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

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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.