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Analytic scattering theory for many-body systems below the smallest three-body threshold

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Abstract

We consider the Schrödinger operatorH=H 0+V of a many-body system, whereV is a sum of dilation-analytic, short range (not necessarily local) two-body interactions, together with the associated self-adjoint analytic familyH(z), |Argz|<a, of complex-dilated operators. For eachz we construct the local wave operators and the S-matrix below the smallest 3-body threshold, using abstract stationary scattering theory and the Weinberg-van Winter equation. The diagonal element of the inverse S-matrix describing scattering within the channel α in the lowest energy range is proved to be the boundary value of a meromorphic functionL α(z)(z), −a<Argz<0, whereL α(z) is the S-matrix forH(z) on the corresponding cut. Generally, the poles ofL α(z) are resolvent resonances, but a resolvent resonance may not be a pole ofL α(z), if it is embedded as an eigenvalue in the continuum ofH(z 0) for a suitablez 0.

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  1. Matematisk Institut, Universitetsparken Ny Munkegade, DK-8000, Aarhus C, Denmark

    Erik Balslev

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  1. Erik Balslev
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Communicated by J. Ginibre

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Balslev, E. Analytic scattering theory for many-body systems below the smallest three-body threshold. Commun.Math. Phys. 77, 173–210 (1980). https://doi.org/10.1007/BF01982716

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  • DOI: https://doi.org/10.1007/BF01982716

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