arXiv subjects

Andrii Khrabustovskyi

Publications and source records attributed to Andrii Khrabustovskyi.

36 records · Page 2Linked to original sources

arXiv2018

Gap control by singular Schrödinger operators in a periodically structured metamaterial

We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of $\varepsilon\mathbb{Z}^n$-periodic Schrödinger operators with $δ'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

math.SP↗

arXiv2018

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

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 $\varepsilon$. Under the assumption that the window is appropriately scaled we prove that the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense as $\varepsilon\to 0$ to a one-dimensional Schrödinger operator corresponding to a $δ'$-interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.

math.SP↗

arXiv2018

Surface Plasmon Polaritons Sustained at the Interface of a Nonlocal Metamaterial

Studying basic physical effects sustained in metamaterials characterized by specific constitutive relation is a research topic with a long standing tradition. Besides intellectual curiosity, it derives its importance from the ability to predict observable phenomena that are, if found with an actual metamaterial, a clear indication on its properties. Here, we consider a nonlocal (strong spatial dispersion), lossy, and isotropic metamaterial and study the impact of the nonlocality on the dispersion relation of surface plasmon polaritons sustained at an interface between vacuum and such metamaterial. For that, Fresnel coefficients are calculated and appropriate surface plasmon polaritons existence conditions are being proposed. Predictions regarding the experimentally observable reflection from a frustrated internal reflection geometry are being made. A different behavior for TE and TM polarization is observed. Our work unlocks novel opportunities to seek for traces of the nonlocality in experiments made with nowadays metamaterials.

physics.optics↗

arXiv2017

Operator estimates for the crushed ice problem

Let $Δ_{Ω_\varepsilon}$ be the Dirichlet Laplacian in the domain $Ω_\varepsilon:=Ω\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $Ω\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $\mathrm{cap}(D_{i \varepsilon})$ the capacity of a single hole. It was known for a long time that $-Δ_{Ω_\varepsilon}$ converges to the operator $-Δ_Ω+q$ in strong resolvent sense provided the limit $q:=\lim_{\varepsilon\to 0} \mathrm{cap}(D_{i\varepsilon}) \varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $Ω$) an estimate for the difference of the $k$-th eigenvalue of $-Δ_{Ω_\varepsilon}$ and $-Δ_{Ω_\varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.

math.SP↗

arXiv2017

Interface conditions for a metamaterial with strong spatial dispersion

Local constitutive relations, i.e. a weak spatial dispersion, are usually considered in the effective description of metamaterials. However, they are often insufficient and effects due to a nonlocality, i.e. a strong spatial dispersion, are encountered. Recently (K.~Mnasri et al., arXiv:1705.10969), a generic form for a nonlocal constitutive relation has been introduced that could accurately describe the bulk properties of a metamaterial in terms of a dispersion relation. However, the description of functional devices made from such nonlocal metamaterials also requires the identification of suitable interface conditions. In this contribution, we derive the interface conditions for such nonlocal metamaterials.

physics.class-ph↗

arXiv2017

Beyond local effective material properties for metamaterials

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, i.e. a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. The isofrequency surfaces of the dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.

physics.optics↗

arXiv2016

Spectrum of a singularly perturbed periodic thin waveguide

We consider a family $\{Ω^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-Δ_{Ω^\varepsilon}$ on $Ω^\varepsilon$. The waveguide $Ω^\varepsilon$ is a union of a thin straight strip of the width $\varepsilon$ and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period $\varepsilon$, along the strip upper boundary. For $\varepsilon\to 0$ we prove a (kind of) resolvent convergence of $-Δ_{Ω^\varepsilon}$ to a certain ordinary differential operator. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of $-Δ_{Ω^\varepsilon}$ is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

math.SP↗

arXiv2016

Example of periodic Neumann waveguide with gap in spectrum

In this note we investigate spectral properties of a periodic waveguide $Ω^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon=-ρ^\varepsilonΔ_{Ω^\varepsilon}$, where $Δ_{Ω^\varepsilon}$ is the Neumann Laplacian in $Ω^\varepsilon$, the weight $ρ^\varepsilon$ is equal to $1$ everywhere except the union of the "rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilon$ has at least one gap as $\varepsilon$ is small enough provided certain conditions on the weight $ρ^\varepsilon$ and the sizes of attached protuberances hold. (Dedicated to Pavel Exner's 70th birthday)

math.SP↗

arXiv2015

Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition

We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $λ$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $λ$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.

math-ph↗

arXiv2015

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

In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain $Ω\subset\mathbb{R}^n$ $(n \geq 2)$ subject to Dirichlet boundary conditions on $\partialΩ$. When $\varepsilon\to 0$ both coefficients $a^\varepsilon$ and $b^\varepsilon$ become high contrast in a small neighborhood of a hyperplane $Γ$ intersecting $Ω$. We prove that the spectrum of $\mathcal{A}^\varepsilon$ converges to the spectrum of an operator acting in $L^2(Ω)\oplus L^2(Γ)$ and generated by the operation $-Δ$ in $Ω\setminusΓ$, the Dirichlet boundary conditions on $\partialΩ$ and certain interface conditions on $Γ$ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when $Ω$ is an infinite straight strip ("waveguide") and $Γ$ is parallel to its boundary. We show that $\mathcal{A}^\varepsilon$ has at least one gap in the spectrum when $\varepsilon$ is small enough and describe the asymptotic behaviour of this gap as $\varepsilon\to 0$. The proofs are based on methods of homogenization theory.

