# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.