Inverse scattering problems for the time harmonic magnetic Schrödinger operator

Résumé

This thesis focuses on analyzing the inverse scattering problems by inhomogeneous medium for the time harmonic magnetic Schrödinger operator. The first inverse problem deals with the stability of the problem of identifying the magnetic and electric potentials from near field and far field patterns using geometrical optics solutions. However, we must first show that the direct scattering problem is well-posed. To do this, we used two different approaches: one is the variational approach and the other is a volume integral equation known as the Lippmann-Schwinger integral equation. Second, we consider the inverse medium scattering problem, we review the sampling methods used to determine the shape of a perturbation from measurements of scattered waves at a fixed frequency, where our focus is on the Linear Sampling Method (LSM) and the Factorization Method (FM). Several validating results are presented in 2D. In addition, it is shown that the shape of a perturbation is uniquely determined from the far field pattern for all incident plane waves. Finally, we investigate the well-posedness of the interior transmission problem and the discreteness of the set of transmission eigenvalues by applying Fredholm theory and the upper triangular Fredholm theory.

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