arXiv subjects

Illya M. Karabash

Publications and source records attributed to Illya M. Karabash.

At least 19 recordsLinked to original sources

arXiv2026

Sobolev multipliers and fractional Gaussian fields on Lipschitz boundaries with applications to deterministic and random acoustic systems

Motivated by Applied Physics and Photonics studies of random resonators, we study in the stochastic part of this paper random acoustic operators in non-smooth bounded domains $G \subset \mathbb{R}^d$ and introduce m-dissipative impedance boundary conditions containing (eigenfunction) fractional Gaussian fields. The deterministic part of the paper constructs and studies the spaces of pointwise multipliers on Lipschitz boundaries $\partial G$, as well as the spaces of Sobolev (distribution-type) multipliers on boundaries $\partial G$ of better regularity. These multipliers are used as generalized impedance coefficients $ζ$ in impedance boundary conditions $ζp = \mathbf{n} \cdot \mathbf{v}$ accompanying the 1st order acoustic system. We study the m-dissipativity of associated acoustic operators and the discreteness of their spectra aiming the main efforts on the weakest possible assumptions on the regularity of $ζ$. In order to pass from deterministic results to the randomization, we introduce fractional Gaussian fields (FGFs) on Lipschitz boundaries $\partial G$ and study their regularity. To this end, we prove that a rough Weyl-type asymptotics takes place for the Laplace-Beltrami eigenvalues on arbitrary compact Lipschitz boundary $\partial G$.

math.PR↗

arXiv2025

M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces

The paper studies properties of acoustic operators in bounded Lipschitz domains $Ω$ with m-dissipative generalized impedance boundary conditions. We prove that such acoustic operators have a compact resolvent if and only if the impedance operator from the trace space $H^{1/2} (\partial Ω)$ to the other trace space $H^{-1/2} (\partial Ω)$ is compact. This result is applied to the question of the discreteness of the spectrum and to the particular cases of damping and impedance boundary conditions. The method of the paper is based on abstract results written in terms of boundary tuples and is applicable to other types of wave equations.

math.AP↗

arXiv2024

Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries

Motivated by engineering and photonics research on resonators in random or uncertain environments, we study rigorous randomizations of boundary conditions for wave equations of the acoustic-type in Lipschitz domains $\mathcal{O}$. First, a parametrization of essentially all m-dissipative boundary condition by contraction operators in the boundary $L^2$-space is constructed with the use of m-boundary tuples (boundary value spaces). We consider randomizations of these contraction operators that lead to acoustic operators random in the resolvent sense. To this end, the use of Neumann-to-Dirichlet maps and Krein-type resolvent formulae is crucial. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. For each particular applied model, one can choose a specific boundary condition from the constructed class either by means of optimization, or on the base of empirical observations. A mathematically convenient randomization is constructed in terms of eigenfunctions of the Laplace-Beltrami operator $Δ^{\partial \mathcal{O}}$ on the boundary $\mathcal{O}$ of the domain. We show that for this randomization the compactness of the resolvent is connected with the Weyl-type asymptotics for the eigenvalues of $Δ^{\partial \mathcal{O}}$.

math.AP↗

arXiv2024

Multi-parameter Hopf bifurcations of rimming flows

In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.

math.AP↗

arXiv2024

Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems

The homogenization of eigenvalues of non-Hermitian Maxwell operators is studied by the H-convergence method. It is assumed that the Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions of the form $n \times E = Z [(n \times H )\times n ] $. We show that, for a wide class of impedance operators $Z$, the nonzero spectrum of the corresponding Maxwell operator is discrete. To this end, a new continuous embedding theorem for domains of Maxwell operators is obtained. We prove the convergence of eigenvalues to an eigenvalue of a homogenized Maxwell operator under the assumption of the H-convergence of the material tensor-fields. This result is used then to prove the existence of optimizers for eigenvalue optimization problems and the existence of an eigenvalue-free region around zero. As applications, connections with the quantum optics problem of the design of high-Q resonators are discussed, and a new way of the quantification of the unique (and nonunique) continuation property is suggested.

math.AP↗

arXiv2021

M-dissipative boundary conditions and boundary tuples for Maxwell operators

For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a translation of the problem to operator-theoretic settings of abstract Maxwell operators. In particular, we show that Calkin reduction operators are naturally connected with Leontovich boundary conditions and provide an abstract version of impedance boundary condition applicable to other types of wave equations. Taking Friedrichs and Krein-von Neumann extensions of related boundary operators, it is possible to associate m-dissipative Maxwell operators to arbitrary non-negative measurable impedance coefficients.

