Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:= -D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family of potentials converges in the sense of distributions to the first derivative of the Dirac delta-function, and then the limit of S_\epsilon might be considered as a "physically motivated" interpretation of the one-dimensional Schr\"odinger operator with potential \delta'.Comment: 30 pages, 2 figure; submitted to Proceedings of the Royal Society of Edinburg

Controlling a resonant transmission across the δ′\delta'-potential: the inverse problem

Recently, the non-zero transmission of a quantum particle through the one-dimensional singular potential given in the form of the derivative of Dirac's delta function, $\lambda \delta'(x)$, with $\lambda \in \R$, being a potential strength constant, has been discussed by several authors. The transmission occurs at certain discrete values of $\lambda$ forming a resonance set ${\lambda_n}_{n=1}^\infty$. For $\lambda \notin {\lambda_n}_{n=1}^\infty$ this potential has been shown to be a perfectly reflecting wall. However, this resonant transmission takes place only in the case when the regularization of the distribution $\delta'(x)$ is constructed in a specific way. Otherwise, the $\delta'$-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence $\Delta'_\varepsilon(x)$ that tends to $\delta'(x)$ in the sense of distributions as $\varepsilon \to 0$. Therefore, from a practical point of view, it would be interesting to have an inverse solution, i.e. for a given $\bar{\lambda} \in \R$ to construct such a regularizing sequence $\Delta'_\varepsilon(x)$ that the $\delta'$-potential at this value is transparent. If such a procedure is possible, then this value $\bar{\lambda}$ has to belong to a corresponding resonance set. The present paper is devoted to solving this problem and, as a result, the family of regularizing sequences is constructed by tuning adjustable parameters in the equations that provide a resonance transmission across the $\delta'$-potential.Comment: 21 pages, 4 figures. Corrections to the published version added; http://iopscience.iop.org/1751-8121/44/37/37530

Quasi-Homogeneous Backward-Wave Plasmonic Structures: Theory and Accurate Simulation

Backward waves and negative refraction are shown to exist in plasmonic crystals whose lattice cell size is a very small fraction of the vacuum wavelength (less than 1/40th in an illustrative example). Such ``quasi-homogeneity'' is important, in particular, for high-resolution imaging. Real and complex Bloch bands are computed using the recently developed finite-difference calculus of ``Flexible Local Approximation MEthods'' (FLAME) that produces linear eigenproblems, as opposed to quadratic or nonlinear ones typical for other techniques. FLAME dramatically improves the accuracy by incorporating local analytical approximations of the solution into the numerical scheme.Comment: 4 pages, 3 figure

An Effective Model for Nematic Liquid Crystal Composites with Ferromagnetic Inclusions

Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrix-ferronematics-as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and simultaneously cause reorientation of the nematic molecules greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring. We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while their total volume fraction tends to zero. The total energy of the mixture is assumed to be the sum of the bulk elastic liquid crystal contribution, the anchoring energy of the liquid crystal on the surfaces of the particles, and the magnetic energy of interaction between the particles and the applied magnetic field. The homogenized limiting ferronematic energy is obtained rigorously using a variational approach. It generalizes formal expressions previously reported in the physical literature

On δ′\delta'-like potential scattering on star graphs

We discuss the potential scattering on the noncompact star graph. The Schr\"{o}dinger operator with the short-range potential localizing in a neighborhood of the graph vertex is considered. We study the asymptotic behavior the corresponding scattering matrix in the zero-range limit. It has been known for a long time that in dimension 1 there is no non-trivial Hamiltonian with the distributional potential $\delta'$, i.e., the $\delta'$ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials \alpha\eps^{-2}Q(x/\eps) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as \eps\to0. We extend these results to star graphs with the point interaction, which is an analogue of $\delta'$ potential on the line. We prove that generically such a potential on the graph is opaque. We also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs. This set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary Sturm-Liouville problem associated with $Q$ on the graph.Comment: 16 pages, 2 figure

Технология устройства ледопородного ограждения при проходке шахтных стволов на примере объектов Петриковского ГОКа

