By Armin Lechleiter

This publication investigates non-destructive checking out difficulties for tough and periodic surfaces, the place the duty is to figure out such buildings from scattered waves. Such difficulties are non-linear and ill-posed. we're attracted to the research of the Factorization process utilized to this classification of inverse scattering difficulties. this technique doesn't try to clear up a non-linear operator equation for the unknown item yet computes a binary criterion characterizing issues inside and out the constitution.

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3) which is required to converge uniformly on compact subsets of {x ∈ R2 : x2 > h}. 3) is deﬁned as the unique holomorphic extension of the square root on the positive numbers to all of C \ [0, −i∞), which is the complex plane slit at the negative imaginary axis. Especially, if ω is positive, βn+ = βn+ (ω) = ω 2 ε+ − α2n , |αn | ≤ ω(ε+ )1/2 , i α2n − ω 2 ε− , |αn | > ω(ε− )1/2 , for n ∈ Z. 4) Our deﬁnition of the square root necessitates to exclude certain frequencies ω, namely those for which ω 2 ε± − α2n ∈ [0, −i∞) for some n ∈ Z.

The upper domain D+ is supposed to be connected. Physical properties of the dielectric Ω± are modelled by the permittivity ε which takes constant values ε+ > 0 in Ω+ and ε− > 0 in Ω− . These two constants are ﬁxed in the entire chapter and determine the piecewise constant function ε+ in Ω+ , ε := ε− in Ω− . 46 The Factorization Method for Photonic Crystals The permeability µ is supposed to be a positive constant whereas the conductivity σ is assumed to vanish in this chapter. As in Chapter I, the number ω > 0 denotes the angular frequency and the wavenumber k± is √ deﬁned as k± = ω ε± µ0 .

Hence, we obtain that the sequence n → en T j=1 Ten j=1 2 2 | φ, ψj | /λj,n is bounded if and only if the sequence n → | φ, ψj | /λj is bounded, that is, if and only if y ∈ D. Our aim is now to perform a similar analysis when both the eigenvectors and eigenvalues enter the Picard series in a perturbed way. To deal with perturbations of the eigenvectors it will be helpful to consider the numbers ρk,n := min{|λk,n − λj,n | : |λk,n − λj,n | > 2dn , j ∈ N}, k, n ∈ N, which indicate the distance of λk,n to that part of the spectrum of F♯n which does not converge to the same eigenvalue of F♯ as n → ∞: Since |λk,n − λj,n | ≤ |λk,n − λk | + |λk − λj | + |λj − λj,n | ≤ 2dn + |λk − λj | for j, k, n ∈ N, we see that if |λk,n − λj,n | > 2dn , then λk,n and λj,n cannot be perturbations of the same eigenvalue.