Sharp uniqueness and stability of solution for an inverse source problem for the Schrodinger equation

This article is concerned with the uniqueness and the stability for an inverse source problem of determining a spatially varying factor f(x) of a source term given by R(t)f(x) with suitably given R(t) of the Schrodinger equation with time independent coefficients. In order to establish these results, for the Schrodinger equation, we prove (i) a logarithmic conditional stability estimate for a Cauchy problem attached with the zero Dirichlet boundary conditions on the whole boundary, (ii) the unique continuation for the Cauchy problem from an arbitrary small part of a lateral boundary. We do not assume any constraints on the geometry of the subboundary and an observation time length. The key is an integral transform with a kernel solving a null controllability problem for the one-dimensional Schrodinger equation, which transforms a solution of the Schrodinger equation to a solution of an elliptic equation.