Scientific-technical aspects of technology of use of the ice wall when shaft sinking in water-bearing rocks, using an example of objects under construction of the Petrikov mining and processing plant, are studied. The algorithm of process of freezing of the mountain massif is described. The method is developed of calculation of the main parameters of the ice barrier, based on classical scientific concepts in the field of geotechnology, geomechanics and mountain thermal physics, containing modified formulas of thickness and time of the ice wall formation. The high efficiency of the developed technique is confirmed by the results of the successful use of the obtained numerical values of the parameters of the fence in the implementation of the project of sinking shafts. It is shown the possibility of choosing a method of defrosting frozen rocks, based on the analysis of a real three-dimensional model of the ice barrier. The most important criterion for this choice is the uniformity of the thickness of the ice barrier along the circumference of the vertical cylinder. The conclusion is made about the high scientific and practical competence of Belarusian mine builders – scientists and specialists. The country has developed a reliable system of scientific and technical support for the technology of sinking vertical mine shafts using temporary ice fences. Effective methods of preliminary geological studies, calculation of parameters of ice-rock fences and freezing equipment, selection of technology of work have been created. The necessary material and technical base for fast and high-quality performance of all complex of works on a construction of mine trunks in the special way is created.Изучены научно-технические аспекты технологии использования ледопородного ограждения при проходке шахтных стволов в водоносных горных породах на примере объектов строящегося Петриковского ГОКа. Описан алгоритм процесса замораживания горного массива. Разработана методика расчета основных параметров ледопородного ограждения, основанная на классических научных представлениях в области геотехнологии, геомеханики и горной теплофизики, содержащая модифицированные формулы толщины и времени образования ледопородной стенки. Высокая эффективность разработанной методики подтверждается результатами успешного использования полученных численных значений параметров ограждения при реализации проекта проходки стволов. Показана возможность выбора способа размораживания мерзлых пород на основе анализа реальной трехмерной модели ледопородного ограждения. Важнейшим критерием такого выбора является равномерность толщины ледопородного ограждения по окружности вертикального ствола. Сделан вывод о достигнутой высокой научной и практической компетенции белорусских шахтостроителей – ученых и специалистов. В стране разработана надежная система научно-технического обеспечения технологии проходки вертикальных шахтных стволов с использованием временных ледопородных ограждений. Созданы эффективные методики проведения предварительных геологических исследований, расчета параметров ледопородных ограждений и замораживающего оборудования, выбора технологии проведения работ. Сформирована необходимая материально-техническая база для быстрого и качественного выполнения всего комплекса работ по сооружению шахтных стволов специальным способом

On instability of a bend Fr eedericksz configuration in nematic liquid crystals

this paper we discuss the stability of a uniform Fr eedericksz configuration in highly anisotropic thin nematic films. We show that the Fr eedericksz transition in a bend geometry is at least two-dimensional, and it occurs at field strengths lower than the known classical threshold. 1. INTRODUCTIO

Risk factors for active trachoma and ocular Chlamydia trachomatis infection in treatment-naïve trachoma-hyperendemic communities of the Bijagós Archipelago, Guinea Bissau.

BACKGROUND: Trachoma, caused by ocular infection with Chlamydia trachomatis, is hyperendemic on the Bijagós Archipelago of Guinea Bissau. An understanding of the risk factors associated with active trachoma and infection on these remote and isolated islands, which are atypical of trachoma-endemic environments described elsewhere, is crucial to the implementation of trachoma elimination strategies. METHODOLOGY/PRINCIPAL FINDINGS: A cross-sectional population-based trachoma prevalence survey was conducted on four islands. We conducted a questionnaire-based risk factor survey, examined participants for trachoma using the World Health Organization (WHO) simplified grading system and collected conjunctival swab samples for 1507 participants from 293 randomly selected households. DNA extracted from conjunctival swabs was tested using the Roche Amplicor CT/NG PCR assay. The prevalence of active (follicular and/or inflammatory) trachoma was 11% (167/1508) overall and 22% (136/618) in 1-9 year olds. The prevalence of C. trachomatis infection was 18% overall and 25% in 1-9 year olds. There were strong independent associations of active trachoma with ocular and nasal discharge, C. trachomatis infection, young age, male gender and type of household water source. C. trachomatis infection was independently associated with young age, ocular discharge, type of household water source and the presence of flies around a latrine. CONCLUSIONS/SIGNIFICANCE: In this remote island environment, household-level risk factors relating to fly populations, hygiene behaviours and water usage are likely to be important in the transmission of ocular C. trachomatis infection and the prevalence of active trachoma. This may be important in the implementation of environmental measures in trachoma control