Inverse Nodal Problems for Diffusion Operators

Preface
This book is devoted to reconstruction problems in spectral theory,
with particular emphasis on nodal structures arising from
eigenfunctions of differential operators. The subject lies at the
intersection of spectral analysis, inverse problems, and the
qualitative theory of differential equations, and has experienced
steady development over the past several decades.
Inverse spectral theory has traditionally focused on the recovery of
operator coefficients from global spectral data such as eigenvalues
and norming constants. While these approaches have led to profound
theoretical results, they are often accompanied by practical and
analytical challenges. Inverse nodal problems offer an alternative
perspective by exploiting geometric information encoded in the zero
sets of eigenfunctions. Nodal points and nodal lengths provide
localized inverse data that reflect both the oscillatory behavior of
eigenfunctions and the influence of underlying operator coefficients.
Despite the growing body of research on inverse nodal problems, the
literature remains scattered across journal articles and specialized
studies. Comprehensive and up-to-date monographs devoted specifically
to nodal-based reconstruction are relatively few. As a result,
researchers and graduate students entering the field often encounter
difficulties in obtaining a unified view of the theory and its modern
extensions. This book has been written with the aim of addressing this
gap by presenting a coherent and systematic exposition of inverse
nodal problems within the broader framework of spectral theory.
The primary focus of the book is on Sturm–Liouville type operators
defined on finite intervals and on graph-based structures,
particularly star graphs. Beginning with classical spectral theory,
the text develops the theoretical foundations of nodal analysis,
including nodal points, nodal lengths, and their asymptotic behavior.
These results are then employed to formulate and solve classical
inverse nodal problems, establishing uniqueness and reconstruction
results under natural assumptions.
A distinguishing feature of the book is its treatment of inverse nodal
problems on graphs. By incorporating matching conditions and edgewise
analysis, the theory is extended beyond interval-based operators to
network-type models. This extension reflects both recent research
trends and the increasing relevance of graph-based differential
operators in applied sciences.
In addition to analytical results, the book addresses numerical
aspects of inverse nodal reconstruction. Rather than focusing on
specific algorithms, the emphasis is placed on the theoretical
foundations of numerical approximation, including truncation effects,
consistency, convergence, and stability. A small number of validation
examples are provided to confirm that the theoretical reconstruction
formulas perform effectively when applied to finite nodal data.
The book is intended for graduate students, researchers, and
mathematicians with an interest in spectral theory and inverse
problems. A basic familiarity with functional analysis and
differential equations is assumed, although the preliminary chapters
are written to make the material accessible to readers entering the
field. The text may serve both as a reference for specialists and as a
basis for advanced graduate courses or independent study.
Throughout the book, the emphasis is placed on clarity of exposition,
consistency of notation, and logical development of ideas. The results
presented are largely classical in nature, though they are organized
and extended in a way that highlights their connections and relevance
to modern research directions. It is hoped that this unified treatment
will facilitate further work in inverse nodal theory and stimulate new
investigations into reconstruction problems in spectral analysis.