The spectral density of a product of spectral projections

Rupert L. Frank; Alexander Pushnitski

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arXiv · 1409.1206

The spectral density of a product of spectral projections

Rupert L. Frank·Alexander Pushnitski

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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.

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BibTeX RIS

Rupert L. Frank, Alexander Pushnitski. 2014-09-03. The spectral density of a product of spectral projections. https://arxiv.org/abs/1409.1206

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