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

@article{Albeverio2017ResonanceFR,
title={Resonance free regions and non-Hermitian spectral optimization for Schr{\"o}dinger point interactions},
author={Sergio Albeverio and Illya M. Karabash},
journal={Operators and Matrices},
year={2017},
pages={1097-1117}
}
• Published 4 August 2017
• Physics
• Operators and Matrices
Resonances of Schrodinger 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…
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## References

SHOWING 1-10 OF 54 REFERENCES
Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures
It is proved that optimal structures are piecewise constant functions taking only two extreme possible values $b_1$ and $b-2$ and this result explains an effect recently observed in numerical experiments.
Kirchhoff's rule for quantum wires
• Mathematics
• 1999
We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general
The low energy expansion in nonrelativistic scattering theory
• Physics
• 1982
We study in detail the low energy behaviour of Schrodinger operators with particular attention to scattering theory. We exploit the fact that the low energy behaviour of 4 + V(x) in L2([R3) is
Spectral isoperimetric inequalities for $\delta$-interactions on open arcs and for the Robin Laplacian on planes with slits
We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction supported on an open arc with two
Long-Lived Scattering Resonances and Bragg Structures
• Mathematics
SIAM J. Appl. Math.
• 2013
The structural design problem: Find a refractive index profile $n_\star({\bf x})$ within an admissible class which has a scattering frequency with minimal imaginary part is studied, defined in terms of the compact suppo...
Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities
• Mathematics
• 2015
We study optimization of quasi-normal-eigenvalues $$\omega$$ω associated with the equation $$y^{\prime \prime } = -\omega ^2 B y$$y″=-ω2By of two-side open optical and mechanical resonators. The
Eigenvalue asymptotics for a star-graph damped vibrations problem
• Mathematics
Asymptot. Anal.
• 2011
A continuity condition and a condition depending on the spectral parameter is imposed at the interior vertex, corresponding to the case of damping in the problem of small transversal vibrations of a star graph of smooth inhomogeneous strings.