Asymptotics of products of spectral projections with applications to Anderson's orthogonality and entanglement entropy

对自伴算子的谱投影的理解在数学中有着悠久的历史，特别是在数学物理中，由于它在薛定谔算子中的应用。我们考虑的差异和产品的光谱投影对薛定谔运营商不同的一个小扰动，更准确地说是一个短程散射潜力。该项目的目的是对这些算子的谱，特别是其迹类性质进行精确的数学分析。这对谱移函数有直接的应用。此外，我们将这些发现应用到两个物理相关的渐近性。首先，我们证明了安德森正交突变的精确渐近性。另一方面，我们希望证明在短程微扰下，准自由费米子系统纠缠熵面积定律的普适性。

The understanding of spectral projections of self-adjoint operators has a long history in mathematics and especially in mathematical physics due to its applications to Schrödinger operators. We consider differences and products of spectral projections of pairs of Schrödinger operators which differ by a small perturbation, more precisely a short-range scattering potential. The aim of this project is a precise mathematical analysis of the spectrum of these operators and especially its trace-class properties. This has immediate applications to the spectral shift function. Moreover, we apply these findings to two physically relevant asymptotics. In the first place, we intend to prove the exact asymptotics in Anderson's orthogonality catastrophe. On the other hand, we wish to show universalities of area laws of entanglement entropy under short-range perturbations for quasi-free fermionic systems.