2022/12/28 by Imanuvilov, Oleg, Yamamoto, M.
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.13650
The manuscript is concerned with uniqueness and stability for inverse source problem of determining spatially varying factor f(x) of a source term given by R(t)f(x) with suitable given R(t) in the right hand side of the Schrödinger equation with time independent coefficients. In order to establish these results we provide a simple proof of a logarithmic conditional stability of the Cauchy problem for the Schrödinger equation with time-independent coefficients and the zero Dirichlet boundary conditions on the whole boundary and a proof of uniqueness of solution to the Cauchy problem for the Schrödinger equation with data on an arbitrary small part of a lateral boundary. We do not assume any geometrical constraints on subboundary and a time interval is arbitrary. The key is an integral transform, with a kernel solving a null controllability problem for the 1-D Schrödinger, which changes a solution of the Schrödinger equation to a solution of an elliptic equation.