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上有很长的历史，特别是在数学物理中，因为它应用于薛定谔算子。我们考虑不同的薛定谔算子对的谱投影的差值和乘积，它们相差一个小的微扰，更准确地说是一个短程散射势。这个项目的目的是对这些算子的谱进行精确的数学分析，特别是它的迹类性质。这对光谱移位函数有直接的应用。此外，我们将这些发现应用于两个物理上相关的渐近性。首先，我们要证明Anderson正交性突变的精确渐近性。另一方面，我们希望证明准自由费米子系统在短程微扰下纠缠熵的面积定律的普适性。

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.