标题： 谱截断谱三元组上的非交换积分，以及与量子遍历性的联系 显示英文标题

标题： A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity

摘要： 我们为Connes--Van Suijlekom范式下的谱截断谱三元组，提出了非交换几何中非交换积分的一个简单近似。 这种近似与量子遍历性和Widom的工作之间的紧密联系立即提供了非交换几何的Szegő极限公式。 然后，我们将其与态密度联系起来。 最后，我们提出了一个关于紧致谱三元组测地流遍历性的定义。 该定义在量子遍历性中被称为$C^*$-动力系统真空态的唯一性，对于满足局部Weyl定律的谱三元组，这表明该谱三元组的狄拉克算子是量子遍历的。 这揭示了量子遍历性与Connes积分公式之间的紧密联系。 显示英文摘要

摘要： We propose a simple approximation of the noncommutative integral in noncommutative geometry for the Connes--Van Suijlekom paradigm of spectrally truncated spectral triples. A close connection between this approximation and the field of quantum ergodicity and work by Widom in particular immediately provides a Szeg\H{o} limit formula for noncommutative geometry. We then make a connection to the density of states. Finally, we propose a definition for the ergodicity of geodesic flow for compact spectral triples. This definition is known in quantum ergodicity as uniqueness of the vacuum state for $C^*$-dynamical systems, and for spectral triples where local Weyl laws hold this implies that the Dirac operator of the spectral triple is quantum ergodic. This brings to light a close connection between quantum ergodicity and Connes' integral formula.