Control for Schrödinger operators on 2-tori: rough potentials. (English) Zbl 1279.35016

Bourgain, Jean; Burq, Nicolas; Zworski, Maciej

J. Eur. Math. Soc. (JEMS) 15, No. 5, 1597-1628 (2013).

Summary: For the Schrödinger equation \((i \partial_t+\Delta)u=0\) on a torus, an arbitrary non-empty open set \(\Omega\) provides control and observability of the solution: \(\| u|_{t=0}\|_{L^2(\mathbb T^2)}\leq K_T\| u \|_{L^2([0,T]\times\Omega)}\). We show that the same result remains true for \(( i\partial_t+\Delta-V)u=0\) where \(V\in L^2(\mathbb T^2)\), and \(\mathbb T^2\) is a (rational or irrational) torus. That extends the results of N. Anantharaman and F. Maciá [“Semiclassical measures for the Schrödinger equation on the torus”, Preprint, arXiv:1005.0296], and the second and third author [Math. Res. Lett. 19, No. 2, 309–324 (2012; Zbl 1281.35011)] where the observability was proved for \(V \in C(\mathbb T^2)\) and conjectured for \(V\in L^\infty(\mathbb T^2)\). The higher-dimensional generalization remains open for \(V\in L^\infty(\mathbb T^n)\).

MSC:

35J10	Schrödinger operator, Schrödinger equation
58J05	Elliptic equations on manifolds, general theory
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

Schrödinger equation; control; observability; two-dimensional tori

References:

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