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It is commonly known that three or more particles interacting via a
two-body potential is an intractable problem. However, similar
systems confined to one dimension yield exactly solvable equations,
which have seeded widely pursued studies of one-dimensional n-body
problems. The interest in these investigations is justified by
their rich and quantitative insights into real-world classical and
quantum problems, birthing a field that is the subject of this
book. Spanning four bulk chapters, this book is written with the
hope that readers come to appreciate the beauty of the mathematical
results concerning the models of many-particle systems, such as the
interaction between light particles and infinitely massive
particles, as well as interacting quasiparticles. As the book
discusses several unsolved problems in the subject, it functions as
an insightful resource for researchers working in this branch of
mathematical physics.In Chapter 1, the author first introduces
readers to interesting problems in mathematical physics, with the
prime objective of finding integrals of motion for classical
many-particle systems as well as the exact solutions of the
corresponding equations of motions. For these studied systems,
their quantum mechanical analogue is then developed in Chapter 2.
In Chapter 3, the book focuses on a quintessential problem in the
quantum theory of magnetism: namely, to find all integrable
one-dimensional systems involving quasiparticles of interacting
one-half spins. Readers will study the integrable periodic chains
of interacting one-half spins and discover the integrals of motion
for such systems, as well as the eigenvectors of their
corresponding Hamiltonians. In the last chapter, readers will study
about integrable systems of quantum particles, with spin and mutual
interactions involving rational, trigonometric, or elliptic
potentials.
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