Inverse Boundary Spectral Problems

Inverse boundary problems are a rapidly developing area of applied mathematics. Inverse boundary spectral problems (IBSP) constitute an important and distinct part of the field because of the special character of the data involved, the specific methods used to deal with that data, and their relevance to applications, including medical imaging and geophysical prospecting. Although the theory of ISBP has been in development for at least 10 years, this book was published, the literature had been scattered throughout various journals. This self-contained monograph summarizes IBSP concepts and the techniques useful for dealing with them.

Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems.

Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following:

"Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?"

Along with this problem, many inverse problems for heat and wave equations are solved.

The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas.

Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.