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Linear and Nonlinear Hyperbolic Equations

CONTROL FOR SCHRÖDINGER OPERATORS ON 2-TORI: ROUGH POTENTIALS

speaker: Nicolas Burq (Universite Paris Sud Orsay)

abstract: Abstract. For the Schrödinger equation, \((i\partial_t + \Delta ) u =0 \) on a torus, an arbitrary non- empty open set Ω provides control and observability of the solution: \(\
u\mid_{t=0}\
_{L^2( T^2)} \leq KT\
u\
_{L^2 ([0,T ]\times \Omega)}\) . We show that the same result remains true for \((i\partial_t + \Delta − V )u = 0\) where \(V\in L^2 (T^2)\) and \(T^2\) is a (rational or irrational) torus. That extends previous results by the two last authors and Anantharaman-Macia where the observability was proved for \(V \in C^0(T^d)\). The higher dimensional generalization remains open. This is a joint work with J. Bourgain and M. Zworski.

timetable:
Wed 3 Jul, 10:00 - 10:50, Aula Dini
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