arXiv subjects

I. M. Karabash

Publications and source records attributed to I. M. Karabash.

4 recordsLinked to original sources

arXiv2012

Nonlinear eigenvalue problem for optimal resonances in optical cavities

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <= e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities e_1 and e_2. To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.

math.OC↗

arXiv2009

A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.

math.SP↗

arXiv2007

Abstract kinetic equations with positive collision operators

We consider "forward-backward" parabolic equations in the abstract form $Jd ψ/ d x + L ψ= 0$, $ 0< x < τ\leq \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \geq 0$, and $\ker L = 0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $ μ\frac {\partial ψ}{\partial x} (x,μ) = b(μ) \frac {\partial^2 ψ}{\partial μ^2} (x, μ)$, $ 0<x<τ$, $ μ\in \R$, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation $ T d ψ/dx = - A ψ(x) + f(x)$, where $T=T^*$ is injective and $A$ satisfies a certain positivity assumption, is considered also.

math.SP↗

arXiv2006

Indefinite Sturm-Liouville operators $ (\sgn x) (- \frac{d^2}{dx^2} +q(x))$ with finite-zone potentials

The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ is studied. It is proved that similarity of $A$ to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components $\Aess$ and $\Adisc$ of $A$ corresponding to essential and discrete spectrums, respectively, are considered. A criterion of similarity of $\Aess$ to a selfadjoint operator is given in terms of Weyl functions for the Sturm-Liouville operator $-d^2/dx^2+q(x)$ with a finite-zone potential $q$. Jordan structure of the operator $\Adisc$ is described. We present an example of the operator $A = (\sgn x)(-d^2/dx^2+q(x))$ such that $A$ is nondefinitizable and $A$ is similar to a normal operator.

math.SP↗