Preface to the First Edition Several years ago, I was working simultaneously on three problems: one con- cerned scattering of a quantum mechanical particle from a very singular repulsive core [88], one involved bounds on the number of negative eigenvalues of —A -f- AV with the correct behavior as A oo [296], and the third involved the structure of the two-dimensional Yukawa quantum field theory [288, 286]. The physics and the fundamental mathematical structure of these problems are quite different. But it turned out that the technical tools needed to solve the problems were remarkably similar, so much so that at times I couldn't keep straight which one I was thinking about. Since that time, I have had a great respect and use for a subject that might be called "the hard analysis of compact operators in Hilbert space." I discovered that many of the ideas that I grew so fond of had already been developed by Rus- sian mathematicians and mathematical physicists, particularly the group around M. S. Birman (e.g., [35, 37, 38, 40, 44, 260]). In these lectures, I wish to describe the main ideas and illustrate the tools in a group of specific problems. I am a firm believer in the principle that ideas in analysis should be valued largely by their applicability to other parts of mathematics, so I have included lots of applications chosen from my own specialty of mathematical physics, especially quantum theory. However, I have sufficient faith in these tools that I don't doubt that I would have lots of applications if I worked in some other area of analysis. I warn the reader that there is some overlap with pedagogical presentations I have given elsewhere of bits and pieces of this material (Section VI.5, 6 of [250] the appendix to Section IX.4 of [251], the second appendix to Section XI.3 in [253], Section XIII. 17 in [254], and my review article on determinants [300]) and that virtually nothing I have to say here is not already in the research literature. For beautiful presentations of some of the material from a somewhat different viewpoint I recommend highly the monograph of Goh'berg-Krein [134] and Ringrose [256]. In particular, much in Chapters 1-3 follows [134]. Many of the results of the Birman school are summarized in the lecture notes of Birman and Solomjak [45] which have recently been translated. Like so much of modern analysis, the material to be described has its roots in the famous paper of Fredholm [115] (this deep paper is extremely readable and I recommend it to those wishing a pleasurable afternoon). One of the responses to this paper was a flurry of activity from Hilbert and his school which led eventually to the abstraction of what we now call Hilbert space and the Hilbert-Schmidt operators. In modern notation, this latter is the family of operators, J2? with TT(A*A) 00. (I should mention that where I use 3P, one often sees Cp, £p, or 23p.) For many years, there were many theorems about operators which are products of two or more Hilbert-Schmidt operators (e.g., [186]) until von Neumann and Schatten [280, 279] formalized the notion of the trace class, 3\. These two vii
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