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Control for Schrödinger operators on 2-tori: rough potentials. (English) Zbl 1279.35016
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)
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[1] Anantharaman, N., Maci‘a, F.: Semiclassical measures for the Schrödinger equation on the torus. · arxiv.org
[2] Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024-1065 (1992) · Zbl 0786.93009 · doi:10.1137/0330055
[3] Bourgain, J., Shao, P., Sogge, C. D., Yao, X.: On Lp-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds. · Zbl 1396.58025 · arxiv.org
[4] Burq, N.: Semi-classical measures for inhomogeneous Schrödinger equations on tori. Anal. PDE, to appear; · arxiv.org
[5] Burq, N., Gérard, P., Tzvetkov, N.: An instability property of the nonlinear Schrödinger equa- tion on Sd . Math. Res. Lett. 9, 323-335 (2002) · Zbl 1003.35113 · doi:10.4310/MRL.2002.v9.n3.a8
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