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...
One property of optimal structures for is that they are piecewise constant structures which achieve the material bounds, , they are bang-bang controls. This property is also realized in a number of optimization problems for eigenvalues of self-adjoint operators @cite_25 @cite_16 @cite_36 @cite_22 @cite_41 as well as for Schr "odinger resonances @cite_5 . In @cite_10 @cite_8 the authors consider the problem of maximizing the lifetime of a state coupled to radiation by an ionizing perturbation. For this class of problems, optimizers are interior points of the constraint set.
@cite_22 @cite_8 @cite_36 @cite_41 @cite_5 @cite_16 @cite_10 @cite_25
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We consider the following eigenvalue optimization problem: Given a bounded domain Ω⊂ℝ and numbers α > 0, A∈[ 0, |Ω|], find a subset D⊂Ω of area A for which the first Dirichlet eigenvalue of the operator −Δ+αχ D is as small as possible. Given an open bounded connected set Ω ⊂R N and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in Ω that extremizes thekth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes toN dimensions the now classical one-dimensional work of M. G. Krein. Abstract Periodic media are routinely used in optical devices and, in particular, photonic crystals to create spectral gaps, prohibiting the propagation of waves with certain temporal frequencies. In one dimension, Bragg structures, also called quarter-wave stacks, are frequently used because they are relatively easy to manufacture and the spectrum exhibits large spectral gaps at explicitly computable frequencies. In this short work, we use variational methods to demonstrate that within an admissible class of pointwise-bounded, periodic media, the Bragg structure uniquely maximizes the first spectral gap-to-midgap ratio. 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 Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point...