math.SP↗

arXiv2015

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

We consider a family of quantum graphs $\{(Γ,\mathcal{A}_\varepsilon)\}_{\varepsilon>0}$, where $Γ$ is a $\mathbb{Z}^n$-periodic metric graph and the periodic Hamiltonian $\mathcal{A}_\varepsilon$ is defined by the operation $-\varepsilon^{-1} {\mathrm{d} ^2\over \mathrm{d} x^2}$ on the edges of $Γ$ and either $δ'$-type conditions or the Kirchhoff conditions at its vertices. Here $\varepsilon>0$ is a small parameter. We show that the spectrum of $\mathcal{A}_\varepsilon$ has at least $m$ gaps as $\varepsilon\to 0$ ($m\in\mathbb{N}$ is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of $Γ$ and of coupling constants involved in $δ'$-type conditions.

math.SP↗

arXiv2015

Neumann spectral problem in a domain with very corrugated boundary

Let $Ω\subset\mathbb{R}^n$ be a bounded domain. We perturb it to a domain $Ω^\varepsilon$ attaching a family of small protuberances with "room-and-passage"-like geometry ($\varepsilon>0$ is a small parameter). Peculiar spectral properties of Neumann problems in so perturbed domains were observed for the first time by R. Courant and D. Hilbert. We study the case, when the number of protuberances tends to infinity as $\varepsilon\to 0$ and they are $\varepsilon$-periodically distributed along a part of $\partialΩ$. Our goal is to describe the behaviour of the spectrum of the operator $\mathcal{A}^\varepsilon=-(ρ^\varepsilon)^{-1}Δ_{Ω^\varepsilon}$, where $Δ_{Ω^\varepsilon}$ is the Neumann Laplacian in $Ω^\varepsilon$, and the positive function $ρ^\varepsilon$ is equal to $1$ in $Ω$. We prove that the spectrum of $\mathcal{A}^\varepsilon$ converges as $\varepsilon\to 0$ to the "spectrum" of a certain boundary value problem for the Neumann Laplacian in $Ω$ with boundary conditions containing the spectral parameter in a nonlinear manner. Its eigenvalues may accumulate to a finite point.

math.SP↗

arXiv2014

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

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 the family of periodic non-compact domains $\{Ω^\varepsilon\subset\mathbb{R}^n\}_{\varepsilon>0}$ such that the spectrum of the Neumann Laplacian in $Ω^\varepsilon$ has at least $m$ gaps when $\varepsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $(α_j,β_j)$ as $\varepsilon\to 0$. The constructed domain $Ω^\varepsilon$ is obtained by removing from $\mathbb{R}^n$ a system of periodically distributed "trap-like" surfaces. The parameter $\varepsilon$ characterizes the period of the domain $Ω^\varepsilon$, also it is involved in a geometry of the removed surfaces.

math.SP↗

arXiv2014

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

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 perpendicular size of the guide tends to zero and the $δ'$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.

math.SP↗

arXiv2013

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

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain $Ω^\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain has the form $Ω^\varepsilon=\mathbb{R}^n\setminus S^\varepsilon$, where $S^\varepsilon$ is an $\varepsilon\mathbb{Z}^n$-periodic family of trap-like screens. We prove that for an arbitrarily large $L$ the spectrum has just one gap in $[0,L]$ when $\varepsilon$ small enough, moreover when $\varepsilon\to 0$ this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D-photonic crystals is discussed.

math.SP↗

arXiv2012

Periodic elliptic operators with asymptotically preassigned spectrum

We deal with operators in $\mathbb{R}^n$ of the form $$\mathbf{A}=-{1\over \mathbf{b}(x)}\sum\limits_{k=1}^n\ds{\partial\over\partial x_k}(\mathbf{a}(x){\partial \over\partial x_k})$$ where $\mathbf{a}(x),\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\mathbf{L}_{\mathrm{per}}$ the set of such operators. The main result of this work is as follows: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(α_j,β_j)\subset[0,L]$, $j=1,...,m$ ($m\in\mathbb{N}$) we construct the family of operators $\{\mathbf{A}^\varepsilon\in \mathbf{L}_{\mathrm{per}}\}_{\varepsilon}$ such that the spectrum of $\mathbf{A}^\varepsilon$ has exactly $m$ gaps in $[0,L]$ when $\varepsilon$ is small enough, and these gaps tend to the intervals $(α_j,β_j)$ as $\varepsilon\to 0$. The idea how to construct the family ${\mathbf{A}\e}_\eps$ is based on methods of the homogenization theory.

math.SP↗

arXiv2011

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

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $δ>0$ and for an arbitrarily large $L>0$ we construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the operator $-Δ_{M}$, moreover the edges of the first $m$ gaps belong to $δ$-neighbourhoods of the edges of the intervals $(\a_j,\b_j)$, while the remaining gaps (if any) are located outside the interval $[0,L]$.

math.SP↗

arXiv2010

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

The paper deals with the asymptotic behavior as $\eps\to 0$ of the spectrum of Laplace-Beltrami operator $Δ\e$ on the Riemannian manifold $M\e$ ($\mathrm{\dim} M\e=N\geq 2$) depending on a small parameter $\eps>0$. $M\e$ consists of two perforated domains which are connected by array of tubes of the length $q\e$. Each perforated domain is obtained by removing from the fix domain $Ω\subset \mathbb{R}^N$ the system of $\eps$-periodically distributed balls of the radius $d\e=\bar{o}(\eps)$. We obtain a variety of homogenized spectral problems in $Ω$, their type depends on some relations between $\eps$, $d\e$ and $q\e$. In particular if the limits $\liml_{\eps\to 0}q\e$ and $\liml_{\eps\to 0}\ds{(d\e)^{N-1}q\e \eps^{-N}}$ are positive then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

math.SP↗