Scattering matrix and functions of self-adjoint operators

2010/08/06 by Pushnitski, Alexander
#47A40 (Primary) #47B25 (Secondary) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1008.1215

Abstract

In the scattering theory framework, we consider a pair of operators H₀, H. For a continuous function ϕ vanishing at infinity, we set ϕ_δ(⋅)=ϕ(⋅/δ) and study the spectrum of the difference ϕ_δ(H-λ)-ϕ_δ(H₀-λ) for δ→0. We prove that if λ is in the absolutely continuous spectrum of H₀ and H, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix S(λ) for the pair H₀, H and (ii) the singular values of the Hankel operator H_ϕ with the symbol ϕ.

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