Regularity of eigenstates in regular Mourre theory

Jacob Schach Møller, Matthias Westrich

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    The present paper gives an abstract method to prove that possibly embedded eigenstates of a self-adjoint operator H lie in the domain of the kth power of a conjugate operator A. Conjugate means here that H and A have a positive commutator locally near the relevant eigenvalue in the sense of Mourre. The only requirement is Ck+1(A) regularity of H. Regarding integer k, our result is optimal. Under a natural boundedness assumption of the multiple commutators we prove that the eigenstate ‘dilated’ by exp(iθA) is analytic in a strip around the real axis. In particular, the eigenstate is an analytic vector with respect to A. Natural applications are ‘dilation analytic’ systems satisfying a Mourre estimate, where our result can be viewed as an abstract version of a theorem due to Balslev and Combes (1971) [3]. As a new application we consider the massive Spin-Boson Model.
    Original languageEnglish
    JournalJournal of Functional Analysis
    Pages (from-to)852-878
    Number of pages27
    Publication statusPublished - 2011


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