Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include: 1) Analysis of the invertibility of the steps in the Gel’fand-Levitan and Marchenko inversion procedures, 2) Theory of the inverse ...

Property C, that is, completeness of the set of products of some solutions to SturmLiouville equations is proved. Several uniqueness theorems for various inverse scattering problems are obtained in a very simple way with the help of property C. Two classical uniqueness results for inverse scattering problem on the half-axis and for the inverse spectral problem are proved in a very short new way...

In the past few years transmission eigenvalues have become an important area of research in inverse scattering theory with active research being undertaken in many parts of the world. Transmission eigenvalues appear in the study of scattering by inhomogeneous media and are closely related to non-scattering waves. Such eigenvalues provide information about material properties of the scattering m...

Inferring internal scattering parameters for general, heterogeneous materials, remains a challenging inverse problem. Its difficulty arises from the complex way in which scattering materials interact with light, as well as the very high dimensionality of the material space implied by heterogeneity. The recent emergence of diverse computational imaging techniques, together with the widespread av...

We compute the single particle inverse lifetime, evaluated in the superconducting state. Within the BCS framework, the calculation can be done nonperturbatively, i.e. poles can be found well away from the real axis. We find that perturbative calculations are in good agreement with these results, even for non-zero impurity scattering. With electron-phonon scattering added to the problem, we use ...

We consider the problem of imaging the optical properties of a highly scattering medium probed by diffuse light. An analytic solution to this problem is derived from the singular value decomposition of the forward-scattering operator, which leads to explicit inversion formulas for the inverse scattering problem with diffusing waves. Computer simulations are used to illustrate these results in m...

We recount some of Ralph Kleinman's scientiic achievements in obstacle scattering for time harmonic waves by reporting on two topics where R.K. and R.K. interacted mathematically. The rst topic is the problem of nonuniqueness for the classical boundary integral equations for the direct scattering problem and the second is the development of numerical methods for the inverse scattering problem.

We generalize the results of [22], [11], [8] to study the inverse scattering problem of the Ward equation with non-small data and solve the Cauchy problem of the Ward equation with a non-small purely continuous scattering data.

We investigate the inverse-scattering problem that arises in total internal reflection microscopy. An analytic solution to this problem within the weak-scattering approximation is used to develop a novel form of three-dimensional microscopy with subwavelength resolution.

In the inverse stationary transport problem through anisotropic attenuating, scattering, and refractive media, the albedo operator stably determines the gauge equivalent class of the attenuation and scattering coefficients.