Chapter 1
INTRODUCTION
1.1 Basic themes
1.2 Classical and quantum mechanics
1.3 Overview
1.4 Notes
1.1. BASIC THEMES
A major goal of this book is understanding the relationships between dy-
namical systems and the behavior of solutions to various linear partial differ-
ential equations (PDE) and pseudodifferential equations containing a small
positive parameter h. We outline in this opening section some of the math-
ematical issues and challenges.
1.1.1. PDE with small parameters. The principal realm of motivation
is quantum mechanics, in which case we informally understand h as related
to Planck’s constant. With this interpretation in mind, we break down our
basic task into these two subquestions:
(i) How and to what extent do classical dynamics determine the behavior
as h 0 of solutions to Schr¨ odinger’s equation
ih∂tu =
−h2Δu
+ V u
and the related Schr¨ odinger eigenvalue equation
−h2Δu
+ V u = Eu?
The name “semiclassical” comes from this interpretation.
1
http://dx.doi.org/10.1090/gsm/138/01
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