Abstract
In this short note, we prove a sharp dispersive estimate \(\|\mathrm {e}^{\mathrm {i} tH} f\|{ }_\infty < t^{-d/3}\|f\|{ }_1\) for any Cartesian product \(\mathbb {Z}^d \mathop {\square } G_F\) of the integer lattice and a finite graph. This includes the infinite ladder, k-strips, and infinite cylinders, which can be endowed with certain potentials.
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Notes
- 1.
In [2], the unitary operator is instead \((\widetilde {U}\psi )_\theta (v_n)=\mathrm {e}^{-2\pi \mathrm {i}\theta \cdot v_n}(U\psi )_\theta (v_n)\), so the fiber operator obtained there is \(\widetilde {H}(\theta ) = \mathrm {e}^{-2\pi \mathrm {i}\theta \cdot }H(\theta )\mathrm {e}^{2\pi \mathrm {i}\theta \cdot }\), where \(\mathrm {e}^{\pm 2\pi \mathrm {i}\theta \cdot }f(v_p)=\mathrm {e}^{2\pi \mathrm {i}\theta \cdot v_p}f(v_p)\). This operator and ours are thus unitarily equivalent; the present version is just more suitable for our computations.
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Ammari, K., Sabri, M. (2023). Dispersion on Certain Cartesian Products of Graphs. In: Ammari, K., Jammazi, C., Triki, F. (eds) Control and Inverse Problems. CIP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-35675-9_11
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