Many Body Schrödinger Equation : Scattering Theory and Eigenfunction Expansions Spectral properties for Schrödinger 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 Schrödinger 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 Schrödinger operators—Time dependent approach and stationary approach (3) Time dependent approach to N body Schrödinger operators (4) Eigenfunction expansion theory for three body Schrödinger operators Compared 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 Schrödinger 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 Schrödinger operators and standard knowledge for future development. . Self Adjoint Operators and Spectra Two Body Problem Asymptotic Completeness for Many Body Systems Resolvent of Multi Particle System Three Body Problem and the Eigenfunction Expansion Supplement.