JournalsjstVol. 1, No. 2pp. 221–236

Scattering matrix and functions of self-adjoint operators

  • Alexander Pushnitski

    King's College London, UK
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Abstract

In the scattering theory framework, we consider a pair of operators H0H_0, HH. For a continuous function φ\varphi vanishing at infinity, we set φδ()=φ(/δ)\varphi_\delta(\cdot)=\varphi(\cdot/\delta) and study the spectrum of the difference φδ(Hλ)φδ(H0λ)\varphi_\delta(H-\lambda)-\varphi_\delta(H_0-\lambda) for δ0\delta\to0. We prove that if λ\lambda is in the absolutely continuous spectrum of H0H_0 and HH, 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)S(\lambda) for the pair H0H_0, HH and (ii) the singular values of the Hankel operator HφH_\varphi with the symbol φ\varphi.

Cite this article

Alexander Pushnitski, Scattering matrix and functions of self-adjoint operators. J. Spectr. Theory 1 (2011), no. 2, pp. 221–236

DOI 10.4171/JST/10