We propose a new iterative method to recover a real compact supported potential of the Schödinger operator from their fixed angle scattering data. The method combines a fixed point argument with a suitable approximation of the resolvent operator by partial sums associated to its Born series. Unlike other iterative methods in the literature, each iteration is explicit and therefore faster. We study the convergence of the numerical method and implementation details.

Broadly speaking, there are two classes of inverse problems — those that are concerned with the analysis of PDEs, and those that are geometric in nature. In this talk, I will introduce the audience to these classes. In the PDE setting, as a motivating example I will provide a brief introduction to Calderón’s problem, in which one aims to determine an electric conductivity in a body from current measurements on the surface. In the geometric setting, I will survey the classical boundary rigidity problem for simply connected, Riemannian manifolds with boundary, in which one seeks to determine the Riemannian metric given the distance between any two points on the boundary. Then, I will present some current results to an inverse problem which uses techniques from both the PDE perspective and the geometric perspective. In particular, I will consider the following question: Given any simple closed curve $\gamma$ on the boundary of a Riemannian 3-manifold \((M,g)\), suppose the area of the least-area surfaces bounded by \(\gamma\) are known. From this data may we uniquely recover the metric \(g\)? In several settings, I will show the answer is yes. In fact, I will provide both global and local uniqueness results given least-area data for a much smaller class of curves on the boundary. The uniqueness for the metric \(g\) will be demonstrated by reformulating parts of the problem as a 2-dimensional inverse problem on an area-minimizing surface. This is joint work with S. Alexakis and A. Nachman.

In this talk, we will first review some classical results on the so-called ’spectral inequalities’, which yield a sharp quantification of the unique continuation of the spectral family associated with the Laplace-Beltrami operator in a compact manifold. In a second part, we will discuss how to obtain the spectral inequalities associated to the Schrodinger operator \(-\Delta_x + V(x)\), in \(\mathbb{R}^d\), in any dimension \(d\geq 1\), where \(V=V(x)\) is a real analytic potential. In particular, we can handle some long- range potentials. This is a joint work with Prof G. Lebeau (Université de Nice-Côte d'Azur, France).