Periodic Elliptic Partial Differential Operators: The Main Structures(Series Number 223 Cambridge Studies in Advanced Mathematics)

The study of periodic partial differential equations has experienced significant growth in recent decades, driven by emerging applications in fields such as photonic crystals, metamaterials, fluid dynamics, carbon nanostructures, and topological insulators. This book provides a uniquely comprehensive overview for mathematicians, physicists, and material scientists engaged in the analysis and construction of periodic media. It describes all the mathematical objects, tools, problems, and techniques involved. Topics covered are central for areas such as spectral theory of PDEs, homogenization, condensed matter physics and optics. Although it is not a textbook, some basic proofs, background material, and references to an extensive bibliography providing pointers to the wider literature are included to allow graduate students to access the content.

Table of Contents:
Preface; Introduction; Tentative contents of the planned sequel; 1. Periodic ordinary differential operators; 2. Multidimensional periodicity: lattices, fundamental domains, Fourier series; 3. Floquet transform and direct integral decomposition; 4. Dispersion relations, Bloch, Fermi and Floquet varieties; 5. Spectral structure of periodic elliptic operators; 6. Localized perturbations of periodic operators; 7. Wannier functions; 8. Operators on Abelian coverings of compact manifolds; Appendix A. Some information from complex analysis; Appendix B. Some information from functional analysis and operator theory; Appendix C. Operator-functions; Appendix D. Banach (locally trivial) vector bundles; Appendix E. The Landis conjecture; Appendix F. Proofs of some technical statements; References; Index.

About the Author :
Peter Kuchment is the Arthur George and Mary Emolene Owen Chair and Distinguished Professor at Texas A & M University. His research interests include partial differential equations, mathematical physics, geometric analysis, inverse problems, tomography and medical imaging, material science, and math education. He is the author of the books `Floquet Theory for Partial Differential Operators' (1993), `Introduction to Quantum Graphs' (2013), 'The Radon Transform and Medical Imaging' (2014) and 'Liouville-Riemann-Roch Theorems on Abelian Coverings' (2021).