Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs

Manifestly non-Hermitian quantum graphs with real spectra are introduced and shown tractable as a new class of phenomenological models with several appealing descriptive properties. For illustrative purposes, just equilateral star-graphs are considered here in detail, with non-Hermiticities introduced by interactions attached to the vertices. The facilitated feasibility of the analysis of their spectra is achieved via their systematic approximative Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting bound-state spectra are found real in a discretization-independent interval of couplings. This conclusion is reinterpreted as the existence of a hidden Hermiticity of our models, i.e., as the standard and manifest Hermiticity of the underlying Hamiltonian in one of less usual, {\em ad hoc} representations ${\cal H}_j$ of the Hilbert space of states in which the inner product is local (at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples of these (of course, Hamiltonian-dependent) hermitizing inner products are offered in closed form. In this way each initial quantum graph is assigned a menu of optional, non-equivalent standard probabilistic interpretations exhibiting a controlled, tunable nonlocality.Comment: 33 pp., 6 figure

q-breathers in Discrete Nonlinear Schroedinger lattices

$q$-breathers are exact time-periodic solutions of extended nonlinear systems continued from the normal modes of the corresponding linearized system. They are localized in the space of normal modes. The existence of these solutions in a weakly anharmonic atomic chain explained essential features of the Fermi-Pasta-Ulam (FPU) paradox. We study $q$-breathers in one- two- and three-dimensional discrete nonlinear Sch\"{o}dinger (DNLS) lattices -- theoretical playgrounds for light propagation in nonlinear optical waveguide networks, and the dynamics of cold atoms in optical lattices. We prove the existence of these solutions for weak nonlinearity. We find that the localization of $q$-breathers is controlled by a single parameter which depends on the norm density, nonlinearity strength and seed wave vector. At a critical value of that parameter $q$-breathers delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into strong mode-mode interaction regime. In particular this breakdown takes place at one of the edges of the normal mode spectrum, and in a singular way also in the center of that spectrum. A stability analysis of $q$-breathers supplements these findings. For three-dimensional lattices, we find $q$-breather vortices, which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.Comment: 19 pages, 9 figure

PT-symmetric deformations of Calogero models

We demonstrate that Coxeter groups allow for complex PT-symmetric deformations across the boundaries of all Weyl chambers. We compute the explicit deformations for the A2 and G2-Coxeter group and apply these constructions to Calogero–Moser–Sutherland models invariant under the extended Coxeter groups. The eigenspectra for the deformed models are real and contain the spectra of the undeformed case as subsystems

A complex chromosomal rearrangement in a child with developmental delay, fractious behavior, and craniofacial anomalies, compatible with Smith-Magenis Syndrome

Smith-Magenis Syndrome (SMS) is a micro-deletion syndrome, and encompasses a picture of dysmorphology, mental defect, and fractious behavior. Evaluation of complex chromosome rearrangements (CCRs) and their potential phenotypic consequences is a common challenge in the genetics clinic and knowledge about the genotype/phenotype relationships are limited. We report the case of a 14-year-old boy who was referred by SMS, presenting developmental delay, fractious behavior, reduced sensitivity to pain, macrocranium and distinctive facial features. Following karyotyping, fluorescence in situ hybridization (FISH) using WCP probes for the chromosomes involved in CCR and 17p11.2 probe for SMS region was performed. Lately, chromosomal Comparative Genomic Hybridization (cCGH) and genomic microarray studies were also performed in order to identify genomic imbalances. The cytogenetic analysis revealed a karyotype: 46,XY,inv(3)(p23q27)t(3;10)(p13.2;p11.2),inv(14)(q13q32)dn.ish inv(3)t(3;10)(wcp10+), der(10)t(3;10)(wcp3+),inv(14)(wcp14+). Parental karyotypes were normal, although the father presented a marked cognitive delay. FISH analyses showed no deletion in 17p11.2 region and confirmed the cytogenetic results, namely the presence of CCR. Additionally, cCGH and genomic microarray studies did not reveal any gains/losses of genetic material in the breakpoints regions. Despite the clinical features of SMS, deletions or duplications in the SMS critical region were not detected in this patient. However, a small number of SMS present a mutation in the RAI1 gene instead of a 17p11.2 deletion, the former cause could not be excluded On the other hand, in CCRs de novo, an apparently balanced karyotype may be associated with an abnormal phenotype, including an increased risk of intellectual delay and congenital malformations. Further studies comprising, e.g., sequencing of the breakpoints, chromatin conformation analysis and refinement of the SMS critical region analysis might be useful to elucidate the phenotypic characteristics. However, the patient has been absent of routine clinical reevaluation; the etiology of the father’s cognitive delay could help shedding some light on the patient phenotypic features

