# I. M. Karabash

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Articles: 3

### Overdamped modes and optimization of resonances in layered cavities

Methods Funct. Anal. Topology 23 (2017), no. 3, 252-260

We study the problem of optimizing the imaginary parts $\mathrm{Im}\, \omega$ of quasi-normal-eigenvalues $\omega$ associated with the equation $y'' = -\omega^2 B y$. It is assumed that the coefficient $B(x)$, which describes the structure of an optical or mechanical resonator, is constrained by the inequalities $0 \le b_1 \le B(x) \le b_2$. Extremal quasi-normal-eigenvalues belonging to the imaginary line ${\mathrm{i}} {\mathbb R}$ are studied in detail. As an application, we provide examples of $\omega$ with locally minimal $|\mathrm{Im}\, \omega|$ (without additional restrictions on $\mathrm{Re}\, \omega$) and show that a structure generating an optimal quasi-normal-eigenvalue on ${\mathrm{i}} {\mathbb R}$ is not necessarily unique.

### On ordinary differential operators of an odd order nonsimilar to normal operators

I. M. Karabash

Methods Funct. Anal. Topology 7 (2001), no. 1, 17-27