谱离域扩展量子图的量子遍历性
Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization
摘要：
我们考虑一个有限量子图序列，它具有很少的圈，使得它们在Benjamini-Schramm意义下收敛到一个随机的无限量子树。我们假设这些量子树在$I$的某个区间上是光谱离域的，在这个意义上，它们在$I$中的光谱是纯绝对连续的，并且它们的格林函数在实轴附近得到很好的控制。我们进一步假设离散图的基本序列是扩展的。我们推导出一个量子遍历性结果，表明本征值位于$I$中的本征函数是空间离域的。
英文摘要：
We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized.
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