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The spectral density of a product of spectral projections

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jun 2015Embargo end date: 01 Jan 2014 United States English Publisher:Elsevier BVJournal:Journal of Functional Analysis, volume 268, pages 3,867-3,894 (issn: 0022-1236,

Authors: Frank, Rupert L.; Pushnitski, Alexander;

doi: 10.1016/j.jfa.2015.03.018 , 10.48550/arxiv.1409.1206

arXiv: http://arxiv.org/abs/1409.1206

The spectral density of a product of spectral projections

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Abstract

We consider the product of spectral projections $$ ��_��(��) = 1_{(-\infty,��-��)}(H_0) 1_{(��+��,\infty)}(H) 1_{(-\infty,��-��)}(H_0) $$ where $H_0$ and $H$ are the free and the perturbed Schr��dinger operators with a short range potential, $��>0$ is fixed and $��\to0$. We compute the leading term of the asymptotics of $\mathrm{Tr}\ f(��_��(��))$ as $��\to0$ 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.

23 pages; minor revisions

Country

United States

Related Organizations

King's College London
United Kingdom
California Institute of Technology
United States
King's College London, University of London

Keywords

330, FOS: Physical sciences, Mathematical Physics (math-ph), 530, 510, Functional Analysis (math.FA), Mathematics - Spectral Theory, Mathematics - Functional Analysis, FOS: Mathematics, Spectral asymptotics, Schrödinger operators, Spectral Theory (math.SP), SchrÃ¶dinger operators, Mathematical Physics, Anderson orthogonality catastrophy, Hankel operators

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