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

• Published in: Inverse problems Vol. 39; no. 10; pp. 105013 - 105031
• Main Authors: Imanuvilov, O Y, Yamamoto, M
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.10.2023
• Subjects: Carleman estimates; inverse source problem; Schrödinger equation; unique continuation
• ISSN: 0266-5611; 1361-6420
• DOI: 10.1088/1361-6420/acf399

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 Schrödinger equation with time independent coefficients. In order to establish these results, for the Schrödinger 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 Schrödinger equation, which transforms a solution of the Schrödinger equation to a solution of an elliptic equation.