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Inverse Spectral and Scattering Theory - An Introduction (Paperback, 1st ed. 2020)
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Inverse Spectral and Scattering Theory - An Introduction (Paperback, 1st ed. 2020)
Series: SpringerBriefs in Mathematical Physics, 38
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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.
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