Preface This book can be considered as the second volume of the author’s monograph “Mathematical Scattering Theory (General Theory)” [I]. It is oriented to applica- tions to differential operators, primarily to the Schr¨ odinger operator. A necessary background from [I] is collected (but the proofs are of course not repeated) in Chapter 0. Therefore it is presumably possible to read this book independently of [I]. Everything said in the preface to [I] pertains also to this book. In particular, we proceed again from the stationary approach. Its main advantage is that, si- multaneously with proofs of various facts, the stationary approach gives formula representations for the basic objects of the theory. Along with wave operators, we also consider properties of the scattering matrix, the spectral shift function, the scattering cross section, etc. A consistent use of the stationary approach as well as the choice of concrete material distinguishes this book from others such as the third volume of the course of M. Reed and B. Simon [43]. The latter course has become a desktop copy for many, in particular, for the author of the present book. However, in view of the broad compass of material, the course [43] was necessarily written in encyclopedic style and apparently cannot replace a systematic exposition of the theory. Hopefully, vol. 3 of [43] and this book can be considered as complementary to one another. There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it. In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book we present both of these trends. The first of them illustrates basic theorems of [I]. Thus, Chapters 1 and 2 are devoted to applications of the smooth method. Of course the abstract results of [I] should be supplemented by some analytic tools, such as the Sobolev trace theorem. The smooth method works well for perturbations of differential operators with constant coefficients. In Chapter 3 applications of the trace class method are discussed. The main advantage of this method is that it does not require an explicit spectral analysis of an “unperturbed” operator. Other chapters are much less dependent on [I]. Chapters 4 and 5 are devoted to the one-dimensional problem (on the half-axis and the entire axis, respectively) which is a touchstone for the multidimensional case because specific methods of ordinary differential equations can be used here. In the following chapters we return to the multidimensional problem and discuss different analytic methods appropriate to differential operators. In particular, in Chapter 6 scattering theory is formulated in terms of solutions of the Schr¨odinger equation satisfying some “boundary conditions” (radiation conditions) at infinity. xi
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