PREFACE vn of Morse's lemma and its extensions to the existence of polynomial normal forms for functions satisfying the Milnor condition. The technique of proof, due to Moser and Palais, will recur several times in various contexts where normal forms of different kinds of geometric objects are required. This chapter is rather short and is intended mainly to motivate the type of asymptotic developments we consider in Chapters II and VII. Chapter II. Differential Operators and Asymptotic Solutions We discuss the Luneburg-Lax-Ludwig technique for constructing asymptotic solutions for differential equations P(x,D9r) involving a large parameter T. This technique involves solving the characteristic or eikonal equation and then inductively solving a series of transport equations. We show how these can be solved using methods of symplectic geometry and in doing so also show how the characteristic and transport equation, viewed symplecticly, make sense in the vicinity of caustics even though the asymptotic expansions blow up at caustics. This enables us, in §6, to analytically "continue" our asymptotic expansions through caustics making the appropriate phase adjustments. In §7, we apply these results to the time independent Schrodinger equation and derive the Bohr- Sommerfeld quantization conditions of Keller and Maslov. We also derive an asymptotic formula for the fundamental solution of the time dependent Schro- dinger equation and show that it can be interpreted, as Feynmann does, as the evaluation of a Feynmann integral by means of stationary phase1. Chapter III. Geometrical Optics We indicate how the techniques of the preceding chapter can be applied to optics. The point of view we emphasize here is that geometrical optics is just the study of the bicharacteristics of Maxwell's equation. In particular we develop the elementary laws of refraction and reflection from this point of view, and discuss focusing and magnification. We spend more time than the reader may think is warranted discussing Gaussian optics. However, this section is partly intended to motivate our study of the linear symplectic group a little later on. We also want to emphasize that the abstract mathematical considerations of Chapters II and IV are quite closely connected to concrete physical applications. However the treatment is not meant to be complete. There are two excellent and highly readable texts on the subject: Born and Wolf, "Principles of Physical Optics" and Luneburg, "Mathematical Theory of Optics", and the reader is referred to them for a detailed treatment. We finally discuss Maxwell's equations themselves. The first three chapters can be viewed as principally motivational. The main formal development starts with Chapter IV. 1 The techniques we describe in this chapter have been further developed by the mathematicians of the NYU school to obtain deep results in physical optics, acoustics and scattering theory. We regret very much that the limitations of this book prevent us from touching on their remarkable work.
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