Semi-Classical Analysis by Victor Guillemin (PDF)

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There are a number of excellent texts available on semi-classical analysis. The focus of the present monograph, however, is an aspect of the subject somewhat less systematically developed in other texts: In semi-classical analysis, many of the basic results involve asymptotic expansions in which the terms can be computed by symbolic techniques, and the focus of this monograph is the “symbol calculus” created thus. In particular, the techniques involved in this symbolic calculus have their origins in symplectic geometry, and the first seven chapters of the present work are a discussion of this underlying symplectic geometry. Another feature which differentiates this monograph from other texts is an emphasis on the global aspects of the subject: A considerable amount of time is spent here showing that the objects studied are coordinate-invariant and hence make sense on manifolds. Wherever possible, intrinsic coordinate-free descriptions of these objects are given. Topics discussed include wave and heat trace formulas for globally defined semi-classical differential operators on manifolds, and equivariant versions of these results involving Lie group actions.

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