Increasing stability in the inverse problem for the Schrödinger equation

Abstract

The Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes with time. It was formulated in late 1925 by the Austrian physicist Erwin Schrödinger. The study of the inverse problem for the Schrödinger equation focuses on finding the potential c from the prescribed boundary condition, which is generally given as Cauchy data containing both solution on the boundary and its normal derivative, or the Dirichlet-to-Neumann operator which maps the solution on the boundary to its normal derivative. The result of research has direct application to optical tomography, which is an inverse problem of reconstructing medical images through transmission of light. More precisely, one can detect cancer by recovering the absorption and scattering coefficients in the transport equation. The paper [30] discussed the simplification of the transport equation into the Schrödinger equation. Optical tomography with partial data is considered extremely valuable, since we do not have access to the full boundary in real application. The research in the dissertation assumes partial data, which can be applied to breast cancer detection. The main result of this dissertation demonstrates the increasing stability phenomenon in the inverse problem for the Schrödinger equation with partial data. We establish the theorem which contains the stability estimate bound for c. The bound decays as the energy k grows in a certain interval, and hence shows a better stability of recovering c there. In addition, we found a numerical algorithm for the linearized (simplified) inverse problem by using the Neumann-to-Dirichlet boundary map. The algorithm gives numerical evidence of increasing stability, which confirmed the theoretical prediction. The proof of uniqueness for this inverse problem was established before. The proof used almost exponential solution for the Schrödinger equation and the Fourier transform of c. A similar technique will be used in this dissertation to obtain the stability bound, but our choice of ζ in the almost exponential solutions is new.