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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