PREFACE This book originated with a course I taught at UC Berkeley during the spring of 2003, with class notes taken by my colleague Lawrence C. Evans. Various versions of these notes have been available on-line as the Evans-Zworski lecture notes on semiclassical analysis and our original intention was to use them as the basis of a coauthored book. Craig Evans’s contributions to the current manuscript can be recognized by anybody familiar with his popular partial differential equations (PDE) text [E]. In the end, the scope of the project and other commitments prevented Craig Evans from participating fully in the final stages of the effort, and he decided to withdraw from the responsibility of authorship, generously allowing me to make use of the contributions he had already made. I and my readers owe him a great debt, for this book would never have appeared without his participation. Semiclassical analysis provides PDE techniques based on the classical- quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel–Kramers–Brillouin (WKB) approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics or effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and non- linear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics. Readers are expected to have reasonable familiarity with standard PDE theory (as recounted, for example, in Parts I and II of [E]), as well as a basic understanding of linear functional analysis. On occasion familiarity with differential forms will also prove useful. xi
Previous Page Next Page