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
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