Inverse transmission problems for magnetic Schrödinger operators (Q2878682)

The present paper generalizes some earlier results of other publications obtained on inverse transmission problems. The proposed transmission problem includes in particular the magnetic Schrödinger operator.NEWLINENEWLINE The main idea is based on a paper by \textit{V. Isakov} [Commun. Math. Phys. 280, No. 3, 843--858 (2008; Zbl 1143.35387)] in which similar problems in the absence of magnetic potentials were studied.NEWLINENEWLINE The goal of this work is to extend such results related to inverse transmission problems for magnetic Schrödinger equations with magnetic potential \(A\) and electric potential \(q\) in the presence of a transparent obstacle \(D\) (plus some suitable conditions on \(A\), \(q\), and the boundary of \(D\)) on bounded domains \(\Omega\subset\mathbb{R}^n\) \((n\geq 3)\) with connected Lipschitz boundary in the self-adjoint case.NEWLINENEWLINE It is shown that, if the magnetic and electric potentials are known outside the obstacle \(D\) in \(\Omega\), the knowledge of the Cauchy data for the transmission problem prescribed on an open nonempty subset of the boundary \(\partial\Omega\) or from complete scattering data at fixed frequency determines uniquely the obstacle, transmission coefficients, magnetic potential and electric potential inside the obstacle.NEWLINENEWLINE The proofs use the so-called ``singular solutions for the transmission problems technique''. This method is distinguished by some estimates for fundamental solutions of the Schrödinger operator, which works in the estimating singular solutions. Also a unique continuations result for the elliptic second-order operators, the mini-max principle to unique solvability of transmission problems by a small perturbation of the boundary \(\partial\Omega\) and finally asymptotic bounds on some volume and surface integrals are required in the reconstruction of the obstacle and of the transmission coefficients. A few nice operational results are added about the relation of the subject to proofs and previous papers. This note is clearly written and provides a valuable contribution to the many aspects of this area.