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Linear operators in Hilbert spaces play a fundamental role in the formulation of quantum theory. With complete proofs and numerous exercises, this book offers a self-contained presentation of the most important tools and methods from Hilbert space theory, with a particular focus on the spectral theory of self-adjoint operators. It also goes further by describing several applications in quantum mechanics, including the analysis of Schrödinger operators and quantum scattering theory. In addition, the book discusses Mourre’s conjugate operator method and its consequences for scattering theory. Based on a one-year course for advanced undergraduates, this text provides a fundamental treatment of the basic ideas of applied Hilbert space theory.
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The conjugate operator method is a powerful recently developed technique for studying spectral properties of self-adjoint operators. One of the purposes of this volume is to present a refinement of the original method due to Mourre leading to essentially optimal results in situations as varied as ordinary differential operators, pseudo-differential operators and N-body Schrödinger hamiltonians. Another topic is a new algebraic framework for the N-body problem allowing a simple and systematic treatment of large classes of many-channel hamiltonians. The monograph will be of interest to research mathematicians and mathematical physicists. The authors have made efforts to produce an essentially self-contained text, which makes it accessible to advanced students. Thus about one third of the book is devoted to the development of tools from functional analysis, in particular real interpolation theory for Banach spaces and functional calculus and Besov spaces associated with multi-parameter C0-groups. Certainly this monograph (containing a bibliography of 170 items) is a well-written contribution to this field which is suitable to stimulate further evolution of the theory. (Mathematical Reviews)