On Novel Geometrical Structures Of Maxwellian Eigenfunctions With Applications In Inverse Scattering Problems

Spectrum theory is the core subject in the field of mathematical physics and applied mathematics.The geometric structure of partial differential operator's eigenfunctions has a very long and colorful history.In this thesis,we shall carry out the research on the geometric structure of Maxwellian eigenfunctions.Specifically,we reveal the mathematical relationship between the vanishing order of the underlying Maxwellian eigenfunction at the edge corner or vertex corner and the dihedral angle of the plane forming the associated edge corner or vertex corner.The findings have important applications in the uniqueness identifiability of electromagnetic obstacle inverse scattering.In this thesis,we develop a local argument to establish the local unique determination of the shape and the corresponding physical boundary parameter by a single far field measurement for the irrational polyhedron with the perfect electric conductor boundary,the perfect magnetic conductor boundary or impedance boundary condition.Furthermore,the convex hull of the aforementioned polyhedron and its impedance boundary parameters can be uniquely determined by a single far-field measurement.We introduce a class of uniformly concave irrational polyhedron and the corresponding global uniqueness of the scatter's shape by a single far-field measurement is established.