Resolvent convergence of Sturm-Liouville operators with singular potentials

In this paper we consider the Sturm-Liuoville operator in the Hilbert space $L_2$ with the singular complex potential of $W^{-1}_2$ and two-point boundary conditions. For this operator we give sufficient conditions for norm resolvent approximation by the operators of the same class.Comment: 6 pages, to appear in Math. Note

Промежуточная оценка степени достижения плановых показателей федеральной целевой программы «Исследования и разработки по приоритетным направлениям развития научно-технологического комплекса России на 2014–2020 годы»

The article summarizes the results of the interim assessment of the degree of achievement of the planned indicators of the Federal target program «Research and development in priority areas of development of the scientific and technological complex of Russia for 2014–2020».В статье приведены обобщенные результаты промежуточной оценки степени достижения плановых показателей федеральной целевой программы «Исследования и разработки по приоритетным направлениям развития научно-технологического комплекса России на 2014–2020 годы»

Spectra of self-adjoint extensions and applications to solvable Schroedinger operators

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.Comment: 81 pages, new references added, subsection 1.3 extended, typos correcte

ДИНАМИКА И СТРУКТУРА ИСПОЛНИТЕЛЕЙ ПРОЕКТОВ ФЕДЕРАЛЬНОЙ ЦЕЛЕВОЙ ПРОГРАММЫ «ИССЛЕДОВАНИЯ И РАЗРАБОТКИ ПО ПРИОРИТЕТНЫМ НАПРАВЛЕНИЯМ РАЗВИТИЯ НАУЧНО-ТЕХНОЛОГИЧЕСКОГО КОМПЛЕКСА РОССИИ НА 2014–2020 ГОДЫ»

The article considers socio-demographic issues of research and development of the «Federal Target Program for Research and Development in Priority Areas of Development of the Russian Scientific and Technological Complex for 2014–2020». Analysis based on project's documents in the Program from 2014 to 2016. We studied the dynamics in both quantitative and qualitative characteristics of project performers – applied research and experimental development, and defined trends of project R&D personnel changes, including sociodemographic characteristics: age, size and composition of performers, the proportion of researchers with academic degree, and the proportion of women-researchers. In the article it is shown that specialists under 40 years of age are the largest part of project performers. The renewal of project personnel in time of projects is 65 percent. In the article it is also shown a strong correlation between project academic personnel and academic personnel of Russia as a whole. We devised proposals for tracking quantitative and qualitative parameters of project performers and for securing growth the skill level of young specialists in projects.В статье рассматриваются социально-демографические аспекты исследований и разработок одной из крупнейших федеральных целевых программ России «Исследования и разработки по приоритетным направлениям развития научно-технологического комплекса России на 2014–2020 годы». Анализ основан на документах проектов Программы 2014–2016 гг. Исследована динамика количественных и качественных характеристик исполнителей проектов – прикладных научных исследований и экспериментальных разработок, определены тенденции развития состава исполнителей проектов (возрастных, квалификационных, гендерных и иных параметров). В статье показано, что специалисты молодого возраста до 40 лет составляют наиболее многочисленную часть исполнителей. Выявлено обновление состава исполнителей в ходе выполнения исследований и разработок в среднем на 65%. Показана тесная связь динамики кадрового обеспечения сферы исследований и разработок с развитием кадрового состава российской науки в целом. Разработаны предложения по мониторингу количественных и качественных характеристик исполнителей проектов и обеспечению роста профессионального уровня молодых специалистов

Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

Sturm-Liouville operators with measure-valued coefficients

We give a comprehensive treatment of Sturm-Liouville operators with measure-valued coefficients including, a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.Comment: 58 page