arXiv subjects

Narek Hovsepyan

Publications and source records attributed to Narek Hovsepyan.

17 recordsLinked to original sources

arXiv2026

On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof

This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $ω$ for which there exists a nontrivial incident $ω$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).

math.AP↗

arXiv2025

Scattering of plane waves

We formulate a problem that can be viewed as a natural variation of the so-called Pompeiu or Schiffer problem in the context of scattering of plane waves for the Linear Helmholtz equation. For the two dimensional version of this variation, we establish conditions on the wave numbers and incident directions, that ensure a non-vanishing scattered field.

math.AP↗

arXiv2025

Scattering from analytic and piecewise analytic inhomogeneities

We study scattering for the linear Helmholtz operator in two dimensions and develop a technique, which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.

math.AP↗

arXiv2024

On the lack of external response of a nonlinear medium in the second-harmonic generation process

This paper concerns the scattering problem for a nonlinear medium of compact support, $D$, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency $ω$, generates additional waves at frequency $2ω$. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency $ω$ and $2ω$. We investigate whether there are situations in which the generated $2ω$ wave is localized inside $D$, that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in $D$ with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.

math.AP↗

arXiv2023

On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in ${\mathbb R}^d$ for $d=2$ or $d=3$, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. We show the discreteness of transmission eigenvalues in the complex plane. Moreover, using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.

math.AP↗

arXiv2023

Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in ${\mathbb R}^d$ for $d=2$ or $d=3$. Using ideas from transformation optics, we construct an approximate cloak by ``blowing up" a small ball of radius $ε>0$ to one of radius $1$. In the anisotropic cloaking layer resulting from the ``blow-up" change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed $ε$) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as $ε$ approaches $0$, the $L^2$-norm of the scattered field outside the cloak, and its far field pattern, approach $0$ uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves.

math.AP↗

arXiv2021

On the extreme rays of the cone of $3\times 3$ quasiconvex quadratic forms: Extremal determinats vs extremal and polyconvex forms

This work is concerned with the study of the extreme rays of the convex cone of $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). We characterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\cal C}_3;$ when the determinant is a perfect square, then the form is either an extreme ray of ${\cal C}_3$ or polyconvex; and finally, when the determinant is identically zero, then the form $f$ must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of ${\cal C}_3,$ and discuss about weak and strong etremals of ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of ${\cal C}_3$ do not hold for ${\cal C}_d$ for $d>3,$ and thus case $d=3$ is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of ${\cal C}_3$ that were proved to be such. Our results also improve the ones proven by the first author and Milton [20].

math.AG↗

arXiv2021

On the commutation properties of finite convolution and differential operators I: commutation

Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to complex-valued and non self-adjoint operators. In addition, we introduce a new kind of commutation relation, which we call sesquicommutation, that also has implications for the spectral properties of the integral operator. In this case we also compute a complete list of sequicommuting pairs of integral and differential operators.

math.CA↗

arXiv2021

On the optimal analytic continuation from discrete data

We consider analytic functions from a reproducing kernel Hilbert space. Given that such a function is of order $ε$ on a set of discrete data points, relative to its global size, we ask how large can it be at a fixed point outside of the data set. We obtain optimal bounds on this error of analytic continuation and describe its asymptotic behavior in $ε$. We also describe the maximizer function attaining the optimal error in terms of the resolvent of a positive semidefinite, self-adjoint and finite rank operator.

math.CV↗

arXiv2020

On feasibility of extrapolation of the complex electromagnetic permittivity function using Kramer-Kronig relations

We study the degree of reliability of extrapolation of complex electromagnetic permittivity functions based on their analyticity properties. Given two analytic functions, representing extrapolants of the same experimental data, we examine how much they can differ at an extrapolation point outside of the experimentally accessible frequency band. We give a sharp upper bound on the worst case extrapolation error, in terms of a solution of an integral equation of Fredholm type. We conjecture and give numerical evidence that this bound exhibits a power law precision deterioration as one moves further away from the frequency band containing measurement data.

math-ph↗

arXiv2020

Rigidity of a thin domain depends on the curvature, width, and boundary conditions

This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A shallow thin domain is a thin domain that has in-plane dimensions of order $O(1)$ and $ε,$ where $ε\in (h,1)$ is a parameter (here $h$ is the thickness of the shell). The problem has been solved in [8,10] for the case $ε=1,$ with the outcome of the optimal constant $C\sim h^{-3/2},$ $C\sim h^{-4/3},$ and $C\sim h^{-1}$ for parabolic, hyperbolic and elliptic thin domains respectively. We prove in the present work that in fact there are two distinctive scaling regimes $ε\in (h,\sqrt h]$ and $ε\in (\sqrt h,1),$ such that in each of which the thin domain rigidity is given by a certain formula in $h$ and $ε.$ An interesting new phenomenon is that in the first (small parameter) regime $ε\in (h,\sqrt h]$, the rigidity does not depend on the curvature of the thin domain mid-surface.

math.AP↗

arXiv2020

Optimal error estimates for analytic continuation in the upper half-plane

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $Γ$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to provide a quantitative version of this statement. Given that $f$ from a unit ball in $H^2$ is small on $Γ$ (say, its $L^2$ norm is of order $ε$), how does this affect the magnitude of $f$ at a point $z$ away from the curve? When $Γ\subset \partial \mathbb{H}_+$, we give a sharp upper bound on $|f(z)|$ of the form $ε^γ$, with an explicit exponent $γ=γ(z) \in (0,1)$ and describe the maximizer function attaining the upper bound. When $Γ\subset \mathbb{H}_+$ we give an implicit sharp upper bound in terms of a solution of an integral equation on $Γ$. We conjecture and give evidence that this bound also behaves like $ε^γ$ for some $γ=γ(z) \in (0,1)$. These results can also be transplanted to other domains conformally equivalent to the upper half-plane.

math.AP↗

arXiv2020

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.

math.CV↗

arXiv2019

Analytic continuation in an annulus and in a Bernstein ellipse

Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In fact, well known theorems, guarantee that every continuous function can be uniformly approximated by analytic functions (polynomials or rational functions, for example). We consider several analytic continuation problems with typical global boundedness constraints. All such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the `worst case' function as a solution of a linear equation with a compact operator. In special geometries, such as the circular annulus this equation can be solved explicitly. The obtained solution is then used to determine the power law exponent for the analytic continuation from an interval between the foci of a Bernstein ellipse to the entire ellipse. In those cases where such exponents have been determined in prior work our results reproduce them faithfully.

math.AP↗

arXiv2019

On the commutation properties of finite convolution and differential operators I: commutation

The commutation relation $KL = LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.

math.AP↗

arXiv2019

On the commutation properties of finite convolution and differential operators II: sesquicommutation

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization of commuting pairs $KL=LK$ and provides an exhaustive list of kernels admitting commuting or sesquicommuting differential operators.

math.AP↗

arXiv2018

Hopf Bifurcation in Structural Population Models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely counter-migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration.

math.AP↗