Spectral Geometry and Inverse Scattering Theory

Inverse scattering problems are a vital subject for both theoretical and experimental studies and remain an active field of research in applied mathematics. This book provides a detailed presentation of typical setup of inverse scattering problems for time-harmonic acoustic, electromagnetic and elastic waves. Moreover, it provides systematical and in-depth discussion on an important class of geometrical inverse scattering problems, where the inverse problem aims at recovering the shape and location of a scatterer independent of its medium properties. Readers of this book will be exposed to a unified framework for analyzing a variety of geometrical inverse scattering problems from a spectral geometric perspective.  This book contains both overviews of classical results and update-to-date information on latest developments from both a practical and theoretical point of view. It can be used as an advanced graduate textbook in universities or as a referencesource for researchers in acquiring the state-of-the-art results in inverse scattering theory and their potential applications.     

Table of Contents:
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.

About the Author :
Hongyu Liu is a Professor and the Associate Head at the Department of Mathematics, City University of Hong Kong. Before taking up the current position, he worked as a Professor and the Associate Head at the Department of Mathematics, Hong Kong Baptist University (2014--2020). Prior to that, he held faculty positions at University of North Carolina, Charlotte, USA (2011--2014), University of Reading, UK (2010/11), and University of Washington, Seattle, USA (2007--2010). He obtained his PhD in Mathematics from The Chinese University of Hong Kong (2007).  His research focuses on the analysis, computations and applications of inverse problems and imaging, wave propagation, partial differential equations, mathematical materials science, scattering theory and spectral theory. He has also been working on the interplay among inverse scattering techniques, bionic learning and artificial intelligence. He has published over peer-reviewed 150 research papers in leading journals, and in addition he has 16 research preprints under review to leading journals. He coauthored one research monograph by Societe Mathematique de France.  Huaian Diao is a Professor at School of Mathematics, Jilin University, China, from October 2021.  He obtained his PhD in Mathematics from City University of Hong Kong (2007). From September 2007 to June 2021, he worked as a lecture and then an associate professor at School of Mathematics and Statistics, Northeast Normal University, China. His research interests include inverse scattering problems, numerical algebra and spectral theory. He has published over 50 peer-reviewed papers in international journals and conferences including J. Math. Pures Appl., Calc. Var. Partial Differential Equations, Comm. Partial Differential Equations, SIAM J. Math. Anal., SIAM J. Appl. Math., J. Differential Equations, Math. Comp., Inverse Problems, and NeurIPS 2019.

Review :
“The book under review provides an overview of certain modern topics in inverse scattering theory studied by the authors and their coauthors. In particular, the book is focussed on the geometric structure of eigenfunctions for several spectral problems … . Researchers working in this area will appreciate this book as a reference for these selected topics with most chapters being relatively self-contained.” (Joe Thomas, zbMATH 1543.58001, 2024)