The spectral density of a product of spectral projections

TL;DR

This paper analyzes the asymptotic behavior of the trace of functions of a product of spectral projections related to Schrödinger operators, revealing connections to Anderson's orthogonality catastrophe and Hankel operators.

Contribution

It computes the leading asymptotic term of the trace of functions of a spectral projection product for Schrödinger operators, highlighting the role of Hankel operators.

Findings

Asymptotic formula for trace of spectral projection products

Connection to Anderson's orthogonality catastrophe

Role of Hankel operators in spectral analysis

Abstract

We consider the product of spectral projections $$ \Pi_\epsilon(\lambda) = 1_{(-\infty,\lambda-\epsilon)}(H_0) 1_{(\lambda+\epsilon,\infty)}(H) 1_{(-\infty,\lambda-\epsilon)}(H_0) $$ where $H_0$ and $H$ are the free and the perturbed Schr\"odinger operators with a short range potential, $\lambda>0$ is fixed and $\epsilon\to0$. We compute the leading term of the asymptotics of $\mathrm{Tr}\ f(\Pi_\epsilon(\lambda))$ as $\epsilon\to0$ for continuous functions $f$ vanishing sufficiently fast near zero. Our construction elucidates calculations that appeared earlier in the theory of "Anderson's orthogonality catastrophe" and emphasizes the role of Hankel operators in this phenomenon.