Spectral Geometry and Inverse Scattering Theory

Introduction. -Geometric structures of Laplacian eiegenfunctions.- Geometric structures of Maxwellian eigenfunctions.-  Inverse obstacle and diffraction grating scattering problems.- Path argument for inverse acoustic and electromagnetic obstacle scattering problems.-  Stability for inverse acoustic obstacle scattering problems. - Stability for inverse electromagnetic obstacle scattering problems.-  Geometric structures of Helmholtz’s transmission eigenfunctions with general transmission conditions and applications.- Geometric structures of Maxwell’s transmission eigenfunctions and applications.- Geometric structures of Lame’s transmission eigenfunctions with general ´ transmission conditions and applications.-  Geometric properties of Helmholtz’s transmission eigenfunctions induced by curvatures and applications. - Stable determination of an acoustic medium scatterer by a single far-field pattern.- Stable determination of an elastic medium scatterer by a single far-field measurement and beyond.