In this work, we study the controllability of the bilinear Schrodinger equation on infinite graphs for periodic quantum states. We consider the equation (BSE) $i\partial_t\psi = −\Delta \psi+ u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathcal{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $−\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u \in L^2 ((0, T), \mathbb{R})$ with $T > 0$. We present the well-posedness of the (BSE) in suitable subspaces of $L^2_p$. In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes.

中文翻译：

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无限图上周期性双线性量子系统的可控性

在这项工作中，我们研究了双线性薛定谔方程在周期性量子态的无限图上的可控性。我们考虑由定义在无限图 $\mathcal{G} 上的函数组成的希尔伯特空间 $L^2_p$ 中的方程 (BSE) $i\partial_t\psi = −\Delta \psi+ u(t)B\psi$ $ 验证无限边上的周期性边界条件。拉普拉斯算子 $−\Delta$ 配备特定边界条件，$B$ 是有界对称算子，$u \in L^2 ((0, T), \mathbb{R})$ with $T > 0$ . 我们在 $L^2_p$ 的合适子空间中展示了 (BSE) 的适定性。在这样的空间中，我们研究了全局精确可控性，并提供了涉及例如蝌蚪图和具有无限辐条的星图的例子。