Several excellent treatments of semiclassical analysis have appeared re-
cently. The book [D-S] by Dimassi and Sj¨ ostrand starts with the WKB-
method, develops the general semiclassical calculus, and then provides high-
tech spectral asymptotics. Martinez [M] provides a systematic development
of FBI transform techniques, with applications to microlocal exponential
estimates and to propagation estimates. This text is intended as a more
elementary, but much broader, introduction. Except for the general symbol
calculus, for which we followed Chapter 7 of [D-S], there is little overlap
with these other two texts or with the influential books by Helffer [He] and
by Robert [R]. Guillemin and Sternberg [G-St1] offer yet another perspec-
tive on the subject, very much complementary to that given here. Their
notes concentrate on global and functorial aspects of semiclassical analy-
sis, in particular on the theory of Fourier integral operators and on trace
The approach to semiclassical analysis presented here is influenced by my
long collaboration with Johannes Sj¨ ostrand. I would like to thank him for
sharing his philosophy and insights over the years. I first learned microlocal
analysis from Richard Melrose, Victor Guillemin, and Gunther Uhlmann,
and it is a pleasure to acknowledge my debt to them. Discussions of semi-
classical physics and chemistry with St´ ephane Nonnenmacher, Paul Brumer,
William H. Miller, and Robert Littlejohn have been enjoyable and valuable.
They have added a lot to my appreciation of the subject.
I am especially grateful to St´ ephane Nonnenmacher, Semyon Dyatlov,
Claude Zuily, Oran Gannot, Xi Chen, Hans Christianson, Jeff Galkowski,
Justin Holmer, Long Jin, Gordon Linoff, and Steve Zelditch for their very
careful reading of the earlier versions of this book and for their many valuable
comments and corrections.
My thanks also go to Faye Yeager for typing the original lecture notes
and to Jonathan Dorfman for TEX advice. Stephen Moye at the AMS pro-
vided fantastic help on deeper TEX issues and Arlene O’Sean’s excellent
copyediting removed many errors and inconsistencies.
I will maintain on my website at the UC Berkeley Mathematics De-
partment http://math.berkeley.edu/~zworski a list of errata and cor-
rections, as well as at the American Mathematical Society’s website
www.ams.org/bookpages/gsm-138. Please let me know about any errors
I have been supported by NSF grants during the writing of this book,
most recently by NSF grant DMS-0654436.