Dispersion on certain Cartesian products of graphs

Abstract

In this short note we prove a sharp dispersive estimate $\Vert {\mathrm{e}}^{\mathrm{i}tH}f{\Vert }_{\infty }<t^{-d/3}\Vert f{\Vert }_1$ for any Cartesian product ${\mathbb{Z}}^d\square G_F$ 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.

5 pages, 4 figures