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 Schrödinger 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 Schrödinger 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.