math.AP↗

arXiv2019

Asymptotics of random resonances generated by a point process of delta-interactions

We introduce and study the following model for random resonances: we take a collection of point interactions $Υ_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schrödinger Hamiltonians $H_Υ= -Δ+ ``\sum \mathfrak{m}(α) δ(x - Υ_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $Σ(H_Υ)$ in the complex plane $\mathbb{C}$. We show that the counting function for the set of random resonances $Σ(H_Υ)$ in $\mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $Υ$, and obtain an explicit formula for the limiting distribution as $m \to \infty$ of the leading parameter of the asymptotic chain of `most narrow' resonances generated by a sequence of uniform binomial processes $Υ^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.

math-ph↗

arXiv2019

On the multilevel internal structure of the asymptotic distribution of resonances

We prove that the asymptotic distribution of resonances has a multilevel internal structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions in $\mathbb{R}^3$, quantum graphs, and 1-D photonic crystals. In the case of $N \ge 2$ point interactions, the set of resonances $Σ(H)$ essentially consists of a finite number of sequences with logarithmic asymptotics. We show how the leading parameters $μ$ of these sequences are connected with the geometry of the set $Y=\{y_j\}_{j=1}^N$ of interaction centers $y_j \in \mathbb{R}^3$. The minimal parameter $μ^{min}$ corresponds to the sequences with `more narrow' and so more observable resonances. The asymptotic density of such narrow resonances is described by the multiplicity of $μ^{\min}$, which occurs to be connected with the symmetries of Y and naturally introduces a finite number of classes of configurations of $Y$. In the case of quantum graphs and 1-D photonic crystals, the decomposition of $Σ(H)$ into a finite number of asymptotic sequences is proved under additional commensurability conditions. To address the case of a general quantum graph, we introduce families of special counting and asymptotic density functions for two types of curved complex strips. The obtained results and effects are compared with those of obstacle scattering.

math-ph↗

arXiv2019

Euler-Lagrange equations for full topology optimization of the Q-factor in leaky cavities

We derive Euler-Lagrange equations for the topology optimization of decay rate in 3-d lossy optical cavities. This leads to a new class of time-harmonic differential or integro-differential equations, which can be written as nonlinear Maxwell systems with switching functions of special types. Our approach is based on the notion of Pareto optimal frontier and on the multi-parameter perturbation theory for eigenfrequencies. Parallels with optimal control theory are discussed.

math.OC↗

arXiv2018

Pareto optimization of resonances and minimum-time control

The aim of the paper is to reduce one spectral optimization problem, which involves the minimization of the decay rate $|\mathrm{Im} \, k |$ of a resonance $k$, to a collection of optimal control problems on the Riemann sphere $\widehat{\mathbb{C}}$. This reduction allows us to apply methods of extremal synthesis to the structural optimization of layered optical cavities. We start from a dual problem of minimization of the resonator length and give several reformulations of this problem that involve Pareto optimization of the modulus $|k|$ of a resonance, a minimum-time control problem on $\widehat{\mathbb{C}}$, and associated Hamilton-Jacobi-Bellman equations. Various types of controllability properties are studied in connection with the existence of optimizers and with the relationship between the Pareto optimal frontiers of minimal decay and minimal modulus. We give explicit examples of optimal resonances and describe qualitatively properties of the Pareto frontiers near them. A special representation of bang-bang controlled trajectories is combined with the analysis of extremals to obtain various bounds on optimal widths of layers. We propose a new method of computation of optimal symmetric resonators based on minimum-time control and compute with high accuracy several Pareto optimal frontiers and high-Q resonators.

math.OC↗

arXiv2018

Generic asymptotics of resonance counting function for Schrödinger point interactions

We study the leading coefficient in the asymptotical formula $ N (R) = \frac{W}π R + O (1) $, $ R \to \infty $, for the resonance counting function $ N (R)$ of Schrödinger Hamiltonians with point interactions. For such Hamiltonians, the Weyl-type and non-Weyl-type asymptotics of $N (R)$ was introduced recently in a paper by J. Lipovský and V. Lotoreichik (2017). In this note, we prove that the Weyl-type asymptotics is generic.

