The connection between the rigged Hilbert space and the complex scaling approaches for resonances. The Friedrichs model

The rigged Hilbert space approach and the complex scaling method are compared for the Friedrichs model. It is shown that they define the same resonance pole. The choice of the test space for the rigged Hilbert space approach is discussed.     

Examples of Gamow vectors

We present three selected physical examples of Gamow vectors to illustrate their physical significance, namely, central potential scattering, one-dimensional lattice with impurities in electric field (Stark) and a simple model for unstable elementary particles (Friedrichs).     

Effect of Freon matrices on the intermediates stabilized in X-ray irradiated 1,7-dioxaspiro[5,5]undecane solutions at 77 K

1. Download : Download high-res image (63KB)
2. Download : Download full-size image

     

[3+2] Cyclodimerization of 2-arylcyclopropane-1,1-diesters. Lewis acid induced reversion of cyclopropane umpolung

A novel Lewis acid catalyzed [3+2] cyclodimerization of 2-arylcyclopropane-1,1-dicarboxylates is reported. It is the first example of a reaction wherein a donor-acceptor cyclopropane provides two carbons in a newly formed ring. The described cyclodimerization represents a general convenient approach to polyfunctionalized cyclopentanes.     

Coefficient of detector collection

An analytical model for the coefficient of collection for an SEM detector field is proposed. The Dirichlet problem for calculating the detector’s electrostatic field is solved. The trajectories of secondary electrons are calculated numerically. Collection efficiency diagrams for different ratios of work length to pole piece diameter are constructed. The results could be useful for linear measurements of nanoscale structures and modeling the reference signal in electron beam lithography.     

Quantum billiards in multidimensional models with branes

A gravitational $D$ -dimensional model with $l$ scalar fields and several forms is considered. When a cosmological-type diagonal metric is chosen, an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed; the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions asymptotic solutions to WDW equation are found in the limit of the formation of the billiard walls which reduce the problem to the so-called quantum billiard on the $(D+ l -2)$ -dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with $9$ -dimensional quantum billiard for $D = 11$ model with $330$ four-forms which mimic space-like $M2$ - and $M5$ -branes of $D=11$ supergravity. The second one deals with the $9$ -dimensional quantum billiard for $D =10$ gravitational model with one scalar field, $210$ four-forms and $120$ three-forms which mimic space-like $D2$ -, $D4$ -, $FS1$ - and $NS5$ -branes in $D = 10$ $II A$ supergravity. It is shown that in both examples wave functions vanish in the limit of the formation of the billiard walls (i.e. we get a quantum resolution of the singularity for $11D$ model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behavior when all $120$ electric branes are present.