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How can one determine the physical properties of the medium or the
geometrical properties of the domain by observing electromagnetic
waves? To answer this fundamental problem in mathematics and
physics, this book leads the reader to the frontier of inverse
scattering theory for electromagnetism.The first three chapters,
written comprehensively, can be used as a textbook for
undergraduate students. Beginning with elementary vector calculus,
this book provides fundamental results for wave equations and
Helmholtz equations, and summarizes the potential theory. It also
explains the cohomology theory in an easy and straightforward way,
which is an essential part of electromagnetism related to geometry.
It then describes the scattering theory for the Maxwell equation by
the time-dependent method and also by the stationary method in a
concise, but almost self-contained manner. Based on these
preliminary results, the book proceeds to the inverse problem for
the Maxwell equation.The chapters for the potential theory and
elementary cohomology theory are good introduction to graduate
students. The results in the last chapter on the inverse scattering
for the medium and the determination of Betti numbers are new, and
will give a current scope for the inverse spectral problem on
non-compact manifolds. It will be useful for young researchers who
are interested in this field and trying to find new problems.
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 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 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.Â
The aim of this book is to provide basic knowledge of the inverse
problems arising in various areas in mathematics, physics,
engineering, and medical science. These practical problems boil
down to the mathematical question in which one tries to recover the
operator (coefficients) or the domain (manifolds) from spectral
data. The characteristic properties of the operators in question
are often reduced to those of Schroedinger operators. We start from
the 1-dimensional theory to observe the main features of inverse
spectral problems and then proceed to multi-dimensions. The first
milestone is the Borg-Levinson theorem in the inverse Dirichlet
problem in a bounded domain elucidating basic motivation of the
inverse problem as well as the difference between 1-dimension and
multi-dimension. The main theme is the inverse scattering, in which
the spectral data is Heisenberg's S-matrix defined through the
observation of the asymptotic behavior at infinity of solutions.
Significant progress has been made in the past 30 years by using
the Faddeev-Green function or the complex geometrical optics
solution by Sylvester and Uhlmann, which made it possible to
reconstruct the potential from the S-matrix of one fixed energy.
One can also prove the equivalence of the knowledge of S-matrix and
that of the Dirichlet-to-Neumann map for boundary value problems in
bounded domains. We apply this idea also to the Dirac equation, the
Maxwell equation, and discrete Schroedinger operators on perturbed
lattices. Our final topic is the boundary control method introduced
by Belishev and Kurylev, which is for the moment the only
systematic method for the reconstruction of the Riemannian metric
from the boundary observation, which we apply to the inverse
scattering on non-compact manifolds. We stress that this book
focuses on the lucid exposition of these problems and mathematical
backgrounds by explaining the basic knowledge of functional
analysis and spectral theory, omitting the technical details in
order to make the book accessible to graduate students as an
introduction to partial differential equations (PDEs) and
functional analysis.
How can one determine the physical properties of the medium or the
geometrical properties of the domain by observing electromagnetic
waves? To answer this fundamental problem in mathematics and
physics, this book leads the reader to the frontier of inverse
scattering theory for electromagnetism.The first three chapters,
written comprehensively, can be used as a textbook for
undergraduate students. Beginning with elementary vector calculus,
this book provides fundamental results for wave equations and
Helmholtz equations, and summarizes the potential theory. It also
explains the cohomology theory in an easy and straightforward way,
which is an essential part of electromagnetism related to geometry.
It then describes the scattering theory for the Maxwell equation by
the time-dependent method and also by the stationary method in a
concise, but almost self-contained manner. Based on these
preliminary results, the book proceeds to the inverse problem for
the Maxwell equation.The chapters for the potential theory and
elementary cohomology theory are good introduction to graduate
students. The results in the last chapter on the inverse scattering
for the medium and the determination of Betti numbers are new, and
will give a current scope for the inverse spectral problem on
non-compact manifolds. It will be useful for young researchers who
are interested in this field and trying to find new problems.
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