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

Karabash, Illya M.

doi:10.3233/asy-2012-1128

Cited by 12 publications

(73 citation statements)

References 20 publications

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“…Our interest in the internal structure of Σ(H) is motivated, in particular, by the necessity to consider the notion of high-energy resonance asymptotics from the point of view of Physics , where only the resonances that are closer to R play role (see [22] and Section 5), and by the recent studies of narrow 'topological' resonances [24,16]. The rigorous definition of structural parameters of Σ(H) can also provide an approach to optimization problems of the type of [12,Section 8], which involve the whole set Σ(H) and are much less studied than problems on optimization of an individual resonance [32,33,34].…”

Section: Main Goals and Related Studiesmentioning

confidence: 99%

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On the multilevel internal structure of the asymptotic distribution of resonances

Albeverio

Karabash

2019

Journal of Differential Equations

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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 R 3 , quantum graphs, and 1-D photonic crystals. In the case of N ≥ 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 } N j=1 of interaction centers y j ∈ 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 can be expressed via 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 asymptotic density functions for two classes of strips in C. The obtained results and effects are compared with those of obstacle scattering.

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Section: Main Goals and Related Studiesmentioning

confidence: 99%

“…Since in this process the derivatives f ′ (x j − 0) are coupled only with the derivatives f ′ (x j + 0), the multiplicative factors iz are eliminated from A(z) (similarly to [14] and the definition of resonances in [12,32,34]). Hence, F (z) takes the form (6.5) with certain C l ∈ R \ {0}.…”

Section: Resonances Of 1-d Photonic Crystalsmentioning

confidence: 99%

On the multilevel internal structure of the asymptotic distribution of resonances

Albeverio

Karabash

2019

Journal of Differential Equations

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“…[45,34,35,40]). The recent progress in fabrication of small size optical resonators [1,26,29] attracted considerable interest to numerical [19,15,3,11,29,31] and analytical [21,20,23] aspects of resonance optimization.…”

Section: Introductionmentioning

confidence: 99%

“…Among other variational problems for eigenvalues, the optimization of resonances can be classified as the nonselfadjoint spectral optimization. It contains essentially new effects and difficulties in comparison with selfadjoint cases (see [7,5,4,21,23]).…”

Section: Introductionmentioning

confidence: 99%

“…Since quasi-eigenvalues ω are complex numbers, one of the ways to formulate related variational problems is to interpret ω as an R 2 -vector and to take the point of view of the optimization theory for vector-valued cost functions (Pareto optimization) [23]. The approach of [23] is a development of that of [21], where the engineering problem of the high-Q design for one-side open multilayer cavities was considered analytically. To provide a simple and simultaneously rigorous formulation, the papers [20,21] restate the problem of high-Q design in terms of the minimization of the decay rate | Im ω| for quasi-eigenvalues with a fixed frequency Re ω = α.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

Karabash

Logachova

Verbytskyi

2017

Integr. Equ. Oper. Theory

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Quasi-normal-eigenvalue optimization is studied under constraints b 1 (x) ≤ B(x) ≤ 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 yis the indicator function of C + , we obtain a variational characterization of Σ nl in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate | 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. MSC-classes: 49R05,

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Pareto optimal structures producing resonances of minimal decay under L1-type constraints

Karabash¹

2014

Journal of Differential Equations

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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 ∈ L 1 (0, ℓ) 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 dM and imposing on dM the condition that its total mass is ≤ m. The problem is to design for a given frequency α ∈ R a medium that generates a resonance ω on the line α + iR 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 dM 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 dM . 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.MSC-classes: 49R05, 58E17, 35B34, 34L15, 32A60, 58C06

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

Cited by 12 publications

References 20 publications

On the multilevel internal structure of the asymptotic distribution of resonances

On the multilevel internal structure of the asymptotic distribution of resonances

Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

Pareto optimal structures producing resonances of minimal decay under L1-type constraints

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