This special issue contains an overview on current trends and up-to-date results on scattering and inverse scattering problems. The topics treated by the contributions include the connection of sampling methods for inverse scattering problems with the transmission eigenvalue problem, scattering and inverse scattering from locally perturbed infinite periodic layers and rough surfaces, approximation techniques for inverse scattering as well as conformal mappings applied to cavity inverse problems, and scattering from chiral media.
The transmission eigenvalue problem has important applications in the inverse scattering theory for non-homogeneous media and received significant attention recently. Bondarenko et al. investigate this eigenvalue problem for an inhomogeneous media with conductive boundary conditions. They prove the discreteness and existence of the transmission eigenvalues and that the first transmission eigenvalue is a monotonic function of the index of refraction and boundary conductivity. Assuming a constant index of refraction. Cakoni and Kress propose a new integral equation formulation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators to compute transmission eigenvalues. Colton and Leung establish the existence of complex transmission eigenvalues for spherically stratified media under suitable conditions for the index of refraction. Peters studies the so-called inside–outside duality for time-harmonic elastic scattering problems, which can be used to determine interior transmission eigenvalues from multi-frequency scattering data.
Based on the Factorization method, Kirsch compares the far-field operator for the full nonlinear inverse scattering problem with the Born approximation as its linearization and proves that the two corresponding inverse scattering problems share the same degree of ill-posedness. Li et al. extend the Kirsch–Kress method to the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. Munnier and Ramdani derive an explicit reconstruction formula for the Calderón inverse problem to reconstruct a cavity included in a domain based on DtN maps and the so-called generalized Pólia–Szegö tensors of the cavity via fairly explicit formulas.
Two papers of this special issue study numerical methods for particularly involved direct scattering problems. Haddar and Nguyen investigate the scattering problem for the case of locally perturbed periodic layers. Using the Floquet-Bloch transform in the periodicity direction, they reformulate this scattering problem as an equivalent system of coupled volume integral equations and apply a spectral method to discretize the obtained system. For the scattering of time-harmonic electromagnetic waves from a chiral medium, Nguyen et al. proposed a Galerkin approximation based on Fourier-type ansatz spaces for an integro-differential equation based on the Drude–Born–Fedorov model.
We hope that this collection of papers gives a timely and useful snapshot of the rapidly growing area of scattering and inverse scattering problems including the transmission eigenvalue problem.
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