter

math.SP2026

Periodic discrete graphs with prescribed spectrum

Andrii Khrabustovskyi, Anna Muranova

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 endpoint…

math.SP2024

Quantum lattice transport along an infinitely extended perturbation

Marzieh Baradaran, Pavel Exner, Andrii Khrabustovskyi

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 stra…

math.SP2023

Creating and controlling band gaps in periodic media with small resonators

Andrii Khrabustovskyi, Evgen Khruslov

We investigate spectral properties of the Neumann Laplacian $A_{ε}$ on a periodic unbounded domain $Ω_{ε}$ depending on a small parameter $ε > 0$ .…

math.SP2022

Spectrum of the Laplacian on a domain perturbed by small resonators

Giuseppe Cardone, Andrii Khrabustovskyi

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 s…

math.SP2021

Singular Schrödinger operators with prescribed spectral properties

Jussi Behrndt, Andrii Khrabustovskyi

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}…

math.SP2021

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

Andrii Khrabustovskyi, Olaf Post

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…

math.SP2020

Periodic quantum graphs with predefined spectral gaps

Andrii Khrabustovskyi

Let $Γ$ be an arbitrary $Z^{n}$ -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $H_{ε}$ on $Γ$ with the action $-\v…

math.SP2019★ 2 cited

Construction of self-adjoint differential operators with prescribed spectral properties

Jussi Behrndt, Andrii Khrabustovskyi

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 con…

math.SP2018

$δ^{'}$ -interaction as a limit of a thin Neumann waveguide with transversal window

Giuseppe Cardone, Andrii Khrabustovskyi

We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order $\varepsil…

math.SP2018

Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

Diana Barseghyan, Andrii Khrabustovskyi

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…

math.SP2018★ 6 cited

Gap control by singular Schrödinger operators in a periodically structured metamaterial

Pavel Exner, Andrii Khrabustovskyi

We consider a family ${H^{ε}}_{ε > 0}$ of $ε Z^{n}$ -periodic Schrödinger operators with $δ^{'}$ -interactions supported on a lattice of clos…

math.SP2017

Operator estimates for the crushed ice problem

Andrii Khrabustovskyi, Olaf Post

Let $Δ_{Ω_{ε}}$ be the Dirichlet Laplacian in the domain $Ω_{ε} := Ω ∖ (\cup_{i} D_{i ε})$ . Here $Ω \subset R^{n}$ and $\{D_{i \varepsil…

math.SP2016

Spectrum of a singularly perturbed periodic thin waveguide

Giuseppe Cardone, Andrii Khrabustovskyi

We consider a family ${Ω^{ε}}_{ε > 0}$ of periodic domains in $R^{2}$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-Δ_…

math.SP2016★ 1 cited

Example of periodic Neumann waveguide with gap in spectrum

Giuseppe Cardone, Andrii Khrabustovskyi

In this note we investigate spectral properties of a periodic waveguide $Ω^{ε}$ ( $ε$ is a small parameter) obtained from a straight strip by attaching an array of…

math.SP2015

Gaps in the spectrum of a periodic quantum graph with periodically distributed $δ^{'}$ -type interactions

Diana Barseghyan, Andrii Khrabustovskyi

We consider a family of quantum graphs ${(Γ, A_{ε})}_{ε > 0}$ , where $Γ$ is a $Z^{n}$ -periodic metric graph and the periodic Hamiltonian $\mathcal…

math.SP2014

Neumann spectral problem in a domain with very corrugated boundary

Giuseppe Cardone, Andrii Khrabustovskyi

Let $Ω \subset R^{n}$ be a bounded domain. We perturb it to a domain $Ω^{ε}$ attaching a family of small protuberances with "room-and-passage"-like geometry ($\varepsi…

math.SP2014

On the spectrum of narrow Neumann waveguide with periodically distributed $δ^{'}$ traps

Pavel Exner, Andrii Khrabustovskyi

We analyze a family of singular Schrödinger operators describing a Neumann waveguide with a periodic array of singular traps of a $δ^{'}$ type. We show that in the limit when perpendi…

math.SP2014

Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

Andrii Khrabustovskyi, Michael Plum

In this paper we study the asymptotic behaviour as $ε \to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varep…

math.SP2013★ 12 cited

Opening up and control of spectral gaps of the Laplacian in periodic domains

Andrii Khrabustovskyi

The main result of this work is as follows: for arbitrary pairwise disjoint finite intervals $(α_{j}, β_{j}) \subset [0, \infty)$ , $j = 1, \dots, m$ and for arbitrary $n \geq 2$ we construct th…

math.SP2013★ 4 cited

Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap

Andrii Khrabustovskyi, Evgeni Khruslov

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain $Ω^{ε}$ depending on a small parameter $ε > 0$ . The dom…

math.SP2012

Periodic elliptic operators with asymptotically preassigned spectrum

Andrii Khrabustovskyi

We deal with operators in $R^{n}$ of the form $$\mathbf{A}=-{1\over \mathbf{b}(x)}\sum\limits_{k=1}^n\ds{\partial\over\partial x_k}(\mathbf{a}(x){\partial \over\partial x_k}…

math.SP2011★ 11 cited

Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator

Andrii Khrabustovskyi

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m \in N$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in th…

math.SP2010★ 5 cited

Homogenization of spectral problem on Riemannian manifold consisting of two domains connected by many tubes

Andrii Khrabustovskyi

The paper deals with the asymptotic behavior as $\eps\to 0$ of the spectrum of Laplace-Beltrami operator $Δ\e$ on the Riemannian manifold $M\e$ ($\mathrm{\dim} M\e=N\geq 2$) depend…

Periodic discrete graphs with prescribed spectrum

Quantum lattice transport along an infinitely extended perturbation

Creating and controlling band gaps in periodic media with small resonators

Spectrum of the Laplacian on a domain perturbed by small resonators

Singular Schrödinger operators with prescribed spectral properties

A geometric approximation of δ-interactions by Neumann Laplacians

Periodic quantum graphs with predefined spectral gaps

Construction of self-adjoint differential operators with prescribed spectral properties

δ′-interaction as a limit of a thin Neumann waveguide with transversal window

Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

Gap control by singular Schrödinger operators in a periodically structured metamaterial

Operator estimates for the crushed ice problem

Spectrum of a singularly perturbed periodic thin waveguide

Example of periodic Neumann waveguide with gap in spectrum

Gaps in the spectrum of a periodic quantum graph with periodically distributed δ′-type interactions

Neumann spectral problem in a domain with very corrugated boundary

On the spectrum of narrow Neumann waveguide with periodically distributed δ′ traps

Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

Opening up and control of spectral gaps of the Laplacian in periodic domains

Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap

Periodic elliptic operators with asymptotically preassigned spectrum

Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator

Homogenization of spectral problem on Riemannian manifold consisting of two domains connected by many tubes

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

$δ^{'}$ -interaction as a limit of a thin Neumann waveguide with transversal window

Gaps in the spectrum of a periodic quantum graph with periodically distributed $δ^{'}$ -type interactions

On the spectrum of narrow Neumann waveguide with periodically distributed $δ^{'}$ traps