Semi-Classical Analysis

The perspective taken here by Guillemin and Sternberg is that of the symbol calculus created by the symbolic techniques required by the asymptotic expansions that are fundamental to the basic results of semiclassical analysis. More precisely, it is the perspective of the relation between the symbol calculus and the underlying symplectic geometry. A central part is taken by the symbol calculus of semiclassical Fourier integral operators. Along with this, the emphasis is also placed, naturally, on the global aspects of this subject: the studied objects are coordinate invariant, and hence make sense on a manifold. The authors’ approach in describing the main objects is consequently as intrinsic as possible. The emphasis on globality therefore is also reflected in the authors’ choice of topics in the later chapters of the book: among others, wave and heat trace invariants for globally defined semiclassical differential operators on manifolds, and equivariant versions of these results involving Lie group actions…

This is a great book. The authors succeed in transferring their wide and versatile view of mathematics, and their elegant taste, into a new contribution to semiclassical analysis. With the same care with which they treated readers of their other remarkable books, the authors lead students (at the graduate level) and researchers (or any reader willing to study this beautiful, deep, and very active field) along a path which offers new landscapes of and geometric insight into semiclassical analysis.