The spectral density of a product of spectral projections

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 Schrödinger operators with a short range potential, $\lambda >0$ is fixed and $\epsilon \to 0$. We compute the leading term of the asymptotics of $\mathrm{Tr}f({\Pi }_{\epsilon }(\lambda ))$ as $\epsilon \to 0$ 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.

23 pages; minor revisions

References in corpus (1)

• Quasi-diagonalization of Hankel operators

Cited by in corpus (1)

• The spectral density of a difference of spectral projections