Approximations of singular vertex couplings in quantum graphs

We discuss approximations of the vertex coupling on a star-shaped quantum graph of $n$ edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the Cheon-Shigehara technique using $\delta$ interactions with nonlinearly scaled couplings yields a $2n$-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the ${n+1\choose 2}$-parameter family of all time-reversal invariant couplings.Comment: LaTeX source file, 33 pages, with 3 eps figure

Towards Rigorous Derivation of Quantum Kinetic Equations

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.Comment: 25 page

The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems

The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear equations is constructed in the form of an expansion over particle clusters whose evolution is described by the corresponding order cumulant (semi-invariant) of evolution operators for the von Neumann equations. For the initial data from the space of sequences of trace class operators the existence of a strong and a weak solution of the Cauchy problem is proved. We discuss the relationships of this solution both with the $s$-particle statistical operators, which are solutions of the BBGKY hierarchy, and with the $s$-particle correlation operators of quantum systems.Comment: 26 page

Bis(dihydrogen norfloxacinium) tri-μ2-chlorido-bis­[trichloridobismuthate(III)] chloride dihydrate

The title compound, {systematic name: (3-carb­oxy-1-ethyl-6-fluoro-7-piperazin-4-ium-1-yl-1H-quinolin-4-yl­idene)oxonium tri-μ2-chlorido-bis­[trichloridobismuthate(III)] chloride dihydrate], (C16H20FN3O3)2[Bi2Cl9]Cl·2H2O, is composed of [Bi2Cl9]3− anions lying on crystallographic twofold rotation axes, Cl− anions also on twofold axes, C16H20FN3O3 2+ cations, and water mol­ecules. The BiIII coordination polyhedron is a distorted octa­hedron and two such octa­hedra share a triangular face to form the complex anion. There are three short terminal Bi—Cl bonds [2.5471 (6)–2.5781(5 Å] and three longer bridging bonds [2.8599 (5)–2.9984 (6) Å] in each octa­hedron. Anions, cations and water mol­ecules are linked by hydrogen bonds to form a three-dimensional network. There are also π–π stacking inter­actions between quinoline ring systems, with an inter­planar distance of 3.27 (1) Å

Исчерпание национальных средств правовой защиты как условие подачи жалобы в Европейский Суд по правам человека

The subject. The article is devoted to the subject of the exhaustion of domestic remediesbefore filing a complaint to the European Court of Human Rights.The purpose. The purpose of this article is to show and reveal the characteristics of suchimportant condition of lodging a complaint before the European Court of Human Rights asthe exhaustion of domestic remedies.The methodology. The following scientific methods have been used to write this article:analysis, comparing and making conclusions.Results, scope of application. The right of individual petition is rightly considered to be thehallmark and the greatest achievement of the European Convention on Human Rights. Individualswho consider that their human rights have been violated have the possibility oflodging a complaint before the European Court of Human Rights. However, there are importantadmissibility requirements set out in the Convention that must be satisfied beforea case be examined. Applicants are expected to have exhausted their domestic remediesand have brought their complaints within a period of six months from the date of the finaldomestic decision. The obligation to exhaust domestic remedies forms part of customaryinternational law, recognized as such in the case – law of the International Court of Justice.The rationale for the exhaustion rule is to give the national authorities, primarily the courts,the opportunity to prevent or put right the alleged violation of the Convention. The domesticlegal order should provide an effective remedy for violations of Convention rights.Conclusions. The rule of exhaustion of domestic remedies is an important part of the functioningof the protection system under the Convention and its basic principle. Цель статьи – раскрыть характеристики такого важного условия подачи жалобы в Европейский Cуд по правам человека, как исчерпание внутренних средств правовой защиты.Право на индивидуальную жалобу справедливо считается отличительной чертой и величайшим достижением Европейской конвенции по правам человека. Правило исчерпания внутренних средств правовой защиты является важной частью функционирования системы защиты, предусмотренной Конвенцией и ее основным принципом. При этом национальное законодательство должно обеспечивать эффективные средства правовой защиты в случае нарушения Конвенции

On the spectrum of a bent chain graph

We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever $\alpha\ne 0$. We consider a "bending" deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling $\alpha$ and the "bending angle" as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.Comment: LaTeX, 23 pages with 7 figures; minor changes, references added; to appear in J. Phys. A: Math. Theo