The spectral density of a product of spectral projections

2014/09/03 by Rupert L. Frank, Frank, Rupert L., Alexander Pushnitski +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.FA #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1409.1206

23 pages; minor revisions

arxiv created 2015/03/10 · arxiv updated 2015/03/11

Abstract

We consider the product of spectral projections Π_ε(λ) = 1_(-∞,λ-ε)(H₀) 1_(λ+ε,∞)(H) 1_(-∞,λ-ε)(H₀) where H₀ and H are the free and the perturbed Schrödinger operators with a short range potential, λ>0 is fixed and ε→0. We compute the leading term of the asymptotics of Tr f(Π_ε(λ)) as ε→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.

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