Operator interpretation of the resonances generated by 2×2 matrix Hamiltonians

Abstract

We consider the analytic continuation of the transfer function for a 2 × 2 matrix Hamiltonian into the unphysical sheets of the energy Riemann surface. We construct a family of non-selfadjoint operators which reproduce certain parts of the transfer-function spectrum including resonances situated on the unphysical sheets neighboring the physical sheet. On this basis, completeness and basis properties for the root vectors of the transfer function (including those for the resonances) are proved.

Graph Subspaces and the Spectral Shift Function

“…Extensive bibliography is devoted to the subject. Not pretending to be complete we refer to [AdL1], [AdL2], [AdLMS], [AdLT], [Adt], [AdMS], [AtLMS], [BhDM], [BhR], [D], [DK1], [DK2], [DR], [LuR], [MalMi], [MeMo1], [MeMo2], [MeS], [Mo1], [Mo2], [Ph], [Ro].…”

Spectral analysis of relativistic atoms – interaction with the quantized radiation field

“…This name is due to Haynsworth [20], who used it in honor of the Schur determinant formula. Also Menniken and Motovilov [34,33] use the Schur complement to treat resonances of 2 × 2-operator matrices. They call it "transfer function", however.…”

“…The author would like to thank his thesis advisor Heinz Siedentop for suggesting the investigation of the lifetime of relativistic excited states and constantly surveying the progress of the work, which is part of his PhD thesis. He would like to thank Ira Herbst and David Hasler for interesting conversations and Christiane Tretter for pointing out references [34,33]. He acknowledges support by the Deutsche Forschungsgemeinschaft (DFG), grant no.…”