Transition Decomposition of Quantum Mechanical Evolution

We show that the existence of the family of self-adjoint Lyapunov operators introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition of the state of a quantum mechanical system into two parts: A past time asymptote, which is asymptotic to the state of the system at t goes to minus infinity and vanishes at t goes to plus infinity, and a future time asymptote, which is asymptotic to the state of the system at t goes to plus infinity and vanishes at t goes to minus infinity. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the past time asymptote captures the behavior of the resonance. In particular, it exhibits the expected exponential decay law and spatial probability distribution.Comment: Accepted for publication in Int. J. Theor. Phy

Adaptation to hummingbird pollination is associated with reduced diversification in Penstemon

This work is licensed under a Creative Commons Attribution 4.0 International License.A striking characteristic of the Western North American flora is the repeated evolution of hummingbird pollination from insect‐pollinated ancestors. This pattern has received extensive attention as an opportunity to study repeated trait evolution as well as potential constraints on evolutionary reversibility, with little attention focused on the impact of these transitions on species diversification rates. Yet traits conferring adaptation to divergent pollinators potentially impact speciation and extinction rates, because pollinators facilitate plant reproduction and specify mating patterns between flowering plants. Here, we examine macroevolutionary processes affecting floral pollination syndrome diversity in the largest North American genus of flowering plants, Penstemon. Within Penstemon, transitions from ancestral bee‐adapted flowers to hummingbird‐adapted flowers have frequently occurred, although hummingbird‐adapted species are rare overall within the genus. We inferred macroevolutionary transition and state‐dependent diversification rates and found that transitions from ancestral bee‐adapted flowers to hummingbird‐adapted flowers are associated with reduced net diversification rate, a finding based on an estimated 17 origins of hummingbird pollination in our sample. Although this finding is congruent with hypotheses that hummingbird adaptation in North American Flora is associated with reduced species diversification rates, it contrasts with studies of neotropical plant families where hummingbird pollination has been associated with increased species diversification. We further used the estimated macroevolutionary rates to predict the expected pattern of floral diversity within Penstemon over time, assuming stable diversification and transition rates. Under these assumptions, we find that hummingbird‐adapted species are expected to remain rare due to their reduced diversification rates. In fact, current floral diversity in the sampled Penstemon lineage, where less than one‐fifth of species are hummingbird adapted, is consistent with predicted levels of diversity under stable macroevolutionary rates

Representation of Quantum Mechanical Resonances in the Lax-Phillips Hilbert Space

We discuss the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips $S$-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the Lax-Phillips $S$-matrix is unitarily related to the $S$-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable $\sigma$ of the Lax-Phillips theory. Analytic continuation in $\sigma$ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example using a Lee-Friedrichs model for the underlying dynamical system.Comment: Plain TeX, 26 pages. Minor revision

Cosmology in massive gravity

We argue that more cosmological solutions in massive gravity can be obtained if the metric tensor and the tensor $\Sigma_{\mu\nu}$ defined by St\"{u}ckelberg fields take the homogeneous and isotropic form. The standard cosmology with matter and radiation dominations in the past can be recovered and $\Lambda$CDM model is easily obtained. The dynamical evolution of the universe is modified at very early times.Comment: 4 pages, 1 figure,add more reference

Constraint and gauge shocks in one-dimensional numerical relativity

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known ``gauge shocks'' associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups ``constraint shocks'' and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.Comment: 19 pages, 8 figures and 1 table, revised version including several amendments suggested by the refere

Approximate resonance states in the semigroup decomposition of resonance evolution

The semigroup decomposition formalism makes use of the functional model for $C_{.0}$ class contractive semigroups for the description of the time evolution of resonances. For a given scattering problem the formalism allows for the association of a definite Hilbert space state with a scattering resonance. This state defines a decomposition of matrix elements of the evolution into a term evolving according to a semigroup law and a background term. We discuss the case of multiple resonances and give a bound on the size of the background term. As an example we treat a simple problem of scattering from a square barrier potential on the half-line.Comment: LaTex 22 pages 3 figure

24^{24}Mg(pp, α\alpha)21^{21}Na reaction study for spectroscopy of 21^{21}Na

The $^{24}$Mg($p$, $\alpha$)$^{21}$Na reaction was measured at the Holifield Radioactive Ion Beam Facility at Oak Ridge National Laboratory in order to better constrain spins and parities of energy levels in $^{21}$Na for the astrophysically important $^{17}$F($\alpha, p$)$^{20}$Ne reaction rate calculation. 31 MeV proton beams from the 25-MV tandem accelerator and enriched $^{24}$Mg solid targets were used. Recoiling $^{4}$He particles from the $^{24}$Mg($p$, $\alpha$)$^{21}$Na reaction were detected by a highly segmented silicon detector array which measured the yields of $^{4}$He particles over a range of angles simultaneously. A new level at 6661 $\pm$ 5 keV was observed in the present work. The extracted angular distributions for the first four levels of $^{21}$Na and Distorted Wave Born Approximation (DWBA) calculations were compared to verify and extract angular momentum transfer.Comment: 11 pages, 6 figures, proceedings of the 18th International Conference on Accelerators and Beam Utilization (ICABU2014