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The Spectral Density of a Difference of Spectral Projections

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Original languageEnglish
Pages (from-to)1153-1181
Number of pages29
JournalCommunications in Mathematical Physics
Volume338
Issue number3
Early online date5 Jun 2015
DOIs
Publication statusPublished - Sep 2015

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Abstract

Let H0 and H be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if λ belongs to the absolutely continuous spectrum of H0 and H, then the difference of spectral projections D(λ)=1(-∞,0)(H-λ)-1(-∞,0)(H0-λ)in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations Dε(λ) of D(λ), given by Dε(λ)=ψε(H-λ)-ψε(H0-λ),where ψε(x)=ψ(x/ε) and ψ(x) is a smooth real-valued function which tends to 1/2 as x → ±∞. We prove that the eigenvalues of Dε(λ) concentrate to the absolutely continuous spectrum of D(λ) as ε → +0. We show that the rate of concentration is proportional to |logε| and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of ψ. The proof relies on the analysis of Hankel operators.

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