math.SP↗

arXiv2017

Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

Resonances of Schrödinger Hamiltonians with point interactions are considered. The main object under the study is the resonance free region under the assumption that the centers, where the point interactions are located, are known and the associated 'strength' parameters are unknown and allowed to bear additional dissipative effects. To this end we consider the boundary of the resonance free region as a Pareto optimal frontier and study the corresponding optimization problem for resonances. It is shown that upper logarithmic bound on resonances can be made uniform with respect to the strength parameters. The necessary conditions on optimality are obtained in terms of first principal minors of the characteristic determinant. We demonstrate the applicability of these optimality conditions on the case of 4 equidistant centers by computing explicitly the resonances of minimal decay for all frequencies. This example shows that a resonance of minimal decay is not necessarily simple, and in some cases it is generated by an infinite family of feasible resonators.

math.OC↗

arXiv2017

Recovery of periodicities hidden in heavy-tailed noise

We address a parametric joint detection-estimation problem for discrete signals of the form $x(t) = \sum_{n=1}^{N} α_n e^{-i λ_n t } + ε_t$, $t \in \mathbb{N}$, with an additive noise represented by independent centered complex random variables $ε_t$. The distributions of $ε_t$ are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., the frequencies $λ_n$, their number $N$, and complex amplitudes $α_n$. For example, one of considered classes of noise is the following: $ε_t$ are independent identically distributed random variables with $\mathbb{E} (ε_t) = 0$ and $\mathbb{E} (|ε_t| \ln |ε_t|) < \infty$. The construction of estimators is based on detection of singularities of anti-derivatives for $Z$-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series. We discuss also decaying signals and the case of infinite number of frequencies.

math.CA↗

arXiv2015

Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when different structures generate the same optimal quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in various senses are in the spectrum $Σ^{nl}$ of the bang-bang eigenproblem $y" = - ω^2 y [ b_1 + (b_2 - b_1) χ_{\mathbb{C}_+} (y^2 ) ]$, where $χ_{\mathbb{C}_+} (\cdot)$ is the indicator function of the upper complex half-plane $\mathbb{C}_+$, we obtain a variational characterization of the nonlinear spectrum $Σ^{nl}$ in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate $| \mathrm{Im} \ ω|$, we study the bang-bang equation and explain how it excludes an unknown optimal $B$ from the optimization process. Computing one of minimal decay structures for 1-side open settings, we show that it resembles gradually size-modulated 1-D stack cavities introduced recently in Optical Engineering. In 2-side open symmetric settings, our example has an additional centered defect. Nonexistence of global decay rate minimizers is discussed.

math.OC↗

arXiv2013

Pareto optimal structures producing resonances of minimal decay under $L^1$-type constraints

Optimization of resonances associated with 1-D wave equations in inhomogeneous media is studied under the constraint $\| B \|_1 <= m$ on the nonnegative function $B \in L^1 (0,\ell)$ that represents the medium's structure. From the Physics and Optimization points of view, it convenient to generalize the problem replacing $B$ by a nonnegative measure $d M$ and imposing on $d M$ the condition that its total mass is $<= m$. The problem is to design for a given frequency $α\in R$ a medium that generates a resonance $ ω$ on the line $α+ i R$ with a minimal possible decay rate $| Im ω|$. Such resonances are said to be of minimal decay and form a Pareto frontier. We show that corresponding optimal measures consist of finite number of point masses, and that this result yields non-existence of optimizers for the problem over the set of absolutely continuous measures $B(x) dx$. Then we derive restrictions on optimal point masses and their positions. These restrictions are strong enough to calculate optimal $d M$ if the optimal resonance $ ω$, the first point mass $m_1$, and one more geometric parameter are known. This reduces the original infinitely-dimensional problem to optimization over four real parameters. For small frequencies, we explicitly find the Pareto set and the corresponding optimal measures $d M$. The technique of the paper is based on the two-parameter perturbation method and the notion of local boundary point. The latter is introduced as a generalization of local extrema to vector optimization problems.

math.OC↗

arXiv2012

Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures

The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $α\in \R$ a medium that generates a resonance on the line $α+ ı\R$ with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values $b_1$ and $b_2$. This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.

math.SP↗

arXiv2012

Optimization of quasi-normal eigenvalues for Krein-Nudelman strings

The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given $α\in \R$ a string that has a resonance on the line $α+ ı\R$ with a minimal possible modulus of the imaginary part. We find optimal resonances and strings explicitly.

math.SP↗

arXiv2010

Bohl-Perron type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) $ł^p$-input $ł^q$-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to $ł^q$ when non-homogeneous terms are in $ł^p$. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted $ł^r$-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and $ł^p$-input $ł^q$-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. $ł^1$-input $ł^\infty$-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.

math.DS↗