We consider a system governed by the wave equation with index of refraction @math , taken to be variable within a bounded region @math and constant in @math . The solution of the time-dependent wave equation with initial data, which is localized in @math , spreads and decays with advancing time. This rate of decay can be measured (for @math , and more generally, @math odd) in terms of the eigenvalues of the scattering resonance problem, a non--self-adjoint eigenvalue problem governing the time-harmonic solutions of the wave (Helmholtz) equation which are outgoing at @math . Specifically, the rate of energy escape from @math is governed by the complex scattering eigenfrequency, which is closest to the real axis. We study the structural design problem: Find a refractive index profile @math within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact suppo...
Results on the existence of optimal scattering resonances and general bounds on the imaginary parts of scattering resonances for Schr "odinger operators can be found in @cite_21 @cite_5 @cite_37 . Very recently, optimal designs have been considered in @cite_26 . Our results for the Helmholtz equation make use of some of the arguments introduced in these papers.
@cite_5 @cite_37 @cite_21 @cite_26
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On considere des potentiels de mecanique quantique constitues d'un fond fixe et un morceau additionnel contraint d'avoir une hauteur finie et d'etre a support dans une region finie donnee en dimension d≤3. On caracterise les potentiels de cette classe qui produisent les resonances les plus etroites Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrodinger operator The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given @math a medium that generates a resonance on the line @math with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints @math with certain constants @math . The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values @math and @math . This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.