Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation

Consider a system governed by the time-dependent Schr\"odinger equation in its ground state. When subjected to weak (size $\epsilon$) parametric forcing by an "ionizing field" (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay-rate of this metastable state is governed by {\it Fermi's Golden Rule}, $\Gamma[V]$, which depends on the potential $V$ and the details of the forcing. We pose the potential design problem: find $V_{opt}$ which minimizes $\Gamma[V]$ (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior points of the constraint set and to be smooth. The multi-scale structures that emerge incorporate the physical mechanisms of energy confinement via material contrast and interference effects. An analysis of locally optimal potentials reveals local optimality is attained via two mechanisms: (i) decreasing the density of states near a resonant frequency in the continuum and (ii) tuning the oscillations of extended states to make $\Gamma[V]$, an oscillatory integral, small. Our approach achieves lifetimes, $\sim (\epsilon^2\Gamma[V])^{-1}$, for locally optimal potentials with $\Gamma^{-1}\sim\mathcal{O}(10^{9})$ as compared with $\Gamma^{-1}\sim \mathcal{O}(10^{2})$ for a typical potential. Finally, we explore the performance of optimal potentials via simulations of the time-evolution.Comment: 33 pages, 6 figure

Significantly reduced CCR5-tropic HIV-1 replication in vitro in cells from subjects previously immunized with Vaccinia Virus

<p>Abstract</p> <p>Background</p> <p>At present, the relatively sudden appearance and explosive spread of HIV throughout Africa and around the world beginning in the 1950s has never been adequately explained. Theorizing that this phenomenon may be somehow related to the eradication of smallpox followed by the cessation of vaccinia immunization, we undertook a comparison of HIV-1 susceptibility in the peripheral blood mononuclear cells from subjects immunized with the vaccinia virus to those from vaccinia naive donors.</p> <p>Results</p> <p>Vaccinia immunization in the preceding 3-6 months resulted in an up to 5-fold reduction in CCR5-tropic but not in CXCR4-tropic HIV-1 replication in the cells from vaccinated subjects. The addition of autologous serum to the cell cultures resulted in enhanced R5 HIV-1 replication in the cells from unvaccinated, but not vaccinated subjects. There were no significant differences in the concentrations of MIP-1α, MIP-1β and RANTES between the cell cultures derived from vaccinated and unvaccinated subjects when measured in culture medium on days 2 and 5 following R5 HIV-1 challenge.</p> <p>Discussion</p> <p>Since primary HIV-1 infections are caused almost exclusively by the CCR5-tropic HIV-1 strains, our results suggest that prior immunization with vaccinia virus might provide an individual with some degree of protection to subsequent HIV infection and/or progression. The duration of such protection remains to be determined. A differential elaboration of MIP-1α, MIP-1β and RANTES between vaccinated and unvaccinated subjects, following infection, does not appear to be a mechanism in the noted protection.</p

Dimension dependent energy thresholds for discrete breathers

Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. We study the existence of energy thresholds for discrete breathers, i.e., the question whether, in a certain system, discrete breathers of arbitrarily low energy exist, or a threshold has to be overcome in order to excite a discrete breather. Breather energies are found to have a positive lower bound if the lattice dimension d is greater than or equal to a certain critical value d_c, whereas no energy threshold is observed for d<d_c. The critical dimension d_c is system dependent and can be computed explicitly, taking on values between zero and infinity. Three classes of Hamiltonian systems are distinguished, being characterized by different mechanisms effecting the existence (or non-existence) of an energy threshold.Comment: 20 pages, 5 figure

Wave operator bounds for 1-dimensional Schr\"odinger operators with singular potentials and applications

Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.Comment: 16 pages, 0 figure