  This book provides a systematic study of spectral and scattering theory for
many-body Schr\"odinger operators at two-cluster thresholds. While the two-body
problem (reduced after separation of the center of mass motion to a one-body
problem at zero energy) is a well-studied subject, the literature on the
many-body problem is sparse. However our analysis covers for example the system
of three particles interacting by Coulomb potentials and restricted to a small
energy region to the right of a fixed nonzero two-body eigenvalue $\lambda_0$.
In general we address the question: How does scattering quantities for the
many-body atomic and molecular models behave in the limit when the total energy
approaches a fixed two-cluster threshold $\lambda_0$? This includes mapping
properties and singularities of the limiting scattering matrix, asymptotics of
the total scattering cross-section and absence of transmission from one channel
to another in the small inter-cluster kinetic energy region. Our principal
tools are the Feshbach-Grushin dimension reduction method and spectral analysis
based on a certain Mourre estimate. Additional topics (of independent interest)
are the limiting absorption principle, micro-local resolvent estimates, Rellich
and Sommerfeld type theorems and asymptotics of the limiting resolvents at
thresholds. While all of these features are fairly well-understood for two-body
Schr\"odinger operators, they are poorly understood in the many-body case even
for two-cluster thresholds. It is the goal of the book to remedy this point.
The mathematical physics field under study is very rich, and there are many
open problems, several of them stated explicitly in the book for the interested
reader.
