Spectral properties for Schr�dinger operators are a major concern in quantum mechanics both in physics and in mathematics. For the few-particle systems, we now have sufficient knowledge for two-body systems, although much less is known about N-body systems. The asymptotic completeness of time-dependent wave operators was proved in the 1980s and was a landmark in the study of the N-body problem. However, many problems are left open for the stationary N-particle equation. Due to the recent rapid development of computer ...
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Spectral properties for Schr�dinger operators are a major concern in quantum mechanics both in physics and in mathematics. For the few-particle systems, we now have sufficient knowledge for two-body systems, although much less is known about N-body systems. The asymptotic completeness of time-dependent wave operators was proved in the 1980s and was a landmark in the study of the N-body problem. However, many problems are left open for the stationary N-particle equation. Due to the recent rapid development of computer power, it is now possible to compute the three-body scattering problem numerically, in which the stationary formulation of scattering is used. This means that the stationary theory for N-body Schr�dinger operators remains an important problem of quantum mechanics. It is stressed here that for the three-body problem, we have a satisfactory stationary theory. This book is devoted to the mathematical aspects of the N-body problem from both the time-dependent and stationary viewpoints. The main themes are:(1) The Mourre theory for the resolvent of self-adjoint operators(2) Two-body Schr�dinger operators-Time-dependent approach and stationary approach(3) Time-dependent approach to N-body Schr�dinger operators(4) Eigenfunction expansion theory for three-body Schr�dinger operatorsCompared with existing books for the many-body problem, the salient feature of this book consists in the stationary scattering theory (4). The eigenfunction expansion theorem is the physical basis of Schr�dinger operators. Recently, it proved to be the basis of inverse problems of quantum scattering. This book provides necessary background information to understand the physical and mathematical basis of Schr�dinger operators and standard knowledge for future development.
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