Quasiperiodic spin-orbit motion and spin tunes in storage rings

We present an in-depth analysis of the concept of spin precession frequency for integrable orbital motion in storage rings. Spin motion on the periodic closed orbit of a storage ring can be analyzed in terms of the Floquet theorem for equations of motion with periodic parameters and a spin precession frequency emerges in a Floquet exponent as an additional frequency of the system. To define a spin precession frequency on nonperiodic synchro-betatron orbits we exploit the important concept of quasiperiodicity. This allows a generalization of the Floquet theorem so that a spin precession frequency can be defined in this case too. This frequency appears in a Floquet-like exponent as an additional frequency in the system in analogy with the case of motion on the closed orbit. These circumstances lead naturally to the definition of the uniform precession rate and a definition of spin tune. A spin tune is a uniform precession rate obtained when certain conditions are fulfilled. Having defined spin tune we define spin-orbit resonance on synchro--betatron orbits and examine its consequences. We give conditions for the existence of uniform precession rates and spin tunes (e.g. where small divisors are controlled by applying a Diophantine condition) and illustrate the various aspects of our description with several examples. The formalism also suggests the use of spectral analysis to ``measure'' spin tune during computer simulations of spin motion on synchro-betatron orbits.Comment: 62 pages, 1 figure. A slight extension of the published versio

Molecular dynamics of a short range ordered smectic phase nanoconfined into porous silicon

4-n-octyl-4-cyanobiphenyl (8CB) has been recently shown to display an unusual sequence of phases when confined into porous silicon (PSi). The gradual increase of oriented short-range smectic (SRS) correlations in place of a phase transition has been interpreted as a consequence of the anisotropic quenched disorder induced by confinement in PSi. Combining two quasielastic neutron scattering experiments with complementary energy resolutions, we present the first investigation of the individual molecular dynamics of this system. A large reduction of the molecular dynamics is observed in the confined liquid phase, as a direct consequence of the dynamical boundary conditions imposed by the confinement. Temperature fixed window scans (FWS) reveal a continuous 'glass-like' reduction of the molecular dynamics of the confined liquid and SRS phases on cooling down to 250 K, where a solid-like behavior is finally reached by a two steps crystallization process

Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be published in J. Phys. A: Mathematical and Genera

Quantum inner-product metrics via recurrent solution of Dieudonne equation

A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.Comment: 21 p

Slaying little dragons: the impact of the Guinea Worm Eradication Program on dracunculiasis disability averted from 1990 to 2016

Background: The objective of this study was to document the worldwide decline of dracunculiasis (Guinea worm disease, GWD) burden, expressed as disability-adjusted life years (DALYs), from 1990 to 2016, as estimated in the Global Burden of Disease study 2016 (GBD 2016). While the annual number of cases of GWD have been consistently reported by WHO since the 1990s, the burden of disability due to GWD has not previously been quantified in GBD.Methods: The incidence of GWD was modeled for each endemic country using annual national case reports. A literature search was conducted to characterize the presentation of GWD, translate the clinical symptoms into health sequelae, and then assign an average duration to the infection. Prevalence measures by sequelae were multiplied by disability weights to estimate DALYs.Results: The total DALYs attributed to GWD across all endemic countries (n=21) in 1990 was 50,725 (95% UI: 35,265–69,197) and decreased to 0.9 (95% UI: 0.5–1.4) in 2016. A cumulative total of 12,900 DALYs were attributable to GWD from 1990 to 2016.Conclusions: Using 1990 estimates of burden propagated forward, this analysis suggests that between 990,000 to 1.9 million DALYs have been averted as a result of the eradication program over the past 27 year