Distribution theory for Schr\"odinger's integral equation

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\"odinger's equation. This paper, in contrast, investigates the integral form of Schr\"odinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schr\"odinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schr\"odinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schr\"odinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schr\"odinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schr\"odinger's integral equation is viable tool for studying singular interactions in quantum mechanics.Comment: 23 page

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

Обоснование возможности выемки оставленных запасов Третьего калийного горизонта Старобинского месторождения, отработанных камерной системой

A technique has been developed of justification of possibility and expediency of extraction of the reserves left, which were mined more than 40 years ago by chamber system with maintenance in a safe condition of underground mining constructions. The methodology includes the identification by studying the geological and mine surveying documentation of districts with reserves in the mine fields; calculation of volumes and quality indicators of reserves left at these areas; assessment of a condition of mine workings that have direct access to the reserves concerned, and the possibility of their re-use, as well as the possibility of new development workings and ensuring their sustainability and maintenance in a safe condition; establishing the features of geomechanical processes in the rock mass and the presence of dangerous violations in the worked-out zone, the degree of the violation and the nature of the deformation of rigid and pliable pillars in the extraction chambers, including the use of specially conducted  mine openings; development of technological schemes for safety extraction of abandoned minerals. With the use of this technique for the first time the technological opportunities and economic feasibility of mining minerals from the treatment units of the First and Second Mines of JSC Belaruskali, worked out more than 40 years ago, has been estimated. The possibility of excavating abandoned reserves in the sylvinite layer 4 and layers 2, 2–3, 3 in inter-chamber pillars in areas previously worked out by the chamber system with total volumes of more than 57 million tons is shown. The results of surveys of the state of capital and development workings are given, according to which for the re-mining of most of the worked-out blocks, new capital and development workings will be required. The results of visual examinations of panel and block drifts, their conjugations with treatment passages, as well as the state and degree of violation of inter-passage and inter-chamber pillars in the treatment chambers are described. Possible methods for re-mining the reserves of the Third potash horizon with treatment longwalls and a chamber system are proposed, which are distinguished by high technical and economic efficiency and mining safety.Разработана методика обоснования возможности и целесообразности выемки оставленных запасов, отработанных более 40 лет назад камерной системой, с поддержанием в безопасном состоянии подземных горнотехнических сооружений. Методика включает выявление путем изучения геолого-маркшейдерской документации участков с оставленными запасами в пределах шахтных полей; расчет объемов и качественных показателей оставленных запасов на этих участках; определение состояния выработок, имеющих непосредственный доступ к рассматриваемым запасам, и возможности их повторного использования, а также возможности проведения новых подготовительных выработок и обеспечения их устойчивости и поддержания в безопасном состоянии; установление особенностей геомеханических процессов в породном массиве и наличия опасных нарушений в зоне выработанного пространства, степени нарушенности и характера деформирования жестких и податливых целиков в очистных камерах, в том числе с использованием специально проведенных вскрывающих исследовательских выработок; разработку технологических схем безопасной отработки оставленных запасов. С использованием этой методики впервые выполнено обоснование технологической возможности и экономической целесообразности выемки полезного ископаемого из отработанных более 40 лет назад очистных блоков Первого и Второго рудников ОАО «Беларуськалий». Показана возможность выемки оставленных запасов в 4-м сильвинитовом слое и слоях 2, 2–3, 3 в междукамерных целиках на участках, отработанных ранее камерной системой, с общим объемом более 57 млн т. Приводятся результаты обследований состояния капитальных и подготовительных выработок в местах возможного доступа на эти участки, согласно которым для повторной отработки большинства отработанных блоков потребуется проведение новых капитальных и подготовительных выработок. Описываются результаты визуальных обследований панельных и блоковых штреков, их сопряжений с очистными ходами, а также состояния и степени нарушенности междуходовых и междукамерных целиков в очистных камерах. Предложены возможные способы повторной отработки оставленных запасов Третьего калийного горизонта очистными лавами и камерной системой, отличающиеся высокой технико-экономической эффективностью и безопасностью ведения горных работ

Access to Water Source, Latrine Facilities and Other Risk Factors of Active Trachoma in Ankober, Ethiopia

Objective: This study aims to determine the prevalence and correlates of active trachoma in Ankober, Ethiopia. Methods: A cross-sectional community-based study was conducted during July 2007. A total of 507 children (ages 1–9 years), from 232 households were included in the study. All children were examined for trachoma by ophthalmic nurses using the WHO simplified clinical grading system. Interviews and observations were used to assess risk factors. Logistic regression procedures were used to determine associations between potential risk factors and signs of active trachoma. Results: Overall, the prevalence of active trachoma was found to be 53.9 % (95%CI 49.6%–58.2%). Presence of fly-eye (fly contact with the eyelid margin during eye examination) (Odds Ratio (OR) = 4.03 95 % CI 1.40–11.59), absence of facial cleanliness (OR = 7.59; 95%CI 4.60–12.52), an illiterate mother (OR = 5.88; 95%CI 2.10–15.95), lack of access to piped water (OR = 2.19; 95%CI 1.14–6.08), and lack of access to latrine facilities (OR = 4.36; 95%CI 1.49–12.74) were statistically significantly associated with increased risk of active trachoma. Conclusion: Active trachoma among children 1–9 years of age in Ankober is highly prevalent and significantly associated with a number of risk factors including access to water and latrine facilities. Trachoma prevention programs that include improved access to water and sanitation, active fly control, and hygiene education are recommended to lower the burde