2013/01/07 by Jean Bourgain, Nicolas Burq, Bourgain, Jean +3
Engineering · Mathematics · #35BXX #35PXX #93BXX #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math.AP #math.OC #msc:35BXX #msc:35PXX #msc:93BXX
paper · pdf · doi:10.48550/arxiv.1301.1282
arxiv created 2013/01/07 · openalex publication_date 2013/01/07 · arxiv updated 2013/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the Schrödinger equation, (i ∂t + Δ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: ‖ u |t = 0 ‖L2 (\T2) ≤ KT ‖ u ‖L2 ([0,T] × Ω) . We show that the same result remains true for (i ∂t + Δ- V) u = 0 where V ∈ L2 (\T2) , and \T2 is a (rational or irrational) torus. That extends the results of \citeAM, and \citeBZ4 where the observability was proved for V ∈ C (\T2) and conjectured for V ∈ L^∞ (\T2) . The higher dimensional generalization remains open for V ∈ L^∞ (\Tn) .