Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

On the systematic position of some Asian enigmatic genera of Asclepiadoideae (Apocynaceae)

The phylogenetic structure of Asclepiadoideae (Apocynaceae) has been elucidated at the tribal and subtribal levels in the last two decades. However, to date, the systematic positions of seven Asian genera, Cosmostigma, Graphistemma, Holostemma, Pentasachme, Raphistemma, Seshagiria and Treutlera, have not been investigated. In this study, we examine the evolutionary relationships among these seven small enigmatic Asian genera and clarify their positions in Asclepiadoideae, using a combination of plastid sequences of rbcL, rps16, trnL and trnL- F regions. Cosmostigma and Treutlera are resolved as members of the non-Hoya clade of Marsdenieae with strong support (maximum parsimony bootstrap support value BSMP = 96, maximum likelihood bootstrap support value BSML = 98, Bayesian-inferred posterior probability PP = 1.0). Pentasachme is resolved as sister of Stapeliinae to Ceropegieae with moderate support (BSMP = 64, BSML = 66, PP = 0.94). Graphistemma, Holostemma, Raphistemma and Seshagiria are all nested in the Asclepiadeae-Cynanchinae clade (BSMP = 97, BSML = 100, PP = 1.0). The study confirms the generally accepted tribal and subtribal structure of the subfamily. One exception is Eustegia minuta, which is placed here as sister to all Asclepiadeae (BSMP = 58, BSML = 76, PP = 0.99) and not as sister to the Marsdenieae + Ceropegieae clade. The weak support and conflicting position indicate the need for a placement of Eustegia as an independent tribe. In Asclepiadeae, a sister group position of Cynanchinae to the Asclepiadinae + Tylophorinae clade is favoured (BSMP = 84, BSML = 88, PP = 1.0), whereas Schizostephanus is retrieved as unresolved. Oxystelma appears as an early-branching member of Asclepiadinae with weak support (BSMP = 52, BSML = 74, PP = 0.69). Calciphila and Solenostemma are also associated with Asclepiadinae with weak support (BSMP = 37, BSML = 45, PP = 0.79), but all alternative positions are essentially without support. The position of Indian Asclepiadoideae in the family phylogeny is discussed. (c) 2014 The Linnean Society of London, Botanical Journal of the Linnean Society, 2014, 174, 601-619.