图量子遍历性条件的必要性及与无限图的笛卡尔积
The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
摘要：
Anantharaman和Le Masson证明了一个图族的邻接算子的任何本征基族都是量子遍历的（一种非定域化的形式），假设图满足扩张和高围长的条件。在本文中，我们证明了这两个条件本身都不足以使量子遍历性成为必然。我们还表明，有条件的膨胀和特定的放松的高围长约束，目前在以后的论文量子遍历是不够的。我们这样做是通过证明新的性质的笛卡尔积的两个图，其中一个是无限的。
英文摘要：
Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.
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